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A Primer and Assessment
of Social Security
Reform in Mexico
M A R C O A . E S P I N O S A - V E G A
A N D TA P E N S I N H A
Espinosa-Vega is a senior economist in the Atlanta
Fed’s research department. Sinha is Seguros Comercial
America Chair Professor of Risk Management and
Insurance at the Instituto Tecnológico Autónomo de
México. They thank Asok Chaudhuri, Frank King,
Dipendra Sinha, Selahattin Imrohoroglu, Steve Russell,
and Steve Smith for insightful comments.
W
HILE A NUMBER OF THEORETICAL ECONOMISTS HAVE ACCEPTED THE NOTION THAT
MOVING FROM A PAY-AS-YOU-GO TO A FULLY FUNDED SOCIAL SECURITY SYSTEM WOULD
IMPROVE A COUNTRY’S WELL-BEING, THERE IS FAR FROM UNIVERSAL AGREEMENT ON
THIS POLICY PRESCRIPTION.1
FOR EXAMPLE, WHILE KOTLIKOFF (1996) CALLS FOR A
move to a fully funded pension system in the United
States, Diamond (1998) presents a number of
caveats for such a move.2 Moreover, the major
thrust of the World Bank (1994) that advocates
moves away from a pay-as-you-go system has been
severely criticized by Orszag and Stiglitz (1999)
from within the World Bank itself. Perhaps these
discrepancies explain why, to date, only a few
economies have switched from a pay-as-you-go to a
fully funded system.3 Recently, however, the economic projections of a number of countries with
pay-as-you-go systems have shown significant
future actuarial imbalances. As a consequence, several of these countries are either contemplating or
are engaged in a significant redesign of their pay-asyou-go systems.
While in the United States the debate about
switching to a fully funded system continues, eight
countries in Latin America claim to have either
abandoned or are in the process of abandoning their
pay-as-you-go systems in favor of fully funded systems.4 Mexico is one of these eight countries, and it
is of particular interest to U.S. analysts because of
both its geographical proximity and close relationships with the United States and the similarities of
its reform program to what many policymakers and
economists advocate for the U.S. system.
The Mexican government claims that it has started a
move to a fully funded system. As proof, proponents of
the new system point out that since 1997 Mexico has
adopted a privately managed defined-contribution
system. It is important to emphasize, however, (as is
done in Espinosa-Vega and Russell 1999) that a
pension system can be privately administered without being fully funded. The new system is seen in
some circles as a great accomplishment. Proponents
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1
of the pension reform (for example, Rodriguez 1999
and Sales-Sarrapy, Solís-Soberón, and VillagómezAmezcua 1998) predict that it will lead to a number
of positive future developments: (1) The system will
be actuarially balanced. (2) It will increase private
(and national) saving. (3) Workers will migrate from
the informal to the formal labor market. (4) More
workers will be covered by the social security system. (5) The new system will create long-term
investment instruments. But before uncorking the
bottle of champagne, it is important to ask a few
questions. Has Mexico started a migration toward a
fully funded system? What are the likely net gains
from the Mexican pension reform? Are predictions
1 through 5 likely to
materialize?
There is voluminous literature on
social security sysMexico is in dire need
tems, both countryof further research to
specific and general.
guide it through its deciA survey on this literature is beyond the
sion on whether and how
scope of this article.
to switch to a fully funded
The objective here
pension system.
instead is to provide
a primer on the
Mexican pension system and to evaluate
it critically. The ultimate intended goal in
analyzing the Mexican experience is to illustrate the
difficulties in assessing the economic significance
of a pension reform. In general the hope is that in
the current environment where every other country
seeking reform claims to be jumping on the fully
funded wagon, this discussion may help to temper
expectations.
The article traces some of the official rationales
for the reform in Mexico and provides a summary of
the new developments leading to it. It reports its
operational rules and the critical elements of the
new pension system. The article also applies the
insight of a companion piece by Espinosa-Vega and
Russell (1999) to assess the significance of the
changes introduced by the reform. It makes clear
that while the reform is likely to bring some benefits, it also has costs (something that has not been
emphasized in the existing literature). Finally, it
calls for further research to appraise the predictions
spelled out above and the net benefits of the reform
for the Mexican society. In the end, the Mexican
case provides a good case study for those countries
that are either considering or have engaged in a
pension reform of their own.
2
Key Features of Mexico’s Old Social
Security System
he next sections introduce the most significant features of the old Mexican public pension system as a point of reference for
discussing the reform. There have in fact been several pension plans in Mexico. Each of these plans is
in turn part of a larger benefits plan. Federal
employees’ accounts are managed by the Instituto de
Seguridad y Servicios Sociales de los Trabajadores
del Estado (ISSSTE). There is a special fund for the
state-owned petroleum-related monopoly, PEMEX.
Private-sector workers’ accounts have been managed by the government-run Instituto Mexicano del
Seguro Social (Mexican Social Security Institute:
IMSS). Furthermore, within each of these institutions, health insurance, housing programs,
and social security programs are bundled together.
Because the first two systems have been left
intact, this discussion focuses solely on IMSS and
more particularly on the old-age security aspect of
IMSS, which is the core of the current Mexican
pension reform.
The IMSS started its operation in 1943–44. Its
social security chapter was designed to cover four
areas: disability, old age, severance, and disability
and life insurance (Invalidez, Vejez, Cesantia en
Edad Avanzada, y Muerte, or IVCM). As stated in
Grandolini and Cerda, “The original IMSS-IVCM
can be characterized as a partially funded defined
benefit scheme. However, since the very beginning, it operated as a pay-as-you-go scheme as the
fund’s actuarial reserves were used to finance
other social insurance activities, particularly
health” (1998, 4).
Restricted to IMSS, the reform affects only the
portion of the economically active population
working in the formal private sector. The fact that
this sector is proportionally smaller than its counterpart in developed economies may be relevant in
assessing the macroeconomic impact of the
reform. More than 40 percent of Mexico’s 33.5 million labor force is outside the formal sector
(Judisman 1997), working in what the International
Labor Organization (van Ginneken 1998) calls the
informal sector: independent workers (excluding
professional, administrative, and technical personnel), domestic workers, and workers in small enterprises (with five workers or fewer).5 Of the total
economically active population, social security
covers less than one-third. To put it differently,
it covers slightly more than half the workers in
the formal sector. Therefore, talk about reform is
talk about directly affecting only half the formal
labor force.
T
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
The discussion that follows reviews other relevant features unique to the Mexican pension system. It starts by looking at the eligibility criteria for
workers to qualify for benefits (called the admission
fee) and the fraction of individuals’ working income
received upon retirement (the so-called replacement rate).
TA B L E 1
Replacement Rates under Mexico’s
IMSS System (Percent)
Years in the System
Salary at
Retirement
10
30
100
100
6.5 x minimum salary
23
25
10 x minimum salary
14
16
1 minimum salary
Benefit-Eligibility Requirements and the
Replacement Rate under the Old System
or retirees the most important aspect of their
benefits is the proportion of wages received
during their active years that is replaced by
their retirement income. This proportion is called
the replacement rate. For example, a replacement
rate of 100 percent would mean that an individual’s
annual income would be the same before and after
retirement. The concept of replacement rate is significant because eligibility requirements are often
quoted in terms of it. It is also important to review
the replacement rate and eligibility criteria here to
be able to contrast the old system with the new.
For most individuals, wages tend to rise with age.
Therefore, it is incorrect to talk about a single
replacement rate. Instead, it is customary to discuss the replacement rate with respect to either
lifetime-average wage or final salary (and in some
cases with respect to an average of a worker’s five
or ten highest-income years).
In the old regime, the system of old age (and disability) benefits was designed so that a worker
became fully eligible to receive benefits after just
500 weeks of contribution. However, benefits did
not increase much with additional contribution,
as Table 1 illustrates. Moreover, if for some reason a
F
Source: Serrano (1999a)
worker stopped contributing for 500 consecutive
weeks, he or she would lose all retirement benefits.
The numbers presented in Table 1 are instructive. The first row states that for a person earning
one minimum salary, the replacement rate would be
100 percent regardless of the number of years the
individual contributed to the pension fund. Things
are not very different for other participants. A person earning ten times the minimum salary, for example, would get 14 percent of his or her wage replaced
after ten years and 16 percent of his annual salary
replaced after thirty years. In this case, although the
incentive to contribute to the pension fund for more
than ten years was not zero, it was minimal. Recent
estimates show that 86 percent of current retirees
get exactly one minimum salary as the retirement
benefit (Sinha 1999a).
Thus there was a fairly low minimum admission
fee. The fact that workers qualified for pension
after only 500 weeks of work created an incentive
to contribute just long enough to become eligible
1. For a detailed description of the key differences between pay-as-you-go and fully funded systems, see a companion article
by Espinosa-Vega and Russell (1999).
2. Feldstein’s (1974) theoretical analysis suggests that privatization of social security would reduce the distortions that payroll
taxes impose on household saving and labor supply decisions. Even in the absence of redistributional considerations or the
presence of market imperfections, Feldstein’s work, as well as that of his successors, is subject to a qualification shown in
Diamond’s (1965) theoretical analysis: a mandated pay-as-you-go defined-contribution social security system would improve
a country’s well-being provided the economy was dynamically inefficient. (Roughly put, a competitive economy is said to be
dynamically inefficient if it saves “too much” relative to the social optimum.)
Imrohoroglu, Imrohoroglu, and Joines (1995) extend Diamond’s general equilibrium work by adding potentially more realistic lifetime structure and market imperfections. They are able to show how the replacement rate (the ratio of retirement
benefits to preretirement wages) varies according to the market structure and specific parameter assumptions. Because
their analysis focuses on economies that are dynamically inefficient, Diamond’s result prevails. Abel and others (1989) provide empirical support for the dynamic efficiency of the U.S. economy. Building on their work and expanding Feldstein’s
analysis to a general equilibrium framework, Kotlikoff (1996) has provided extensive simulation analysis for the U.S. economy that supports Feldstein’s conclusion. In a framework that allows for intra- and intergenerational redistribution, these
authors show that in a competitive economy privatization of the social security system would—after intragenerational lumpsum transfers if necessary—improve the well-being of the country. A recent example of a serious critique of a fully funded
scheme is found in Sinn (2000).
3. See Schwarz and Demirguc-Kunt (1999) for a complete list of countries engaged in pension reform.
4. The countries are Argentina, Bolivia, Chile, Colombia, El Salvador, Mexico, Peru, and Uruguay.
5. In Latin America, more than half of the economically active population work in the informal sector.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
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and then either drop out to the informal sector or
“unregister” with the IMSS and continue working
without being officially on the payroll. This awkward
eligibility requirement was another factor contributing to the actuarial imbalance of the IMSS-IVCM
under the old system, which was funded essentially
by a payroll tax. At the same time, because employers were responsible for paying part of this tax,
many of them understated the wage rate of workers
just to avoid paying the payroll tax.
In addition, the government had relaxed eligibility
by, for example, relaxing the age of retirement, by
using broader definitions of disability or poor health,
and so forth. One manifestation of this problem,
which was severe in the Mexican system, is that an
increasing number of people were getting a disability
pension. Since the middle of the 1980s, the proportion of people drawing a disability pension has stayed
at more than 40 percent (see Table 2), a very high
figure compared with Organisation for Economic
Cooperation and Development (OECD) countries,
whose population is generally much older.
As is described below, one can identify two opposing factors affecting the balance of the pension portion of the IMSS fund. Even though the replacement
rate appears generous at first glance, it is quoted in
terms of the minimum salary. The minimum salary in
Mexico at the time of the reform was roughly $24 a
day. The World Bank (2000) considers that in developing nations an “adequate” standard of living can
be maintained with $40 a day. This low level of disbursement (in combination with the large proportion of young to old people described below) worked
to boost the coffers of the IMSS. On the other hand,
the low admission fee made it unattractive to stay in
the system for more than ten years and thus constituted a strain on the coffers of the system.
At the same time, there were other strains on the
retirement account of the IMSS. The Mexican benefit system has historically been tied to the minimum wage (that is, it has always been calculated as
a multiple of the minimum wage), which is adjusted
only by legislation. Indexing of retirement benefits
was first introduced in the Mexican system in 1989,
when Congress passed a law stating that for calculation of IMSS benefits the minimum wage would
be indexed to the consumer price index. The government thereby increased the benefits of the
retired population by indexing benefits to inflation
but added to strains on the IMSS because it did not
at the same time index revenue to inflation.
In spite of these idiosyncrasies, from its inception
the private pension system in Mexico operated with
surpluses because of favorable demographic factors. For example, behind these surpluses lay a
4
TA B L E 2
IMSS-IVCM Disbursements by Old-Age
Retirement and Disability Categories (Percent)
Year
Old Age
Disability
1981
64.95
35.05
1985
58.86
41.14
1990
56.47
43.53
1994
57.01
42.99
Source: IMSS (1997)
large base of contributors relative to benefit recipients. However, for most of those years, instead of
building reserves these surpluses were used to subsidize IMSS’s other programs such as its health
insurance component. According to the IMSS, this
status quo was sustainable without any changes
until the year 2007. However, as the next section
illustrates, in recent years Mexico has experienced
dramatic changes in mortality rates and demographic trends, changes that would have reduced
and even eliminated the surpluses on the IMSS
pension accounts.
The Demographic Angle
n recent years Mexico has experienced a significant drop in its fertility and mortality rates,
which has led to a relatively rapid aging of its
population. For example, the proportion of population above age sixty in France was 5 percent in
1750. Mexico reached the same milestone in 1985.
However, by 1985 the proportion of French population older than sixty rose to 15 percent. It took
France 235 years to get to that point. Mexico will
reach this number by 2025, in only 35 years. France
had the opportunity to change its social institutions
slowly to cope with the problems associated with
population aging. Mexico, on the other hand, has had
to expedite its social security reform.
Table 3 presents a clearer picture of how rapidly
population changes are occurring in Mexico. The
table shows actual population proportions for 1970
and 1990. In addition, it includes projected population proportions in 2010, 2030, and 2050. As the
numbers clearly show, over a period of eighty years
(between 1970 and 2050), the proportion of population older than sixty rises from 6.13 percent to
24.35 percent.
One reason for such a dramatic change in population structure is a rapid decline in fertility rates.
In 1970 the mean fertility rate of women was 6.5
children per lifetime. This figure is projected to fall
I
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
T A B L E 3 Actual and Projected Changes in Age Distribution, 1970–2050
Age
1970
1990
2010
2030
2050
0–4
18.59
13.20
9.38
7.38
6.43
5–9
15.14
12.65
9.47
7.31
6.42
10–14
12.77
12.70
9.59
7.37
6.45
15–19
10.38
12.18
9.39
7.33
6.43
20–24
8.22
9.86
8.80
7.19
6.30
25–29
6.78
7.97
8.32
7.21
6.18
30–34
5.53
6.72
8.28
7.26
6.20
35–39
4.69
5.49
7.99
7.16
6.22
40–44
3.90
4.40
6.60
6.86
6.24
45–49
3.33
3.62
5.38
6.54
6.31
50–54
2.44
2.92
4.51
6.48
6.33
55–59
2.13
2.41
3.60
6.14
6.14
60–64
1.87
1.91
2.76
4.89
5.71
65–69
1.53
1.50
2.14
3.79
5.21
70–74
1.20
0.97
1.56
2.92
4.80
75–79
0.85
0.71
1.11
2.06
4.08
80+
0.68
0.77
1.11
2.11
4.55
Total
100
100
100
100
100
Source: Data from United Nations (1998, table 3)
to 2.1 by 2050. At the same time, the infant mortality rate fell from sixty-nine per thousand live births
to eleven per thousand live births. These two
trends have opposite effects with the decline in fertility leading to fewer people entering the workforce and the improved infant mortality rate
somewhat alleviating this problem. At the same
time, the mortality rate of people in higher age
groups has also fallen, contributing further to the
aging of the population structure.
All these changes can be summarized in what is
called the dependency ratio of the population. The
dependency ratio is usually defined as the number
of people in 0–14 and 65+ age groups (the dependent group) divided by the number of people in the
15–64 age group (because the labor force usually
consists of the latter age group).
Over the eighty-year period from 1970 to 2050,
the dependency ratio changes dramatically (from
1.03 to 0.52) in the first forty years. As Table 4
shows, it drops from 1.03 to 0.52. Thus, the number
of people dependent on the working-age population
by 2010 will have fallen by 50 percent. Then it is
projected to rise somewhat. This rise is somewhat
TA B L E 4
Actual and Projected Dependency Ratios,
1970–2050
Indicator
1970
2010
2050
Dependency Ratio
1.03
0.52
0.61
Old-Young Ratio
0.09
0.21
0.97
Note: The dependency ratio is the number of people in age groups
0–14 and 65+ divided by the number of people in the 15–64 age
group. The old-young ratio is the number of people in the 65+ age
group divided by the number of people aged 0–14.
Source: Atlanta Fed calculation using data from United
Nations (1998)
deceptive, however, hiding the composition of the
dependent population. The change in composition
of the dependent population is evident in the ratio
of old to young in the population, which moves from
9 percent to 97 percent over the total period. This
scale of change in age composition has been witnessed by very few countries over such a short time.
In view of these demographic changes, policymakers have had to face a pressing question: How
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
5
onerous would maintaining the status quo be? The
discussion now turns to this question.
Estimating the Actuarial Imbalance
o far, the discussion has identified and
described the strains to the Mexican private
pension system without actually reporting
what it would have cost the government to maintain
the status quo under IMSS-IVCM. Without trying to
evaluate their accuracy, this section reports three
such estimates.
Table 5 contains a projection attributed to IMSS
by Grandolini and Cerda (1998). The table reports
that the present value of IMSS commitments
through 2058 as of December 31, 1994, (the year
Congress started to consider a second reform) was
142 percent of the 1994 gross domestic product
(GDP) present-value deficit.
To get a sense of how the time path of actuarial
deficit would play out had there been no changes in
the system, IMSS itself calculated the projected
deficit. The IMSS figures are reproduced here as
Table 6. The table shows that the IMSS would have
run a surplus until 2005 (a positive number in the
table indicates a surplus) had the old system not
been changed. Therefore, the situation in Mexico was
not like that of Argentina or Uruguay (where the governments were already filling up the deficits of their
pay-as-you-go pension systems with current government budgetary resources). On the other hand, after
2020 the deficit would have mounted rapidly.
Sales-Sarrapy, Solís-Soberón, and VillagómezAmezcua (1998) present an alternative estimate of
the cost of maintaining the IMSS-IVCM status quo.
The least costly of their scenarios has the cost going
from 1.55 percent of GDP in 1997 to 3.59 percent in
2022 and 6.69 percent in 2047.
Why do these estimates of the deficits differ? For
example, according to the IMSS figures, for 1997
there was a surplus in the pension fund. On the
other hand, Sales-Sarrapy, Solís-Soberón, and
Villagómez-Amezcua (1998) report a deficit for
1997. Given the information provided by the different authors, it is impossible to identify explicitly
the reason for most of these differences, and it is
therefore impossible to adequately compare the
different estimates.
An additional challenge is that not all estimates
consider the same concepts. The concept of
implicit pension debt measures the stock of debt
today. If all the taxes to finance the pay-as-you-go
system are set to zero, the implicit pension debt
shows how much the government owes (implicitly)
to the current generation as of today. There is no
liability for future generations in this calculation.
S
6
The calculation is the exact analog of government
debt with one difference: explicit government
debt does not depend on the mortality experience
of the current generation. On the other hand,
implicit pension debt does because most governments promise pensions for widows (and sometimes to other dependents).
This concept should be contrasted with that of
the present value of cash flow deficits. As the name
suggests, cash flow deficits are calculated as the
difference between expected contributions at every
future date, which in most cases represents a
deficit. Then, the present value of the stream of
numbers is calculated. If contribution rates and
benefits rates do not change but the underlying
demographics do, the deficit will be altered.
Specifically, aging of the population will make
deficits worse. The period over which the deficit is
calculated also matters. The larger the period, of
course, the bigger the deficit.
The issue is further complicated because authors
may not explicitly identify the concepts with which
they are working. For example, the Grandolini and
Cerda (1998) and Sales-Sarrapy, Solís-Soberón, and
Villagómez-Amezcua (1998) studies do not always
clarify whether they are talking about implicit pension debts or cash flow deficits.
Additional problems arise from the fact that there
is no universal standard for the discount rate chosen to calculate the present value. For example,
Grandolini and Cerda (1998) chose to use a 3 percent discount rate. The advantage of Table 6 is that
it allows avoiding taking an arbitrary discount rate.
Instead of all the numbers being lumped by being
added up, they remain a vector of values. The significance of such confusions is that they can lead to
vastly different conclusions (see Sinha forthcoming, chap. 3, especially table 3.33.) Nonetheless,
without attempting to homogenize the different
estimates of the cost of maintaining the status quo
under IMSS-IVCM, it is clear that, according to
these studies, maintaining the status quo would
have been very costly for the country.
The New Mexican Social Security System
n December 1995, the Mexican Congress passed
the new Social Security Law (Ley de Seguro
Social), paving the way for the current system.
A second set of laws (Ley de los Sistemas de Ahorro
para el Retiro) was passed in April 1996. These laws
allowed privatized management of the country’s
pension system. They approved operation of investment management companies (Administradores de
Fondos de Ahorro, or AFOREs) to manage individual retirement funds (Sociedades de Inversion
I
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
TABLE 5
Present Value of Future Pension Deficits (in Billions of Pesos) as of December 31, 1994
Assets
Reserves
Liabilities
3.25
PV of old pension
96.93
PV of future contributions
683.67
PV of future liability
2,390.61
(Affiliates now)
179.74
(This generation)
1,017.40
(Future generations)
503.93
(Future generations)
1,373.21
Total
683.92
Total
2,487.54
Source: Grandolini and Cerda (1998)
T A B L E 6 Actuarial Deficit Projection of IMSS If the Old System Had Continued
Year
Millions of 1994 Pesos
Percent of 1996 GDP
2000
9,916
0.39
2005
667
0.03
2010
–23,407
–0.93
2015
–63,950
–2.55
2020
–122,827
–4.89
2025
–200,741
–8.00
2030
–264,501
–10.54
Note: Some of the estimates that went into computing Tables 5 and 6 include (1) Demographics: sizes of workers and retirees of every
generation in the future. These numbers will in turn depend on fertility and mortality projections (ignoring migration). (2) Estimates of growth
rates of real wages in the future. (3) Retirement pattern of the elderly in the future. (4) Participation rate of women and other part-time workers in the labor force. (5) Proportion of economically active population participating in the formal sector. (6) Inflation rate projection. Under
the old regime, the benefits are calculated on the basis of the average nominal salary of the last five working years. It also required a choice
of a discount rate to convert these figures to a single number. Although the authors reveal that these are partial equilibrium computations
the exact methodology is not spelled out in the document.
Source: IMSS (1997, table 18)
Especializadas en Fondos para el Retiro, or
SIEFOREs). In addition, the Mexican government
set up a separate division to oversee all activities of
the AFOREs: Comisión Nacional del Sistema de
Ahorro para el Retiro (CONSAR). To clarify the
roles of the AFOREs, CONSAR has set out general
rules of operation for the companies (see Banco de
Mexico 1996).
The stated objectives of AFOREs include the following: (1) To open, administer, and manage the individual retirement accounts in agreement with
provisions in social security laws. Regarding housingpromotion subaccounts, the AFOREs will register
each worker’s contributions and the interest paid
thereon, using information provided by social security institutions.6 (2) To receive from social security
institutions the contributions made, in accordance
with the law, by the government, employers, and
workers, as well as voluntary contributions by workers and employers. (3) To itemize the amounts
received periodically from social security institutions and deposit them into each worker’s individual
6. The housing subaccount requires a contribution of 5 percent of wages. This amount is substantial (the retirement contribution is 6.5 percent of wages). In the past, this housing subaccount has earned a negative real rate of return. All future estimates assume that it will earn a zero real rate of return. One interesting question is, Why is the government so keen on getting
the house in order for the retirement account but not touch the housing subaccount?
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
7
retirement account as with the returns obtained on
the investment of these funds. (4) To provide administrative services to mutual investment funds (the
SIEFOREs). These are direct subsidiaries of the
AFORES. In fact, at present, each AFORE is allowed
to have one SIEFORE.
Contribution Structure. The contribution
structure of the new system is as follows: Each
individual pays a compulsory 6.5 percent of wages
into an individual retirement account. The government contributes a “social quota” (called cuota
social) of 5.5 percent of minimum wage (regardless
of the wage rate of the worker). This social quota is
funded from the government’s general revenue
every year; thus the funding mechanism is taxes on
the current generation of workers. In addition,
workers must contribute 5 percent to a housing
subaccount (INFONAVIT) that will be consolidated
with the AFORE account upon retirement. Also,
4 percent of wages go to IMSS for disability and survivors insurance. Workers can also make additional
voluntary contributions. The AFOREs started to
collect compulsory and voluntary contributions in
February 1997. Contribution to the new system
became compulsory for all private-sector workers
in September 1997.
AFOREs are allowed to charge management fees
either as a percentage of contribution, a percentage
of value accumulated, or any combination thereof.
Most AFOREs charge fees as a percentage of contribution. All are required to inform affiliates about
their accounts at least once a year with statements
that include information about accumulated value,
contributions during the year, and any charges the
account has incurred.
Contribution Requirements: A Comparison.
In order to gain some perspective on the differences
between the required contributions and on eligibility requirements under the old and the new social
security systems, the following information is provided. The box on page 19 is a compilation of information provided by CONSAR, IMSS, and SHCP
(Secretaría de Hacienda y Crédito Público) and
reported in Sales-Sarrapy, Solís-Soberón, and
Villagómez-Amezcua (1998, 146) and Grandolini
and Cerda (1998, 13).
The box allows identifying at a glance some of
the idiosyncratic features of the old system mentioned above that have been eliminated. For one
thing, the minimum ten-year contribution necessary to qualify for retirement benefits has been
replaced by a minimum twenty-five-year contribution. Also, because there is only a minimum defined
benefit under the new regime, the asymmetric
inflation-indexing problem described above should
8
be eliminated. And because the notion of a social
security surplus has been eliminated, the funds
can no longer be a source of subsidy for other
IMSS activities. At the same time, because the
IVCM has been separated from the health care
and maternity benefits provided by IMSS, deficits
in these areas will be directly reflected in government deficits.
Issues Involving the Fund Managers. Workers
can choose any AFORE for contribution. Once an
AFORE is chosen, no change can be made for one
year, though it is possible to choose a different
AFORE every year without any financial penalty.
In Mexico, fund-hopping has been very low. In
1999, less than 0.01 percent of workers changed
funds. This stability stands in sharp contrast with
Chile, where fund-hopping has exceeded 25 percent per year.
By the end of 1997 CONSAR had licensed seventeen AFOREs (listed in Table 7). Some of the
AFOREs are fully owned by Mexican companies,
and others are partly owned by foreign companies.
For example, AFORE Bancomer is 51 percent
owned by the second-largest banking group in
Mexico, and the remaining 49 percent is owned by
Aetna, one of the largest insurance companies in
the United States. Garante has a particularly interesting ownership structure with majority shareholding by a Mexican group, part ownership by
Citibank, and part by a pension fund from Chile,
AFP Habitat. Ownership structure of Siglo XXI is
also notable: half of it is owned by the IMSS, the
government organization that continues to run
health care and disability and death insurance for
the entire system.
Three of the AFOREs established in 1997 have
merged with others. Confia bought Atlantico,
Santander bought Genesis, and Profuturo bought
Previnter. Consequently, as of August 1999, fourteen AFOREs are left in the market. All of these
mergers had to be approved by CONSAR.
Market Share. There were two very distinct
waves of membership in the new social security
scheme. The first was the initial rapid expansion
until the number of affiliates hit around 10,000,000
within a span of ten months (see Chart 1). Then
came a second, slower stage of expansion over the
next fourteen months. At the end of August 1999,
about 14,900,000 workers had signed up for one
AFORE or another.
It should be noted that of the approximately fifteen million workers who belonged to some AFORE
in August 1999, about 87 percent are active contributors. The fact that an individual signs up and
becomes an affiliate does not necessarily mean that
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
T A B L E 7 AFOREs Authorized by CONSAR, 1997
AFORE
Main Shareholders with Percentage Holding
Atlántico Promex
Banca Promex, 50; Banco del Atlántico, 50
Banamex
Grupo Financiero Banamex-Accival, 100
Bancomer
Grupo Financiero Bancomer, 51; Aetna Internacional, Inc., 49
Bancrecer-Dresdner
Grupo Financiero Bancrecer, 51; Dresdner Pension Fund Holdings, 44;
Allianz México, S.A., 5
Bital
Grupo Financiero Bital, 51; ING America Insurance Holding
Inc., 49
Capitaliza
General Electric Capital Assurance Co., 100
Confia-Principal
Abaco Grupo Financiero, 51; Principal International, 49
Garante
Grupo Financiero Serfín, 51; Grupo Financiero Citibank, 40;
Hábitat Desarrollo Internacional, 9
Génesis
Seguros Génesis, S.A., 100
Inbursa
Grupo Financiero Inbursa, 100
Previnter
Boston AIG Company, 90; Bank of Nova Scotia, 10
Profuturo GNP
Grupo Nacional Provincial, 51; Banco Bilbao Vizcaya-México,
S.A., 25; Provida Internacional, S.A., 24
Santander Mexicano
Grupo Financiero Inverméxico, 75; Santander Investment,
S.A., 25
Siglo XXI
Instituto Mexicano del Seguro Social, 50; IXE Grupo
Financiero, 50
Sólida Banorte
Grupo Financiero Banorte, 99
Tepeyac
Seguros Tepeyac, 99
Zurich
Zurich Vida, Compañía de Seguros, 77; Gabriel Monterrubio
Guasque, 10
Note: No mention is made of shareholders with equity participation under 5 percent of the total capital of the respective AFORE.
Source: Banco de Mexico (1997)
he or she will contribute to the system regularly. In
addition, each may have more than one account,
inflating the number of affiliates. SAR (Sistema de
Ahorro para el Retiro) accounts provide one classic
example: by the end of 1995, there were 65 million
accounts in SAR but less than twelve million workers in the formal sector.
The amount of contributions in the system has
also increased steadily. Between July 1997 and July
1998, investment in the system equaled US$3 billion
(at an exchange rate of 10 pesos per U.S. dollar as
of January 1999). Over the next seven months (July
1998 to January 1999), investment grew another
US$3 billion. If this trend continues, in twenty-five
years AFOREs will hold an amount equal to 40 percent of GDP (assuming a real GDP growth rate of
2 percent a year and real rate of return of funds at
6 percent a year).
Table 8 presents a summary of compulsory and
voluntary contributions to the existing AFOREs as
of the end of 1998. As is evident, the market is
highly concentrated, a feature common to other
Latin American countries such as Chile and
Argentina (see Queisser 1998).
CONSAR has explicitly prohibited any AFORE
from holding more than 17 percent of market share
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
9
CHART 1
Systemwide Take-Up Rates of AFOREs
through January 1999
Affiliates, Millions
14
10
6
2
0
10
20
Months
Source: CONSAR (1999)
in terms of the number of affiliates. However, it does
not restrict market share in terms of total value of
assets in portfolios. For example, an AFORE may
have 20 percent market share in terms of its portfolio’s value but less than 17 percent in terms of the
number of affiliates. The value of investments in
both Banamex and Bancomer has exceeded 17 percent during most of the past two years, for instance.
As Table 8 shows, the level of concentration in terms
of total investment portfolio is far higher than in
terms of the number of affiliates: six companies have
around 77 percent of the market share in terms of
investments. Contrary to what the regulation intended, with more market consolidation, this proportion
is likely to rise in the future.
The Portfolios of the Fund Managers. An
integral component of any pension system is the
composition of the portfolio held on behalf of its
contributors. As Table 9 shows, AFOREs’ portfolios
are heavily concentrated in government bonds.
This portfolio composition results from CONSAR’s
stipulation that a minimum of 51 percent of an
AFORE portfolio be held in the form of inflationindexed bonds and at least 65 percent in assets with
a maturity of no more than 183 days. CONSAR’s reasons for this portfolio requirement are to build trust
in the system and avoid volatility in the portfolio (see
CONSAR 1999). On January 31, 1999, more than 66
percent of AFOREs’ portfolios were in inflationindexed bonds (called BONDE91 and UDIBONOS).
Another 22 percent were in CETES (Mexican
10
Treasury bills). The average maturity of investment
portfolios is 111 days, well below CONSAR’s cap.
CONSAR specifies that an AFORE can hold up to
35 percent of its portfolio (disposicion quinta, 5) as
private debts (CONSAR 1997). Given this range,
why do private debt holdings amount to only 2.83
percent of all portfolio assets? CONSAR’s qualifications on the type of debt that can be included probably account for the low figure. Regulations specify
that only private short-term debt meeting Standard
and Poor’s mxA-3 grade or equivalent and longterm private debt meeting Standard and Poor’s
mxAA+/mxAA grade would make the cut (CONSAR
1997, chap. 3 and app. A), and only a very small
fraction of Mexican private debt meets these eligibility requirements.
Recent Proposals for Portfolio Changes.
Recently there has been criticism about the need
for Mexico to move forward with privatizing its pension system (for example, Rodriguez 1999).
Specifically, criticism has focused on the fact that
the AFOREs’ portfolios consist mostly of government debt. Even though, as explained in EspinosaVega and Russell (1999), there would be no
guarantee that the new system would be fully funded
if the government relaxed its high government-debt
requirement for AFOREs’ portfolios, questions
remain about the economic impact of allowing
AFOREs more flexibility in the composition of their
portfolios.7 Recently, the Mexican federal government has been considering a proposal to allow
AFOREs greater flexibility on how retirement savings are invested. CONSAR has proposed to
increase investment options for the fund administrators, including the right to buy debt sold by
Mexican corporations in overseas markets. It is
not hard to foresee a trend in the direction of
allowing AFOREs to have larger holdings of stocks.
As explained in the next paragraph, this move,
together with the issuance of inflation-indexed
long-term government bonds, may represent net
gains for the generation of active workers when
they retire.
In the shift from defined benefits under the old
system to defined contributions under the new system, risk is shifted from workers’ active years to
their years of retirement. The reason for this shift
involves risk created by the possibility of increases
in the inflation rate. Under Mexico’s old definedbenefits social security system, the nominal (peso)
value of the benefits was indexed to the inflation
rate. Thus, workers did not have to fear that the
purchasing power of their social security benefits
would be reduced by higher inflation. However, during periods when the government had budget prob-
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
T A B L E 8 Money Invested and Number of Affiliates in AFOREs
Fund
Amount in Pesosa
Percent
Affiliatesb
Banamex
10,209,603,059
18.16
1,742,930
Bancomer
13,183,450,811
23.45
2,364,074
Bancrecer
1,981,244,448
3.52
619,789
Banorte
2,628,350,064
4.68
1,260,762
Bital
4,772,054,894
8.49
1,499,758
Confía
1,027,806,121
1.83
332,999
Garante
4,824,234,812
8.58
1,633,528
Génesis
302,320,757
0.54
140,957
Inbursa
5,072,806,294
9.02
378,376
Profuturo
5,155,845,595
9.17
1,998,211
Santander
3,534,727,403
6.29
2,026,656
Siglo XXI
3,041,053,960
5.41
462,473
Tepeyac
285,687,001
0.51
228,621
Zurich
199,941,483
0.36
185,576
56,219,126,705
100
14,874,710
Total
a
As of the end of 1998
As of the end of August 1999
Source: CONSAR (1999)
b
T A B L E 9 Asset Allocation of AFOREs at the End of 1998
Type
Amount in Pesos
Percent
Nominal government
13,699,239,431
22.54
Real government
40,139,931,258
66.04
Repurchase agreements
1,810,442,216
2.98
Private papers
1,722,866,789
2.83
Deposit in Banco de Mexico
3,404,749,272
5.60
60,777,228,966
100
Total
Note: Nominal government means government bonds denominated in nominal terms. Real government means government bonds denominated
in real (inflation-adjusted) terms.
Source: CONSAR (1999)
lems—for example, if weak performance of the
economy reduced tax revenues or high interest
rates increased the burden of debt service—the
government had to increase taxes or borrow to
maintain the level of benefits.
Under the new system, an increase in the inflation
rate will reduce the purchasing power of the govern-
ment or private bonds with fixed nominal values held
by the social security system. This inflation will also
reduce the purchasing power of benefits paid to
retirees. The government may be tempted to take
advantage of this fact and increase the inflation rate
during periods when it has budget problems, rather
than increasing taxes or borrowing. Thus, the new
7. For a fuller explanation of the related issues, see Espinosa-Vega and Russell (1999).
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
11
system may increase the risk facing retirees, but it
will reduce the risk facing active workers.
On the other hand, there are reasons to expect that
the amount of inflation risk facing retirees under the
new system may not be very large. First, some of the
assets held by the Mexican social security system will
consist of stock, and the rate of return on stock tends
to increase when the inflation rate increases so that
some of the reduction in purchasing power from
retiree benefits from government or private bonds
will be offset. Second, most of the bonds held by the
system are likely to be short-term bonds. The longterm bond market in Mexico currently has a low volume of issues and little trading, and it will probably
stay that way unless or until the government convinces the public that it is unlikely to indulge in high
inflation. As the inflation rate increases, maturing
short-term bonds can be replaced with new shortterm bonds yielding higher interest rates, so the
losses from inflation will be limited. Finally, the
interest rates on many of the longer-term bonds purchased by the system may be indexed to the inflation
rate. The Mexican government has recently begun to
issue indexed bonds in substantial quantities.
Transactions Costs and Commissions. A key
source of dissatisfaction and confusion with most
newly privatized pension systems is the fees
charged by the fund managers (also called commissions or costs of transactions). The main concerns
are that the management fees are high, that the fees
are lumped with insurance premiums for the life
insurance component of the pension plans (see
Sinha forthcoming, chap. 3), that the management
fees are obfuscated because in most cases they are
presented as a fraction of a worker’s salary, and that
sometimes it is not clear whether the commissions
are expressed as a proportion of flow into the fund
on a yearly basis or as a proportion of balance in the
fund at a given point in time. As pointed out by
Diamond (1998), some of these concerns are widespread. For that reason, this section takes a close
look at the commission structure for the Mexican
pension system.
Table 10 gives the details of the commissions
charged by the 17 AFOREs that started out in 1997.
The three of them that have since merged with others have assumed the names of the companies
under which they are operating.
Most of the commissions charged apply to the flow
of contributions. However, some companies charge
on the balance in the fund as well as on flows. One
company (Inbursa) charges commissions exclusively
on the real (inflation-adjusted) rate of return of the
fund. (Inbursa charges no fee if the real rate of
return is not positive.) In addition to the different fee
12
structures, the way charges are expressed is somewhat misleading because they are expressed as a
percentage of wages and not as a percentage of contribution every year.8 For example, 1.7 percent
charges on a person contributing to the system a
mandatory 6.5 percent of her $100.00 wage will be
effectively paying a $26.15 (26.15 percent) commission charge.
Given that it is somewhat difficult to compare
charges across different AFOREs using the figures
in Table 10, CONSAR has worked out commission
“equivalents.” The idea is that all charges are converted to a charge only on flow of funds. CONSAR
publishes these estimates, shown here in Table 11.
The results assume that the real rate of return is 5
percent (with no inflation), the charges are applied
to a person with three times the minimum wage, and
that person has the same income throughout life.
Even after the conversion into equivalent charges
on flow, comparisons of different AFOREs are difficult. Among other things, the commission charged is
effectively a function of years in the system. It is also
a function of factors such as the expected rate of
return and the level of wages (some of these results
can be seen in Sinha, Martinez, and Barrios-Muñoz
1999). In addition, as Table 11 shows, in some cases
(Siglo XXI, Tepeyac, Profuturo, Santander) charges
actually go up monotonically without falling over
time for the same AFORE. The reason is simple.
These companies charge fees on contributions as
well as on the account balance. As time passes, more
money is accumulated in the accounts, and the bite
of charges on the account balance gets bigger.
Assessing the Reform
he focus of the analysis thus far has been
the pension system of the formal private
sector in Mexico. The discussion has examined the factors that have accounted for surpluses
on this pension system and the strains that, if
unchanged, would transform these surpluses into
deficits in the near term. It also describes key
aspects of the new pension system such as the new
managers of the system (AFOREs), their portfolios, and recent proposals to modify their portfolios
and looks at the new eligibility criteria, the “admission fee,” and the replacement rate under the new
system. The next sections establish what it would
take to assess the significance of the reform and
identify a research agenda.
Most economists agree that the old Mexican pension system was a pay-as-you-go system. Many of the
predictions about the state of the economy under the
new pension system (for example, a sharp increase
in national saving) have the economy resembling one
T
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
T A B L E 1 0 Fee Structure of AFOREs
AFOREs
Atlántico Promex
Charges on Flow Each
Year (Percent of Wages)
Charge on Account
Balance (Percent)
1.40
Charge on Real Rate
of Return (Percent)
20.00
Banamex
1997
0.20
January 1998
0.85
March 1998 onward
1.70
Bancomer
1.70
Bancrecer-Dresdner
1.60
Banorte
1.00
Bital
1.68
Capitaliza
1.60
Confia Principal
0.90
Garante
1.68
Génesis
1.65
1.50
1.00
Inbursa
33.00
Previnter
1.55
Profuturo GNP
1.70
0.50
Santander
1.70
1.00
Siglo XXI
1.50
0.20
Tepeyac
1.17
1.00
Zurich
0.95
Variable
Note: In addition, Bancomer, Banamex, Bital, Garante, and Génesis have discounts for people who stay with their funds for long periods
of time. These are not shown in the table above.
Source: CONSAR (1999)
under a fully funded system. Thus, the old and new
social security systems seem to be two radically different systems. An important question to ask, however, is whether the new system is truly a fully
funded system or simply represents a change in the
form of the pay-as-you-go system.
Fully Funded: Is It There Yet?9 There are a
number of ways to engage in genuine reform, that
is, to go from a given modality of a pay-as-you-go
system to a fully funded system. However, choosing
the way to reform is not trivial. Each alternative
social security scheme implies different costs for
different generations of workers, and their implementation thus can be subject to the forces of political discourse. The simplest way to carry out a
genuine reform would be to have the current workers
pay (on top of their regular pension contributions)
for the benefits of the current generation of retirees.
8. The following illustrates the difference: Expressing the commission in terms of a fraction (z) of a person’s wage (w) would
mean that her total commission payments t = z × w. The total commission payments by an individual who is required to contribute a fraction (ss) of her wage, subject to a commission (cc) on her contribution would be t = ss × w × cc. This means that
one can extract the effective commission fee (cc) by noticing that cc = z × w/(ss × w).
9. This section relies heavily on Espinosa-Vega and Russell (1999). Readers should consult that companion article for a fuller
understanding of the issues.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
13
T A B L E 1 1 Equivalence of Commissions (Percentage of Wages)
Fund
One Year
Two Years
Five Years
Ten Years
Twenty Years
Banamex
1.70
1.70
1.69
1.65
1.58
Bancomer
1.68
1.68
1.66
1.65
1.64
Bancrecer
1.60
1.60
1.60
1.57
1.51
Banorte
1.16
1.18
1.23
1.29
1.44
Bital
1.68
1.68
1.68
1.64
1.61
Garante
1.68
1.68
1.68
1.68
1.68
Génesis
1.65
1.65
1.65
1.65
1.65
Inbursa
0.36
0.43
0.64
1.00
1.73
Principal
1.43
1.43
1.44
1.44
1.41
Profuturo
1.75
1.76
1.79
1.84
1.95
Santander
1.81
1.82
1.86
1.98
2.19
Siglo XXI
1.52
1.53
1.54
1.56
1.60
Tepeyac
1.28
1.30
1.34
1.47
1.69
Zurich
1.09
1.09
1.14
1.19
1.14
Source: CONSAR (1999)
Clearly, though, this approach would place an
unbearable burden on current workers. An alternative would be to issue debt to pay off the current
retirees at the time of the reform and then retire the
debt, through time, by taxing the current and future
workers for a number of years. Under either scenario,
the government’s actions at the beginning of the transition process would be the same. Bonds must be
issued to obtain funds needed for social security payments to current and near-future retirees.
The actions that will distinguish a transition to a
fully funded system from a transition to a pay-as-yougo system of the bond/tax-or-transfer type will occur
in the future. If the government switches to a fully
funded system, then over the next few generations it
must collect enough revenue, via new taxes, to retire
the aforementioned bonds. If it is switching to a payas-you-go, bond/tax-or-transfer system, however,
then it may not have to change its total social security tax collections because it will roll the bonds
over indefinitely without retiring any of them.10
How can it be known today whether the Mexican
government will retire the bonds in the future? That
is to say, how can the government’s intentions to
switch to a fully funded social security system be
known at this point? Although the government has
announced that it does plan to switch to a fully
funded system, it has not announced any plans to
14
increase future taxes and it has not announced any
schedule for retiring the bonds. Even if the government did make such announcements, how credible
would they be? Future Mexican governments
might feel free to ignore them, either by explicitly
reversing the decision to retire the debt or by postponing the beginning of the debt-retirement
process. Future governments would have plenty of
incentive to not follow through. Beginning the bondretirement process would require increasing taxes, a
move likely to be opposed by the voting public.
Viewed in this light, there are good reasons to
question the likelihood of Mexico’s ultimate success
in switching to a fully funded social security system
as well as the motivations for the reform. On the one
hand, the government may wish to get the credit for
initiating a switch to a fully funded system—a system, which, in the view of most economists, would
be better for Mexico in the long run. On the other
hand, the government may be slow to take any concrete steps to begin the transition to such a system
because, as mentioned above, they would be politically costly in the short run.11 It seems more likely,
then, that the switch will turn out to be one from a
tax-transfer pay-as-you-go system to a bond-based
pay-as-you-go system.
Why Change Systems? A switch of this sort may
have some significant economic effects, but perhaps,
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
more importantly, it creates the appearance of
reform. It does so in several different ways. First,
since switching to a bond-based system could (but
does not necessarily) represent the very first step in
a transition to a fully funded system, this switch
allows the government to claim that it has begun the
transition process. Second, the switch to a bondbased system allows the government to privatize a
number of aspects of the administration of the social
security system—a step that might have some benefits in its own right and that many people are likely to
misinterpret as representing more effectual reform.
Third, all the idiosyncratic features of the old system
discussed above represent economic distortions for
the decisions of firms and workers in the system.
And, as mentioned earlier, other than demographics,
it is these idiosyncratic features that would have contributed to the actuarial imbalance of the Mexican
pay-as-you-go system. The adoption of a pay-as-yougo system does not require indexing benefits against
inflation while leaving contributions unchanged, awkward qualification criteria (lax eligibility requirements), or allowing the government to use the
system as an outright source of government revenue.
Elimination of these features could go a long way in
improving the operation of the old pay-as-you-go system and solving its actuarial imbalance problems.
Asymmetric Indexation and the Admission
Fee. Indexing benefits against inflation while leaving
contributions unchanged insulates social security
recipients, but it represents higher taxes in other
sectors or a higher government deficit. This asymmetric indexation became a serious problem when
inflation rose to triple digits in 1994–95. This problem
will be avoided by the move to defined contributions.
As discussed above, the low admission fee under
the old system created the incentive to stop pension
contributions altogether after the initial 500 weeks
required for becoming eligible for retirement benefits. There is also clear evidence that many employers understated the wage rate of workers just to
avoid paying the payroll tax. Under this admission
scheme, contributing workers in the formal sector
subsidized those in the informal sector that had
stopped contributing once they had paid their low
admission fee. Under the new system, a minimum
twenty-five year contribution is required to qualify
for any benefits, and at retirement a worker gets the
market return to his contributions. This tightening
of the eligibility requirements will eliminate the
strain that the old requirements had on the IMSS
pension accounts.
Management Fees. The new system introduces a
feature only implicit in the old system: transaction
costs (or commissions) imposed by AFOREs on their
contributors. It is not straightforward to compare the
transaction fees charged by the different AFOREs.
However, careful review (see Sinha 1999a) reveals
that most charges (by private pension companies in
other Latin American countries) are in the order of
20–25 percent of contributions—ten times higher
than charges in defined-contribution plans
of Singapore or Malaysia. The usual defense offered
by proponents of Mexico’s system has been that
(1) Chile has a similar cost structure and (2) mutual
funds in the United States have similar cost structures. In fact, CONSAR has recently used the data in
Table 12 to argue that the commission in Mexico is
on the average lower than in other Latin American
countries, a claim originally made by Solís-Soberón
(1997).
There are potential problems with reaching such
a conclusion, however, as comparing Chile and
Mexico’s commission structures illustrates. Table 12
states commission as a percentage of covered pay,
not as a percentage of contribution. In Chile each
worker contributes 10 percent of the salary into the
system whereas in Mexico the contribution is only 6.5
percent of salary. Thus, as a percentage of contribution, average charges in Mexico would be more than
29 percent (1.919/6.5) while in Chile it is under 23
percent (2.291/10). Another factor is that the government contribution to the system has been
ignored. It is difficult to evaluate the government
contribution because it varies with the wage rate of
the worker. For an average worker earning three
times the minimum salary, it amounts to 1.83 percent
of the salary. If that factor is added to 6.5 percent, the
total is 8.33 percent. Computed this way, the commission charges amount to 23 percent (1.919/8.33),
exactly the same as charges imposed in Chile. This
complexity illustrates the need for a careful review of
the claim that Mexico has managed to reduce the
charges that Chile could not.
In contrasting transaction costs of pensions in
Mexico with those charged by mutual funds in the
United States, the following has to be considered.
First, charges for the majority of mutual funds in the
United States are on the order of 10 percent
(of contribution) and not 25 percent (see Mitchell
10. The government will have to pay the interest on the bonds, but it can do so without increasing its social security tax
collections.
11. Cooley and Soares (1999) discuss how a pay-as-you-go system may in fact represent the rational politico-economic outcome
in a democratic regime.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
15
TABLE 12
Commission Structure Comparison
Country
Percent of Covered Pay
Argentina
2.410
Chile
2.291
Mexico
1.919
Peru
2.294
Uruguay
2.070
Source: Solís-Soberón (1997)
1996). In fact, there are mutual funds that charge
1 percent or less. Second, unlike in the case of
mutual funds, in Mexico (as in Chile) membership in
an AFORE is compulsory. Thus, even if the transaction costs were the same, comparing the two would
be like comparing apples and oranges.
The most pertinent question for this discussion
concerns how the transaction costs under the new
system compare with costs under the old pay-as-yougo system. Under the old system, administrative costs
(the analog of transaction costs) as a percentage of
total social security expenditures hovered around
17 percent in the 1980s for Mexico (International
Labor Organization 1991). As mentioned above, on
average the current transaction cost is around 24 percent, implying a cost increase of roughly 7 percent.
It appears that, at least for the time being,
AFOREs are acting as bookkeeping entities with few
true portfolio manager functions. As mentioned
above, only a very small fraction of Mexican private
debt meets CONSAR’s eligibility requirements. As is
clear in Table 11, the 35 percent ceiling that an
AFORE is allowed to hold in the form of private
debt is far from binding. In this context, it might
make sense for the government to solicit bids and
choose the single bookkeeper charging the lowest
management fee. While a monopoly in the private
management of widely diversified pension portfolios is not desirable, a monopoly in bookkeeping
may very well be, as Bolivia’s experience demonstrates. Bolivia awarded a monopoly right to an
international consortium to manage its pension system.12 The commission charges are 0.5 percent of
average salary (10.5 percent of salary for retirement and 2 percent for life insurance).13 In contrast, under the Mexican system, the average
charges are 1.6 percent of salary with only 6.5 percent going into the retirement fund. In addition, in
Bolivia the disability and death insurance payment
is equal to 4 percent of salary.
16
A potential justification for leaving bookkeeping
and portfolio management together in the AFOREs
could be that the AFOREs will be active when
CONSAR relaxes its portfolio restrictions. The
problem with this justification is that separating
bookkeepers and fund managers may be more efficient even under these conditions. The best
bookkeepers may not be the best managers of a
wider-spectrum portfolio.
The Replacement Rate. It is also important to
recognize that because of differences in how the
replacement rate is determined—under the new system it will be market-determined—contrasting alternative replacement rate projections under the new
system against the replacement rate under the old
system will be challenging. The ultimate concern, of
course, is whether the new replacement rate will represent an improvement for future retirees. A complete answer requires a sophisticated analysis. The
replacement rate under the new regime is the
endogenous result of changes in factors such as wage
levels (because the value of the social quota depends
on the level of wages; that is, a low-wage earner gets
a relatively high social quota), real interest rates,
mortality rates (because mortality rates are going to
change over the next sixty years), the discount rate
at which the whole life annuity is calculated, and
commissions (for example, some commissions
depend on the inflation rate, as for one company that
does not charge if there are no gains in real terms).
Table 13 illustrates the sensitivity of the replacement rate to alternative economic scenarios. It
reports the resulting replacement rates, other
things being equal, under alternative real rates of
return of AFOREs’ portfolios. These simulations are
subject to criticism on a number of grounds. Among
other things, they take a flat lifetime wage profile,
assume a flat interest rate profile with zero inflation
rates, take mortality projections from United
Nations (1998), and assume that commissions stay
put. However, the simulations suffice to focus attention on the sensitivity of the replacement rate to the
relevant economic variables.
Table 13 also has calculations of replacement rates
under different scenarios when three elements are
altered: (1) The replacement rate is calculated with
different real rates of return. However, the effects of
inflation are not taken into account, and since earnings of managed funds depend on the nominal interest rate, inflation would have an impact. (2) For each
panel of the table, the wage rate varies. Wage rates
are expressed as multiples of the minimum wage. So
calculations are made with one, two, three, four, five,
six, and ten multiples of minimum wages. (3) The
number of years in the labor force also varies (from
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
T A B L E 1 3 Replacement-Rate Calculations for Whole Life Annuity Starting at Age 65a
Salary in Times Minimum Wage
Time (Years)
1
2
3
4
5
6
10
Real Rate of Return, 3 Percent; Real Wage Growth Rate, 0 Percent
5
3.53
2.63
2.32
2.17
2.08
2.02
1.90
10
7.41
5.51
4.87
4.56
4.37
4.24
3.99
15
11.83
8.79
7.78
7.28
6.97
6.77
6.37
20
16.85
12.53
11.09
10.37
9.94
9.65
9.07
25
22.54
16.76
14.84
13.87
13.30
12.91
12.14
30
29.01
21.57
19.09
17.85
17.11
16.61
15.62
35
36.35
27.03
23.92
22.37
21.44
20.82
19.57
40
44.69
33.23
29.41
27.50
26.36
25.59
24.06
45
54.17
40.28
35.65
33.33
31.94
31.02
29.16
Real Rate of Return, 10 Percent; Real Wage Growth Rate, 0 Percent
a
5
4.20
3.12
2.76
2.58
2.47
2.40
2.26
10
10.55
7.84
6.94
6.49
6.22
6.04
5.67
15
20.51
15.24
13.48
12.61
12.08
11.73
11.03
20
36.09
26.82
23.73
22.18
21.25
20.64
19.40
25
60.49
44.94
39.75
37.16
35.61
34.57
32.50
30
98.74
73.33
64.87
60.63
58.09
56.40
53.01
35
158.78
117.89
104.27
97.45
93.36
90.64
85.19
40
253.13
187.90
166.16
155.28
148.76
144.41
135.72
45
401.56
298.00
263.48
246.22
235.86
228.96
215.15
Calculated with a flat lifetime wage profile and no consideration of inflation.
Source: Sinha (1999b)
five to forty-five years). Note that the wage profile
does not vary; the wage rate for every year in the
labor force is assumed to be the same.
Consider, for example, the first entry of the top panel
of the table—3.53 percent. A person earning the minimum wage for five years and retiring at the age of 65
will get 3.53 percent of his or her wage replaced if he
or she earns one minimum wage. Each entry has two
other elements built into it. One is the assumption
about the real interest rate that an annuity would earn.
The other is the (conditional) mortality rate of the population (after retiring at 65). The 3.53 percent results
from calculating the replacement rate that would be
the average of the rates obtained under each of the seventeen AFOREs. Therefore, each calculation explicitly
takes into account management or commission fees.
For understanding the significance of the different replacement rates, it is helpful to compare these
replacement rates to the U.S. average. For instance,
in the United States in 1998 the average wage was
around $28,000. The retirement benefit after fortythree years of service was $11,256. This amounts,
roughly, to a 40 percent (11,256/28,000) replacement rate. In Mexico, the average salary is slightly
more than three times minimum salary. In U.S. dollars, this amount is around $10 a day. Thus, under
the scenario with 3 percent real rate of return, to
get a replacement rate of 40 percent the average
worker has to work for more than forty-five years.
Under the assumption of a 10 percent real rate of
return, the 40 percent replacement rate can be
achieved in twenty-five years.
12. The consortium consists of Banco Bilbao Vizcaya S.A. and Invesco-Argentaria.
13. See von Gersdorff (1997) for a summary of the Bolivian system.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
17
How does the new system compare with the old in
terms of replacement rates? For a person earning one
minimum salary, under the old system a 100 percent
replacement could be had in ten years (Table 2).
Under the new system, if the real rate of return is
3 percent, a person with one minimum salary after
forty-five years of service would be able to achieve
only a 54.17 percent replacement rate. With a real
rate of return of 10 percent, the same person after
thirty years of service will get 98.74 percent of the
salary replaced. Only if a worker stayed in the labor
force for at least thirty-five years and the rate of
return was high (10 percent or more) would the
retirement benefits under the new system be higher
(for almost all wage levels) than those under the old
system. One other important observation is that for
low-income workers the replacement rate is always
higher than for high-income workers. The reason is
that as a percentage of income the social quota is
financially more important for low-income workers.
What Are the Net Gains from Switching across
Pay-As-You-Go Pension Systems in Mexico?
he last section identified some of the potential gains and costs of the reform. However,
what needs to be established are the net
gains or overall effects on well-being for current and
transitional retirees as well as for future generations
of Mexican citizens. Economic outcomes are the
result of complex simultaneous interactions among
different economic variables in both the short and
long runs. To say, for example, that the government
contributes a “cuota social” to the retirement fund
of a worker (as is currently the case under the
reform) is to say that the government commits to
borrowing or taxing in the future (from either the
same or future generation of workers) to meet the
obligation of an accounting entry. Thus, contrasting
the well-being of Mexican citizens under the alternative pay-as-you-go systems would require a
sophisticated analysis. At a minimum, the analysis
should recognize the intertemporal nature of individuals’ decision making, that individuals’ expectations are based on all available information, including
government policies, and that economic variables
interact with one another. Thus, to determine the
consequences of a reform, one must consider saving
decisions, taxing, and government debt policies
simultaneously— a general equilibrium framework.
The following example illustrates how such an
approach could make a difference. De Nardi,
Imrohoroglu, and Sargent (1999) look at the effects
of projected U.S. demographics on its current payas-you-go system. They use projected increases in
the dependency ratio and analyze the economic
T
18
consequences of several alternative fiscal adjustment packages. One of their experiments consists of
leaving the social security system unfunded (perhaps an analog of the Mexican case). They conclude
that back-of-the-envelope accounting calculations
made outside a general equilibrium framework differ significantly from those obtained in a general
equilibrium context. One may therefore wonder
about the accuracy of the projected actuarial imbalances—discussed earlier—arising from sticking to
the old Mexican pension system.
Another finding in De Nardi, Imrohoroglu, and
Sargent is that even when a country sticks to a payas-you-go system, “reducing retirement benefits
through taxation of benefits and consumption or
through postponing the retirement eligibility age
results in a significant reduction of the fiscal adjustment necessary to cope with the aging of the population” (1999, 578). As discussed above, under the new
Mexican system, there is a new minimum twenty-five
year contribution required to qualify for any benefits.
This regulation amounts to a reduction of retirement
benefits. Also, as discussed above, the Mexican
reform may not represent a departure from its payas-you-go system. Thus, just as in the comparisons
performed in De Nardi, Imrohoroglu, and Sargent,
the relevant comparison in Mexico may be across two
pay-as-you-go regimes rather than between a pay-asyou-go and a fully funded system. Because that is the
case, the relevant analysis would, for example, have
to contrast the net benefits of maintaining a pay-asyou-go system while changing the minimum contribution requirement from ten to twenty-five years and
introducing the cuota social and other features of the
new system reviewed above. It would not be surprising if, just as in De Nardi, Imrohoroglu, and Sargent,
a reduction of retirement benefits resulted in a significant reduction of the adjustments necessary to
cope with the aging of the population in Mexico.
However, De Nardi, Imrohoroglu, and Sargent’s findings are specific to the structure and parameters of
the U.S. economy, and it seems essential to perform
the same experiment for the specific parameters of
the Mexican economy. To date, there has not been a
general equilibrium analysis of the net benefits of
modifying the pay-as-you-go system in Mexico. Thus
the answer to the question of what the net gains of a
modified pay-as-you-go system might be is that
nobody knows.
What Are the Net Gains from Switching to a
Fully Funded Pension System?
A
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
s discussed above as well as more thoroughly
in Espinosa-Vega and Russell (1999), the
theoretical literature suggests that, with
B O X
Comparison of Mexico’s Pay-As-You-Go
and Reformed Old-Age Security Systems
Area
Institutional responsibilities:
Old age and severance (RCV)
Disability and life insurance (IV)
Contributions (percent of wage)a:
Contribution by employer
and employee
Government contribution
Eligibility requirements:
Old age
Severance
Pay-As-You-Go
IMSS
a
b
c
d
New entrant picks
AFORES or IMSS
retirement
(transition
generation only)
IMSS
IMSS
10.075
10.075
0.425
2.425
500 weeks’ (10 years’)
contribution; 65 years old
25 years’ contribution;
65 years old
500 weeks’ contribution;
60 years old
25 years’ contribution;
60 years old
Old age: Withdrawalsb
Minimum pension guarantee (MPG)
Reformed
Gradual withdrawals
from individual
account
in AFORE,c or
annuity bought from
an insurance company
Equivalent to one Mexico City
minimum-wage level indexed to
actual minimum wage
Equivalent to one
Mexico City minimum
wage on 7/1/97
indexed to the CPId
Under IVCM, contributions could not exceed ten times the minimum wage, and under the new system the limit is twenty-five
times. The column listing the after-reform structure includes Life and Disability Assurance.
Lump withdrawal at retirement permitted only for balances in excess of 130 percent of the cost of an annuity equal to the
minimum pension guarantee (MPG).
Only gradual withdrawals are allowed in order to reduce the risk that recipients will outlive their accumulated balances.
Currently average wage for IMSS affiliates is 2.6 minimum wages; thus MPG is approximately 38 percent of average wage.
Sources: Grandolini and Cerda (1998) and Sales-Sarrapy, Solís-Soberón, and Villagómez-Amezcua (1998)
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
19
some qualifications, moving to a fully funded system
or abolishing a pension system altogether should
improve the well-being of society at large. The qualifications include a country having a dynamically
efficient economy and the absence of any significant imperfections in capital and labor markets. A
number of students of pension systems agree with
Abel and others (1989) that the U.S. economy is
dynamically efficient but acknowledge that the
United States may experience some potentially significant market failures. Huang, Imrohoroglu, and
Sargent (1997) start by explicitly incorporating
what is generally considered a standard market failure. In their model, there are uninsurable uncertainties about lifetimes and labor income. This type
of market failure has indeed been used to justify
pay-as-you-go social security systems.
Huang, Imrohoroglu, and Sargent perform two
experiments similar to the actions some analysts
claim Mexico has started to take. In the first, the
government eliminates its pay-as-you-go system,
issuing a large quantity of bonds to buy out the transitional generation of retirees. At the same time, the
government raises labor income taxes for the next
forty years in order to pay back this debt (this tax
represents the transition cost). The second experiment is motivated by some recent proposals in the
United States that call for investing the current
social security surpluses in the stock market. In their
second experiment, the government raises labor
income taxes. And the proceeds are used to acquire
equity. Future retirement benefits are then paid out
of the returns to this equity. The advantage of adopting this government-sanctioned fully funded scheme
is that it allows the government to redistribute benefits to those citizens who because of, for example,
extended layoff periods were unable to accumulate
enough savings for their retirement or those citizens
who outlived their retirement benefits. They show
that in the long run, the second experiment provides
the larger net gain. While the study was performed
for the U.S. economy parameters (and, as the
authors note, there are a number of directions in
which it could be improved), its importance for
Mexico consists of recognizing that the size of the
net gains is itself a function of key features of the
economy, including the specific type of market failures in place (such as the large size of the informal
sector in Mexico). For example, a common theme
throughout most analyses of Mexico’s reform is the
expected migration from the informal to the formal
sector of the economy. In order to understand the
impact of the reform on the informal-sector workers’
decisions to migrate, one has to endogenize informality. Stated differently, there is no clear idea of the
20
sensitivity of workers’ decisions about where to work
and the role of changes in payroll taxes associated
with alternative pension schemes in those decisions.
Neither is it clear how workers’ decisions about
migrating in turn affect the ultimate impact of government policies in the economy.
At the same time, it may be of significant relevance
where the revenues needed to pay for the transition
come from. The macroeconomic implications of
either taxing wage income or issuing government
debt to finance the transition from a pay-as-you-go to
a fully funded system may be quite different from
each other. Also, it is important to specify the timing
and the type of tax to be used to finance the deficit
that arises from financing the reform; otherwise, the
analysis will be at best incomplete.
Serrano (1999b) attempts to estimate the net benefits of switching to a funded system in Mexico in the
context of a general equilibrium analysis.14 He finds
that the gains from doing so significantly outweigh
the present value of the transition costs, which in his
worst-case scenario represent 59.3 percent of 1997
GDP. Should CONSAR view Serrano’s work as their
endorsement to press ahead with a reform to a fully
funded pension in Mexico? In an effort to answer
this question, it is important to outline Serrano’s
study. As in Huang, Imrohoroglu, and Sargent
(1997), Serrano incorporates a market failure in his
analysis. The market failure consists of a minimumdenomination restriction in the formal financial
intermediation sector under the pay-as-you-go system. That is, he assumes that when such a pension
system is in operation, poor savers are unable to participate in the formal banking sector. In his analysis,
once a fully funded system is adopted, these poor
savers will have access to the formal financial system. In his words, “Our thesis is that many poor
workers will obtain access to the formal financial
system through the privatized social security system. . . . The introduction of an obligatory FF [fully
funded] system may give these people access to the
financial system (1999b, 3).” In other words, adopting a fully funded system would have the same
benefits, in his analysis, as eliminating the
minimum-denomination restriction in the formal
banking sector. The question, of course, would be
why the poor savers did not have access to the formal financial system in the first place. The answer
may be that the low level of savings by the poor was
insufficient to justify the necessary maintenance or
transaction costs incurred by financial intermediaries when opening a new account. If this is the case,
what leads Serrano to believe that things would be
different under a fully funded system? The government could, of course, subsidize these transaction
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
costs, but it could do that regardless of the pension
system in place. In addition, the overall impact of
such subsidies would have to be carefully analyzed.
It is also true that a large number of poor savers
are part of the informal labor market. A governmentsanctioned fully funded system is relevant to them
only if they migrate to the formal labor market.
Serrano would have to explain why such a pension
system would lead to migration to the formal sector.
Following the steps of Serrano’s analysis, one cannot disentangle where the gains come from. Do they
come from eliminating the market failure, or
do they come from switching from a pay-as-you-go
to a fully funded system? In short, it seems best to
think of Serrano’s work as a work in progress.
Another important point to note is that Serrano’s
estimate of the transition cost (59.3 percent of 1997
GDP) serves as a reminder that even when there
are net gains from switching to a funded system the
transition imposes a hefty cost on any society.
Agreeing to when and how to pay for it would be the
subject of difficult political discourse. It is no coincidence that while there are other estimates of the
transition costs, there is no mention (other than
Serrano’s) about when and how such costs would be
taken care of.
In addition to Serrano (1999b), other articles estimate the transition cost. By the authors’ own admission, these estimates are far from perfect; for
example, Sales-Sarrapy, Solís-Soberón, and
Villagómez-Amezcua acknowledge that their model
“is a partial equilibrium framework that treats relevant macroeconomic variables as given” (1998,
158). Partial equilibrium estimates may drastically
err in either direction, as discussed above and
shown by De Nardi, Imrohoroglu, and Sargent
(1999). For completeness, this discussion includes
transition costs estimates by Sales-Sarrapy, SolísSoberón, and Villagómez-Amezcua (1998) and
Grandolini and Cerda (1998) with a reminder that
these costs are only part of the information needed
for estimating the net gains of the reform.
Sales-Sarrapy, Solís-Soberón, and VillagómezAmezcua (1998) and Grandolini and Cerda (1998)
acknowledge that moving away from the old
pay-as-you-go system will impose on the Mexican
economy two types of costs with certainty and
another two potential costs. First, the government
has to pay the so-called social quota to every participant, and that payout affects the government
deficit. As can be seen in the box, this amount on
average equals an additional 2 percent of wages.
Second, since all contributions of private-sector
workers went to AFOREs starting September 1,
1997, one has to consider the resources necessary
to provide payments to pensioners existing prior to
that date.
The first group of retirees under the new system
will emerge in about twenty-five years. What exactly
will they get? There are two possible scenarios. In
the first, if the funds in the individual’s account at
retirement do not exceed an annual income stream
equivalent to one minimum wage for the actuarial
remainder of his life, then the government will
guarantee benefits equivalent to one minimum
wage for the duration. This approach may affect
the government deficit. In the second scenario, if
the individual’s account exceeds an annual income
stream equivalent to one minimum wage for the
actuarial remainder of the worker’s life, he or she
can choose between withdrawing funds on a
monthly basis or purchasing a lifetime private contingent annuity. Under this scenario there is no
impact on the deficit. Transition workers have a
pension-guaranteed switch option. At retirement,
they will be able to choose between their benefits
under the old or the new system. Their choices may
have an impact on the government deficit.
Grandolini and Cerda (1998, table 11, 24) report
the fiscal cost of the transition as calculated by the
Ministry of Finance (Secretaría de Hacienda y
Crédito Público, or SHCP). The present value of the
total cost of the transition from 1997 to 2024, as a
fraction of 1994 GDP, is 17.76 percent. SalesSarrapy, Solís-Soberón, and Villagómez-Amezcua
(1998) compute the fiscal deficit arising from the
compensation to current pensioners and transition
workers from 1997 to 2047 under the new social
security scheme. For this period, they estimate that
the total cost of the transition would be 82.6 percent of GDP.
After 2047 (the year most transitional workers
cease to exist) the only social security cost for the
government would be the social quota. If, as estimated in Grandolini and Cerda (1998), this cost were
less than 0.1 percent of GDP after 2025, the total
social quota cost from 2047 to 2058 would be about
1.1 percent of GDP. Based on the computations presented earlier, the highest estimated cost of the transition from 1997 to 2047 would be 82.6 percent of
GDP. Thus, in the worst-case scenario the cost of the
transition from 1997 to 2058 would be roughly 84
percent of GDP (1.1 percent plus 82.6 percent).
The message from these computations seems
to be that given the fiscal cost savings—even for
the worst-case scenario, the reform represents a
14. As of this writing, Serrano’s is apparently the only study of its kind.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
21
44 percent savings over the estimated 141 percent
of GDP cost of holding on to the status quo—the
country should press ahead with the transition to a
funded system.
But is this interpretation of these estimates correct? First, as mentioned earlier, detailed attention
needs to be given to the different assumptions
regarding the type and time of taxes necessary to
pay for the transition cost as well as other assumptions about the relevant market imperfections and
redistributional considerations. Second, as the discussion above implies, it may be that at best the
studies represent estimates of the cost of changing
between types of pay-as-you-go systems and not of
transitioning to a fully funded system. Since both
the estimates of the cost of maintaining the status
quo and transitioning to a funded system are partial
equilibrium estimates, it is hard to place confidence
in them. Finally, in order to evaluate the desirability
of either changing the form of the pay-as-you-go
system or moving to a fully funded system, it is
important to emphasize that what matters is the net
benefit of the change in pension system. General
equilibrium analyses modeling Mexican unique features are essential if comparative policy analysis is
going to be meaningful. Informality in the labor market of the economy and emerging financial markets
are two examples of such features that must be
accounted for. In closing, on the basis of the information reviewed here, it seems safe to say that
nobody really knows what the net gains from
switching to a fully funded system might be.
Conclusion
his article describes some of the factors that
if left unchanged very likely would have led
to an actuarial imbalance of the Mexican pension system in the near term. This article contends
that after reviewing the key aspects of the new pension system, one cannot tell whether the government intends to switch to a fully funded social
security system. An interesting question is whether
the Mexican public believes the government will
carry out such a switch. If the public believes a gen-
T
uine transition will begin in the future, then they will
expect interest rates to be lower in the future: stated
differently, the public will begin to view current
interest rates as above their long-run levels. This situation should cause them to increase their saving to
take advantage of the temporarily high level of interest rates. As they do, the level of private saving
should begin to rise and the level of market interest
rates should begin to fall.
Unfortunately, the more gradual the public
expects the transition to a fully funded system to be,
the smaller the expectational effect on saving and
interest rates is likely to be. Since Mexico’s current
financial and political situation would seem to favor
a very gradual transition, the effect might not be
large enough to be identified easily. It might take
many years, or even a generation or more, for the
effect on saving and interest rates to be noticeable.
If the conjecture in this article and in EspinosaVega and Russell (1999) is correct, the current pension reform is another pay-as-you-go system. It
would be interesting, then, to try to estimate the net
gains from the switch from the old pay-as-you-go
system to a new version of it. Unfortunately, to date
there are no solid studies with such estimates. More
research is needed to better assess how the impending demographic changes and the changes in the eligibility criteria, the replacement rate, and other
aspects of the new system will affect the country’s
overall economic welfare.
The commitment to switch to a fully funded system is not trivial. It requires decisions about how
and when to pay for hefty transition costs.
Unfortunately, while there are a number of studies
about the effects of switching to a fully funded pension system for the United States, there is little
solid information for Mexico. Mexico is in dire need
of further research to guide it through its decision
on whether and how to switch to a fully funded
pension system.
It must be said that Mexico is not alone in this
respect. Mexico’s experience should be viewed as an
illustration of the difficulties in assesssing the net
benefits of a pension reform.
REFERENCES
ABEL, ANDREW B., N. GREGORY MANKIW, LAWRENCE H.
SUMMERS, AND RICHARD J. ZECKHAUSER. 1989. “Assessing
Dynamic Efficiency: Theory and Evidence.” Review of
Economic Studies 56:1–20.
BANCO DE MEXICO. 1996, 1997. “The Mexican Economy.”
<http://www.banxico.org.mx/gPublicaciones/
FSPublicaciones.html>.
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CONSAR. 1997. “Circular CONSAR 15-1.” Diario Oficial
de la Federacion, June 30.
———. 1999. <http://www.consar.gob.mx>.
COOLEY, THOMAS F., AND JORGE SOARES. 1999. “Social
Security Based on Reputation.” Journal of Political
Economy 107 (February): 135–60.
Federal Reserve Bank of Atlanta E C O N O M I C R E V I E W First Quarter 2000
DE NARDI, MARIACRISTINA, SELAHATTIN IMROHOROGLU, AND
THOMAS J. SARGENT. 1999. “Projected U.S. Demographics
and Social Security.” Review of Economic Dynamics 2
(July 1999): 575–615.
SALES-SARRAPY, CARLOS, FERNANDO SOLÍS-SOBERÓN, AND
ALEJANDRO VILLAGÓMEZ-AMEZCUA. 1998. “Pension System
Reform: The Mexican Case.” In Privatizing Social
Security, edited by Martin Feldstein. Chicago: University
of Chicago Press.
DIAMOND, PETER A. 1965. “Government Debt in a
Neoclassical Growth Model.” American Economic
Review 55:1126–50.
———. 1998. “The Economics of Social Security
Reform.” National Bureau of Economic Research Working
Paper no. 6719, September.
ESPINOSA-VEGA, MARCO A., AND STEVEN RUSSELL. 1999.
“Fully Funded Social Security: Now You See It, Now
You Don’t?” Federal Reserve Bank of Atlanta Economic
Review 84 (Fourth Quarter): 16–25.
SERRANO, CARLOS. 1999a. “Social Security Reform—How
Much Will It Cost and Who Will Pay for It: The Mexican
Case.” World Bank. Unpublished paper.
———. 1999b. “Social Security Reform, Income
Distribution, Fiscal Policy, and Capital Accumulation.”
World Bank. Unpublished paper.
FELDSTEIN, MARTIN S. 1974. “Social Security, Induced
Retirement, and Aggregate Capital Accumulation.”
Journal of Political Economy 82 (September/October):
905–26.
GRANDOLINI, GLORIA, AND LUIS CERDA. 1998. “The 1997
Pension Reform in Mexico.” World Bank Policy Research
Working Paper no. 1933, June.
SINHA, TAPEN. 1999a. “Lessons from Privatization of
Pension Plans.” Paper presented at the Canadian
Institute of Actuaries special conference on retirement.
———. 1999b. “We Are Not in Kansas Anymore: Risks of
Privatizing Pension.” Instituto Tecnológico Autónomo de
México. Unpublished paper.
———. Forthcoming. Privatization of Social Security
in Latin America. Norwell, Mass.: Kluwer Academic
Publishers.
HUANG, HE, SELAHATTIN IMROHOROGLU, AND THOMAS J.
SARGENT. 1997. “Two Computations to Fund Social
Security.” Macroeconomic Dynamics 1, no. 1:7–44.
IMROHOROGLU, AYSE, SELAHATTIN IMROHOROGLU, AND DOUGLAS
JOINES. 1995. “A Life Cycle Analysis of Social Security.”
Economic Theory 6, 83–114.
IMSS (Instituto Mexicano del Seguro Social). 1997. La
Seguridad Social ante el Futuro. Mexico.
SINHA, TAPEN, FELIPE MARTINEZ, AND CONSTANZA BARRIOSMUÑOZ. 1999. “Publicly Mandated Privately Managed
Pension in Mexico: Simulations with Transactions Cost.”
Society of Actuaries, Actuarial Research Clearing
House, no. 1:323–54.
SINN, HANS-WERNER. 2000. “Why a Funded Pension
System Is Useful and Why It Is Not Useful.” National
Bureau of Economic Research Working Paper no.
W-7592, March.
INTERNATIONAL LABOR ORGANIZATION. 1991. “The Cost of
Social Security.” Geneva, March.
JUDISMAN, CLARA. 1997. “La Informalidad en Mexico:
Características y Tendencias.” Secretaria del Trabajo.
Unpublished document.
SOLÍS SOBERÓN, FERNANDO. 1997. “Análisis comparativo de
las comisiones que cobrarán las AFORES.” CONSAR.
Unpublished paper.
KOTLIKOFF, LAURENCE J. 1996. “Privatization of Social
Security: How It Works and Why It Matters.” National
Bureau of Economic Research Working Paper no. 5330,
October.
UNITED NATIONS. 1998. Demographic Bulletin. Santiago,
Chile, July.
MITCHELL, OLIVIA S. 1996. “Administrative Costs in Public
and Private Retirement Systems.” National Bureau of
Economic Research Working Paper no. 5734, August.
ORSZAG, PETER R., AND JOSEPH E. STIGLITZ. 1999.
“Rethinking Pension Reform: Ten Myths about Social
Security Systems.” World Bank Conference on New Ideas
about Old-Age Security. September 14–15, 1999.
Washington, D.C.: World Bank.
QUEISSER, MONIKA. 1998. “The Second Generation Pension
Reforms in Latin America.” Development Centre Studies,
Organisation for Economic Cooperation and Development, Paris.
RODRIGUEZ, L. JACOBO. 1999. “In Praise and Criticism of
Mexico’s Pension Reform.” Cato Institute Policy
Analysis, no. 340, April 14.
SCHWARZ, ANITA M., AND ASLI DEMIRGUC-KUNT. 1999.
“Taking Stock of Pension Reforms around the World.”
World Bank. Unpublished paper. May.
VAN GINNEKEN, WOUTER. 1998. “Social Security for the
Informal Sector: Investigating the Feasibility of Pilot
Projects in Benin, India, El Salvador, and Tanzania.”
Issues in Social Protection Discussion Paper No. 5. Social
Security Department, International Labor Office, Geneva,
Switzerland.
VON GERSDORFF, HERMANN. 1997. “Pension Reform in
Bolivia: Innovative Solutions to Common Problems.”
World Bank Policy Research Working Paper no. 1832,
September.
WORLD BANK. 1994. Averting Old Age Crisis. New York:
Oxford University Press.
———. 2000. “Understanding and Responding to
Poverty.” <http://www.worldbank.org/poverty/
mission/up1.htm> (April 4).
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Issues in Hedging
Options Positions
S A I K AT
D A N I E L
N A N D I A N D
F. W A G G O N E R
Nandi is a senior economist and Waggoner is an
economist in the financial section of the Atlanta Fed’s
research department. They thank Lucy Ackert, Jerry
Dwyer, and Ed Maberly for helpful comments.
M
ANY FINANCIAL INSTITUTIONS HOLD NONTRIVIAL AMOUNTS OF DERIVATIVE SECURITIES
IN THEIR PORTFOLIOS, AND FREQUENTLY THESE SECURITIES NEED TO BE HEDGED
FOR EXTENDED PERIODS OF TIME.
OFTEN
THE RISK FROM A CHANGE IN VALUE OF A
DERIVATIVE SECURITY, ONE WHOSE VALUE DEPENDS ON THE VALUE OF AN UNDERLYING
asset—for example, an option—is hedged by transacting in the underlying securities of the option.
Failure to hedge properly can expose an institution
to sudden swings in the values of derivatives resulting from large unanticipated changes in the levels or
volatilities of the underlying assets. Understanding
the basic techniques employed for hedging derivative securities and the advantages and pitfalls of
these techniques is therefore of crucial importance
to many, including regulators who supervise the
financial institutions.
For options, the popular valuation models developed by Black and Scholes (1973) and Merton
(1973) indicate that if a certain portfolio is formed
consisting of a risky asset, such as a stock, and a
call option on that asset (see the glossary for a definition of terms), then the return of the resulting
portfolio will be approximately equal to the return
on a risk-free asset, at least over short periods of
time.1 This type of portfolio is often called a
hedge/replicating portfolio. By properly rebalancing the positions in the underlying asset and the
24
option, the return on the hedge portfolio can be
made to approximate the return of the risk-free
asset over longer periods of time. This approach is
often referred to as dynamic hedging. However,
forming a hedge portfolio and then rebalancing it
through time is often problematic in the options
market. There are two potential sources of errors:
The first is that the option valuation model may not
be an adequate characterization of the option
prices observed in the market. For example, the
Black-Scholes-Merton model says that the implied
volatility should not depend on the strike price or
the maturity of the option.2 In most options markets, though, the implied volatility of an option does
depend on the strike price and time to maturity of
the option, a phenomenon that runs contrary to the
very framework of the Black-Scholes-Merton model
itself. The second potential source of error is that
many option valuation models, such as the BlackScholes-Merton model, are developed under the
assumption that investors can trade and hedge continuously through time. However, in practice,
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investors can rebalance their portfolios only at discrete intervals of time, and investors incur transaction costs at every rebalancing interval in the form
of commissions or bid-ask spreads. Rebalancing too
frequently can result in prohibitive transaction
costs. On the other hand, choosing not to rebalance
may mean that the hedge portfolio is no longer
close to being optimal, even if the underlying option
valuation model is otherwise adequate.
This article examines some strategies often used
to offset limitations in the Black-Scholes-Merton
model, describing how the risk of existing positions
in options can be hedged by trading in the underlying asset or other options. It shows how certain
basic hedge parameters such as “deltas,” which are
defined and discussed later, are derived given an
option pricing model. Subsequently, the discussion
notes some of the practical problems that often
arise in using the dynamic hedging principles of
the Black-Scholes-Merton model and considers
how investors and traders try to circumvent some
of these problems. Finally, the hedging implications of the simple Black-Scholes-Merton model
are tested against certain ad hoc pricing rules that
are often used by traders and investors to get
around some of the deficiencies of the BlackScholes-Merton model. The Standard and Poor’s
(S&P) 500 index options market, one of the most
liquid equity options markets, is used to compare
the hedging efficacies of various models. This study
suggests that ad hoc rules do not always result in
better hedges than a very simple and internally
consistent implementation of the Black-ScholesMerton model.
How Are Option Payoffs Replicated and
Deltas Derived?
o hedge an option, or any risky security, one
needs to construct a replicating portfolio of
other securities, one in which the payoffs of
the portfolio exactly match the payoffs of the
option. Replicating portfolios can also be used to
price options, but this discussion will be limited to
T
their hedging properties. Before considering the
hedging aspects of the Black-Scholes-Merton model,
a few simple examples will illustrate how such portfolios are constructed.
One-Period Model.3 The first example is a
European call option on a stock, assuming that the
stock is currently valued at $100.4 In this example,
the option expires in one year and the strike or
exercise price is $100, and the annual risk-free
interest rate is 5 percent so that borrowing $1 today
will mean having to pay back $1.05 one year from
now. For simplicity, the assumption is that there are
only two possible
outcomes when the
option expires—the
stock price can be
either $120 (an up
Because of the simplicity
state) or $80 (a
and tractability of the
down state). Note
Black-Scholes-Merton
that the value of the
call option will be
model for valuing options,
$20 if the up state
the model is widely used
occurs and $0 if the
by options traders and
down state occurs as
shown below (see
investors.
Chart 1).5
Since there are
only two possible
states in the future,
it is possible to replicate the value of the option in
each of these states by forming a portfolio of the
stock and a risk-free asset. If ∆ shares of the stock
are purchased and M dollars are borrowed at the
risk-free rate, the stock portion of the portfolio is
worth 120 × ∆ in the up state and 80 × ∆ in the
down state while 1.05 × M will have to be paid back
in either of the states. Thus, to match the value of
the portfolio to the value of the option in the two
states, it must be the case that
120 × ∆ – 1.05 × M = 20 (up state)
(1)
80 × ∆ – 1.05 × M = 0 (down state).
(2)
and
1. For the purposes of this article, the risk-free asset is a money market account that has no risk of default.
2. Implied volatility in the Black-Scholes-Merton model is the level of volatility that equates the model value of the option to the
market price of the option.
3. The fact that results reported in this article have been rounded off from actual values may account for small differences when
the computations are recreated.
4. The general principle of hedging discussed here applies not only to stock options but also to interest rate options and currency options. Although not discussed here, deltas for American options can be similarly derived for the example shown here.
See Cox and Rubinstein (1985) for American options.
5. Note that the risk-free interest rate of 5 percent lies between the return of 20 percent in the up state and –20 percent in the
down state. For example, if the interest rate were above 20 percent, then one would never hold the risky asset because its
returns are always dominated by the return on the risk-free asset.
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CHART 1
Stock and Option Values in the One-Period Model
Stock Values
Option Values
$20 (up state)
$120
$?
$100
$0 (down state)
$80
Today
One Year
The resulting system of two equations with two
unknowns (∆ and M) can be easily solved to get
∆ = 0.5, and M is approximately 38.10. Therefore,
one would need to buy 0.5 shares of the stock and
borrow $38.10 at the risk-free rate in order for the
value of the portfolio to be $20 and $0 in the up
state and down state, respectively. Equivalently,
selling 0.5 shares of the stock and lending $38.10 at
the risk-free rate would mean payoffs from that
portfolio of –$20 and $0 in the up and down state,
respectively, which would completely offset the payoffs from the option in those states.6 It is also worth
noting that the current value of the option must
equal the current value of the portfolio, which is 100
× ∆ – M = $11.90.7 In other words, a call option on
the stock is equivalent to a long position in the stock
financed by borrowing at the risk-free rate.
The variable ∆ is called the delta of the option. In
the previous example, if Cu and Cd denote the values of the call option and Su and Sd denote the price
of the stock in the up and down states, respectively,
then it can be verified that ∆ = (Cu – Cd)/(Su – Sd).
The delta of an option reveals how the value of the
option is going to change with a change in the stock
price. For example, knowing ∆, Cd, and the difference between the stock prices in the up and down
state makes it possible to know how much the
option is going to be worth in the up state—that is,
Cu is also known.
Two-Period Model. A model in which a year
from now there are only two possible states of the
world is certainly not realistic, but construction of a
multiperiod model can alleviate this problem. As for
the one-period model, the example for a two-period
model assumes a replicating portfolio for a call
option on a stock currently valued at $100 with a
26
Today
One Year
strike price of $100 and which expires in a year.
However, the year is divided into two six-month
periods and the value of the stock can either
increase or decrease by 10 percent in each period.
The semiannual risk-free interest rate is 2.47 percent, which is equivalent to an annual compounded
rate of 5 percent. The states of the world for the
stock values are given in Chart 2. Given this structure, how does one form a portfolio of the stock and
the risk-free asset to replicate the option? The calculation is similar to the one above except that it is
done recursively, starting one period before the
option expires and working backward to find the
current position.
In the case in which the value of the stock over
the first six months increases by 10 percent to $110
(that is, the up state six months from now), the
value of the option in the up state is found by forming a replicating portfolio containing ∆u shares of
the stock financed by borrowing Mu dollars at the
risk-free rate. Over the next six months, the value of
the stock can either increase another 10 percent to
$121 or decline 10 percent to $99, so that the option
at expiration will be worth either $21 or $0. Since
the replicating portfolio has to match the values of
the option, regardless of whether the stock price is
$121 or $99, the following two equations must be
satisfied:
121 × ∆u – 1.0247 × Mu = 21
(3)
99 × ∆u – 1.0247 × Mu = 0.
(4)
and
Solving these equations results in ∆u = 0.9545 and
Mu = 92.22. Thus the value of the replicating portfolio is 110 × ∆u – Mu = $12.78. If, instead, six months
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CHART 2
Stock Values in the Two-Period Model
CHART 3
Option Values in the Two-Period Model
$21
$121
$12.78
$110
$100
$7.77
$99
$0
$0
$90
$0
$81
Today
Six Months
Today
One Year
from now the stock declines 10 percent in value, to
$90 (the down state), the stock price at the expiration of the option will either be $99 or $81, which is
always less than the exercise price. Thus the option
is worthless a year from now if the down state is
realized six months from now, and consequently the
value of the option in the down state is zero. Given
the two possible values of the option six months
from now, it is now possible to derive the number of
shares of the stock that one needs to buy and the
amount necessary to borrow to replicate the option
payoffs in the up and down states six months from
now. Since the option is worth $12.78 and $0 in the
up and down states, respectively, it follows that
110 × ∆ – 1.0247 × M = 12.78,
(5)
90 × ∆ – 1.0247 × M = 0.
(6)
and
Solving the above equations results in ∆ = 0.6389
and M = 56.11. Thus the value of the option today is
100 × ∆ – M = $7.77. The values of the option are
shown graphically in Chart 3.
A feature of this replicating portfolio is that it is
always self-financing; once it is set up, no further
external cash inflows or outflows are required in the
future. For example, if the replicating portfolio is set
up by borrowing $56.11 and buying 0.6389 shares of
the stock and in six months the up state is realized,
Six Months
One Year
the initial portfolio is liquidated. The sale of the
0.6389 shares of stock at $110 per share nets $70.28.
Repaying the loan with interest, which amounts to
$57.50, leaves $12.78. The new replicating portfolio
requires borrowing $92.22. Combining this amount
with the proceeds of $12.78 gives $105, which is
exactly enough to buy the required 0.9545 (∆u)
shares of stock at $110 per share. Replicating portfolios always have this property: liquidating the current portfolio nets exactly enough money to form
the next portfolio. Thus the portfolio can be set up
today, rebalanced at the end of each period with no
infusions of external cash, and at expiration should
match the payoff of the option, no matter which
states of the world occur.
In the replicating portfolio presented above, the
option expires either one or two periods from now,
but the same principle applies for any number of
periods. Given that there are only two possible
states over each period, a self-financing replicating
portfolio can be formed at each date and state by
trading in the stock and a risk-free asset. As the
number of periods increases, the individual periods
get shorter so that more and more possible states of
the world exist at expiration. In the limit, continuums of possible states and periods exist so that the
portfolio will have to be continuously rebalanced.
The Black-Scholes-Merton model is the limiting case
of these models with a limited number of periods.
6. In other words, a long position in one unit of the option can be hedged by holding a short position in 0.5 shares of the stock
and lending $38.10 at the risk-free rate: the value of the total position is $0 in both states.
7. If the current value of the option were higher/lower than the value of the replicating portfolio, then an investment strategy
could be designed by selling/buying the option and forming the replicating portfolio such that one will always make money at
no risk, often called an arbitrage opportunity.
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Thus the Black-Scholes-Merton model must assume
that investors can trade, or rebalance, continuously
through time.8 Another assumption of the BlackScholes-Merton model concerns the volatility of the
stock returns over each time period. Volatility
is related to the up and down movements in the
limited-period models. The Black-Scholes-Merton
model assumes that the volatility of the stock
returns is either constant or varies in such a way
that future volatilities can be anticipated on the
basis of current information.9
Although the continuous trading assumption may
seem unrealistic, the Black-Scholes-Merton model
nevertheless provides traders and investors with a
very convenient formula in which all the input variables but one are observable. The only unobservable
input variable is the implied volatility, that is, the
average expected volatility of the asset returns until
the option expires. A reasonable guess about the
expected future volatility is not very difficult, however, because one can estimate the prevalent volatility from the history of asset prices to the present
time. From a trader’s or investor’s perspective, using
the Black-Scholes-Merton formula, then, requires
only guessing the implied volatility.10 A more sophisticated option pricing model, in contrast, may
require the trader to guess values of model variables
more difficult to obtain in real time, such as the
speed of mean reversion of volatility and others. In
fact, the simplicity of the Black-Scholes-Merton
model largely explains its widespread use regardless
of some of its glaring biases from a theoretical perspective. Despite the Black-Scholes-Merton model’s
very convenient pricing formula, it seems to have
serious constraints: it does not allow forming a selffinancing replicating portfolio with the provision
that one can trade only at discrete intervals of time
with nonnegligible transaction costs such as commissions or bid-ask spreads.
Delta Hedging under the Black-ScholesMerton Model. Considering a European call option
on a nondividend paying stock will illustrate some
of the shortcomings of the Black-Scholes-Merton
model.11 This example assumes that the option has a
strike price of $100 and expires in 100 days; that the
current stock price is $100 and the implied volatility
is 15 percent annually; and that the current annual
risk-free rate, continuously compounded, is 5 percent. If 100 call options have been written (100
options typically constitute an options contract), a
delta-neutral portfolio will have to be formed to
hedge exposure to stock price movements. A deltaneutral portfolio is one that is insensitive to small
changes in the price of the underlying stock. Using
the Black-Scholes-Merton option valuation formula
28
given in Box 1, the value of each option is approximately $3.8375, so that $383.75 is received by selling
or writing the option. Since the portfolio should be
self-financing, the proceeds from the options are
invested in the stock and risk-free asset. Thus
$383.75 is invested in a portfolio of N shares of the
stock and in M dollars of the risk-free asset.
Let ∆ denote the delta of the option and, in accordance with the formula for ∆ for the Black-ScholesMerton model given in Box 1, ∆ = 0.5846. The delta of
the total position (option, stock, and risk-free asset)
is a linear combination of the deltas of the options,
the stock, and the risk-free asset. The delta of a long
(short) position in the option is Λ (–Λ), the delta of a
long (short) position in the stock is 1 (–1), and the
delta of the risk-free asset is zero. As 100 options
have been sold and N shares have been bought, the
delta of the portfolio is –100 × ∆ + N.
In order for the portfolio to be delta-neutral, the
following equation must be satisfied:
–100 × ∆ + N = 0.
(7)
Similarly, for the portfolio to be self-financing, it has
to be the case that
N × 100 + M = 383.75.
(8)
In solving the two equations above for N and M,
N = ∆ × 100 = 58.46 and M = –5,462.25. Thus 100
options have been sold for a total of $383.75, 58.46
units of the share have been bought, and $5,462.25
has been borrowed at an annual interest rate of
5 percent. The total value of the portfolio is zero
when it is formed because the portfolio is selffinancing. What happens, though, to the portfolio
value on the next trading day for three different levels of the stock prices? Borrowing $5,462.25 has
incurred interest charges of approximately
$5,462.25 × 0.05/365.0 = $0.748. Thus the value of
the portfolio on the next day (denoted as t + 1) is
V(t + 1) = 58.46 × S(t + 1) – 100
× C(t + 1) – (5,462.25 + 0.748),
(9)
where S(t + 1) and C(t + 1) denote the values of the
stock and the call option, respectively, on the next
day. Table 1 gives the value of the option and thereby the value of the delta-neutral portfolio for various
values of the stock price, assuming that everything
else (including the implied volatility) is the same.
The value of the delta-neutral portfolio is not zero
in any of these cases, even though in one the stock
price did not change from its initial value of $100.
The reason is that the delta has been derived from a
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model that assumes continuous trading and thus
requires continuous rebalancing for the delta-neutral
portfolio to retain its original value. Transactions
costs, like broker commissions and margin requirements, would further deteriorate the performance
of the delta-neutral portfolio.12
Other Dynamic Hedging Procedures Using
the Black-Scholes-Merton Model. The previous
example assumed that the underlying BlackScholes-Merton model generated the option prices
so that the implied volatility was the same on both
days. However, in reality the implied volatility is not
constant but changes through time in almost all
options markets. The following example demonstrates the outcome if the implied volatility changes
on the next day, assuming that the implied volatility on the next day (t + 1) is 15.5 percent, 15 percent, and 14.5 percent, corresponding to three
different stock prices of $99, $100, and $101. The
fluctuation of implied volatility suggested here corresponds to stock price, increasing as the stock
price goes down and decreasing as it goes up—a
feature of many equity and stock index options
markets. Table 2 shows the values of the portfolio
corresponding to three different levels of stock
prices and implied volatilities.
Thus, with a change in the implied volatility of
around 0.5 percent (frequently observed in options
markets), the hedging performance of the BlackScholes-Merton model deteriorates quite sharply.
The hedge portfolios constructed on the previous
day are quite poor primarily because the model’s
assumption of constant variance is violated.
Extensive academic literature documents how
implied volatilities in the options market change
through time (Rubinstein 1994; Bates 1996; and
many others).13 Further, volatility often varies in
ways that cannot always be predicted with current
information. How could traders or investors set up
hedge portfolios that would account for the random
variation in volatilities? One alternative is to derive
TABLE 1
The Delta-Neutral Portfolio on the Next Day
with No Change in Implied Volatility
Stock Price
Option Price
Portfolio Value
$ 99
$3.26
–$0.96
$100
$3.82
$1.53
$101
$4.42
–$0.84
TABLE 2
The Delta-Neutral Portfolio on the Next
Day When Implied Volatility Changes
Stock Price
Implied Volatility
(Percent)
Portfolio Value
$ 99
15.5
–$11.26
$100
15.0
$ 1.50
$101
14.5
$ 9.06
the hedge portfolio from a more sophisticated (and
more complex) option pricing model such as a stochastic volatility model (to be discussed later).
However, estimating and implementing such a
model can be difficult for an average trader or
investor. Practitioners may be better served by finding ways to circumvent the hedging deficiencies of
the Black-Scholes-Merton model stemming from
implied volatilities that change through time but
sticking to the model as much as possible.
One way to get around the problem of time-varying
volatility that occurs with the Black-Scholes-Merton
model is to form a hedge portfolio that is insensitive
to both the changes in the price of the underlying
asset and its volatility. The sensitivity of an option
price with respect to the volatility is often
referred to as vega. In order to hedge against
changes in both the asset price and volatility, one
can form a portfolio that is delta-neutral as well as
8. This replication with continuous trading is possible due to a special property known as the martingale representation property of Brownian motions (see Harrison and Pliska 1981).
9. However, with continuous trading, one can form a self-financing portfolio by trading in the stock and the risk-free asset even
if the volatility of the stock is random. All that is needed is that the Brownian motions driving the stock price and the volatility are perfectly correlated (see Heston and Nandi forthcoming).
10. Given the existence of multiple implied volatilities from different options (on the same asset), this task is a little more
complicated.
11. If the stock pays dividends, then the present value of the dividends that are to be paid during the life of the option must be
subtracted from the current asset price; the resulting asset price is used in the option pricing formula.
12. It is also worth noting that the portfolio is not self-financing on the next day because rebalancing would incur an external
cash flow in each of the three states.
13. One can also go to the Web site www.cboe.com/tools/historical/vix1986.txt to see the daily history of the implied volatility
index on the Standard and Poor’s 100, called the VIX. VIX captures the implied volatilities of certain near-the-money options
on the Standard and Poor’s 100 index (ticker symbol, OEX).
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B O X
1
Black-Scholes Price and Deltas
he Black-Scholes-Merton formula gives the current value of a European call/put option in
terms of (a) S(t), the price of the underlying asset;
(b) K, the strike or exercise price; (c) τ, the time to
maturity of the option; (d) r(τ), the risk-free rate or
the equivalent yield of a zero-coupon bond (that
matures at the same time as the option); and (e) σ ,
the square root of the average per period (for example, daily) variance of the returns of the underlying
asset that will prevail until the option expires.1
Assuming that the underlying asset does not pay
any dividends until the option expires, the call and
put values are at time t.
T
C(t) = S(t) N(d1)
– K exp[–r(τ)τ] N(d2),
(B1)
P(t) = K exp[–r(τ)τ] N(–d2)
– S(t) N(–d1),
(B2)
and
where N() is the standard normal distribution function
and
d1 = {ln(S/K) + [r(τ)+ 0.5σ 2]τ}/σ τ
(B3)
d2 = d1 – σ τ.
(B4)
(The tables for computing the function are found in
almost all basic statistics books.) If the underlying
asset pays known dividends at discrete dates until the
option expires, then the present value of the dividends
must be subtracted from the asset price to substitute for S(t) in the above formulas.2 Of the abovementioned variables that are required as inputs to the
Black-Scholes-Merton formula, only σ is not readily
observable.
The delta of the option is the partial derivative of
the option price with respect to the asset price, that is,
dC/dS for call options and dP/dS for put options. An
important property of the Black-Scholes-Merton
formula is that the option price is homogeneous of
degree 1 in the asset price and the strike price. Hence it
follows from Euler’s theorem on homogeneous functions
(see Varian 1984) that the delta of the call option is
N(d1) and that of the put option is N(d1) – 1.
The vega of a call or put option is dC/dσ or dP/dσ.
Hull (1997, 329) gives the formula for vega in terms
of the same variables that appear in the valuation
formula.
and
1. Actually the Black-Scholes (1973) model assumes that the risk-free rate is constant. However, Merton (1973) shows that even
if interest rates are random, the appropriate interest rate to use in the Black-Scholes formula for a stock option is the yield
of a zero-coupon bond that expires at the same time as the option. In that case, the simple Black-Scholes (1973) formula
serves as an extremely good approximation because the volatility of interest rates is relatively low compared with the volatility of the underlying stock.
2. The corresponding exact valuation formula for American put options (or call options on dividend paying assets) and deltas
are not known explicitly. However, there are good analytical approximations as in Carr (1998), Ju (1998), and, Huang,
Subrahmanyam, and Yu (1996).
vega-neutral. The formation of such a portfolio is
indeed ad hoc: in fact, it is theoretically inconsistent because under the Black-Scholes-Merton
model volatility is constant (or deterministic) and
therefore does not need to be hedged. Forming a
delta-vega-neutral portfolio would require trading
two options, the underlying asset and the riskfree asset.
Adding to the previous example, in which an
option contract has been sold (with 100 days to
expire) and in which all other variables such as the
stock price and the strike price are the same as
before, N2 units of a second option, N3 units of the
stock, and M dollars of the risk-free asset are
30
required. The current values of the first and second
option are denoted as C(1) and C(2), respectively,
whereas the current stock price is denoted as S(t).
Since the second option can be chosen freely, an
option of the same strike ($100) but a maturity of
150 days is selected. Given these, C(1) = $3.8375
and C(2) = $4.898. The current deltas of the two
options are denoted as ∆(1) and ∆(2), and the
vegas, as vega(1) and vega(2) (see Hull 1997 for the
formula for vega).
For the delta of the portfolio to be zero, it is necessary that
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–100 × ∆(1) + N2 × ∆(2) + N3 = 0.
(10)
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For the vega of the portfolio to be zero, it is necessary that
–100 × vega(1) + N2 × vega(2) = 0.14
(11)
For the portfolio to be self-financing, it is necessary
that
–100 × C(1) + N2 × C(2)
+ N3 × S(t) – M = 0.
TABLE 3
The Delta-Vega-Neutral Portfolio on the Next
Day When Implied Volatility Changes
(12)
Solving the equations in this system of three equations with three unknowns (N2, N3, and M) shows
that N2 = 82.59, N3 = 8.64, and M = $884.96. Thus
82.59 units of the second option and 8.64 units of
the stock must be bought, and $884.96 must be borrowed at the risk-free rate. Table 3 shows the value
of the delta-vega-neutral portfolio on the next day.
The terms C(1)t + 1 and C(2)t + 1 denote the prices of
the first and second option on the next day.
The delta-vega-neutral hedge portfolio performs
much better than a delta-neutral hedge portfolio
that uses just one option, especially if the implied
volatilities change. The only disadvantage in using
this kind of hedge is that the portfolio requires two
options, and options markets tend to be less liquid
than the market on an underlying asset, such as a
stock. On average, options have much higher bidask spreads (relative to their transaction prices)
than those on an underlying asset such as a stock.
Using a second option to hedge the volatility risk
therefore could increase transaction costs, especially for a retail investor.
Similar to delta-vega hedging is what is known as
delta-gamma hedging. The gamma of an option
measures the rate of change of its delta with
respect to a change in the price of the underlying
asset. The more the delta of the option changes
with the asset price, the more a portfolio will have
to be rebalanced to remain delta-neutral. The purpose of delta-gamma hedging is to create a portfolio that is both delta-neutral and gamma-neutral.
Thus, ceteris paribus, the amount of rebalancing
required in a delta-gamma-neutral portfolio would
tend to be lower than that in a delta-neutral portfolio over short periods of time, and lower rebalancing could be used to offset higher transactions
costs. Constructing a delta-gamma-neutral portfolio
also requires two options; the number of units of
the second option can be found by solving a similar
set of equations to those applied to the delta-veganeutral portfolio discussed previously. A deltavega-gamma-neutral portfolio can also be created,
Stock
Price
Implied
Volatility
(Percent)
C(1)t + 1
C(2)t + 1
Portfolio
Value
$ 99
15.5
$ 3.36
$ 4.42
$ –0.30
$100
15
$ 3.81
$ 4.88
$ 0.51
$101
14.5
$ 4.32
$ 5.38
$ –0.34
but forming such a portfolio requires positions in
three options.
The hedging problems discussed thus far fall
under the rubric of dynamic hedging in that they
require a portfolio formed of the underlying asset
and a risk-free asset or options that must be rebalanced through time. Since the number of units of
the underlying asset and the risk-free asset or other
options are derived from an option pricing model,
such as the Black-Scholes-Merton model, the formation of the hedge portfolio is prone to model misspecifications; that is, the underlying options
valuation model is not consistent with the option
prices observed in the market. An alternative to
dynamic hedging is static hedging in which a portfolio is formed as of today and requires no further
trading in the underlying asset and options.
Let S, K, P, and C denote the underlying asset
price, strike price, put price, and call price, respectively. (Note that both the put and call have the
same strike price.) If r and τ denote the risk-free
rate and time to expiration, then the put-call parity
relationship for European options states that the
following has to hold exactly at any given point in
time (in the absence of transactions costs):
P = C – S + Ke–rτ.
(13)
Thus, to replicate the payoff of a put option with
the strike price, K, and time to maturity, τ, a synthetic portfolio must be constructed containing a call
option of the same strike and maturity as that of the
put, a short sell of the asset, and a long position on K
units of a discount bond (that pays off $1 at maturity) that matures at the same time as the options.
Once the synthetic portfolio has been set up for the
put option, rebalancing is no longer necessary
because the price of the put option is identical to
that of the synthetic portfolio if put-call parity is to
be preserved. Since the put-call parity relationship is
14. The vega of a portfolio of options is a linear combination of the vegas of the individual options, and the vega of the underlying asset is zero. Vega(1) = 20.41; vega(2) = 24.71; ∆(1) = 0.585; ∆(2) = 0.603.
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independent of any option valuation model, static
hedging may seem to be the preferable path.
However, static hedging is also prone to some of the
same drawbacks that occur when options are
hedged with options—namely, that options markets
are relatively illiquid, and the second option may not
be available in the right quantity. For example, in the
Standard and Poor’s 500 index options, a market
maker may have to satisfy huge buy order flows in
deep out-of-the-money put options—those with strike
prices substantially below the current S&P 500
level—from institutional investors who want to hedge
their positions against sharp downturns in the index.
However, the volume of deep-in-the-money call
options that would be required in the hedge/replicating portfolio (as per put-call parity) is relatively low,
and hedging deep-out-of-the-money puts via deep-inthe-money calls may not be readily feasible.
Static hedging has often been advocated as a useful tool for certain types of exotic options known as
barrier options.15 Barrier options tend to have
regions of very high gammas; that is, the delta
changes very rapidly and thus requires frequent
rebalancing in certain regions (for example, if the
asset price is close to the barrier). Dynamic hedging
may therefore turn out to be quite difficult and
costly for barrier options. Nevertheless, liquidity
issues concerning static hedging discussed previously also apply to barrier options. A further difficulty is that some options needed as part of the
static hedge portfolios for barrier options may not
be traded at all, so close substitutes must be chosen.
In hedging exotic options such as barrier options, a
trade-off between the pros and cons of static and
dynamic hedging is thus inevitable.
Smile, Smirk, and Hedge. Because of its simplicity (traders have to guess only one unobservable
variable—the average expected volatility of the
underlying asset over the life of the option) the
Black-Scholes-Merton model continues to be very
popular with most traders. However, from a theoretical perspective, the model always exhibits certain
biases. One very prevalent and widely documented
bias is that the implied volatilities in the BlackScholes-Merton model depend on the strike price
and maturity of an option. Chart 4 shows the
implied volatilities in the Standard and Poor’s 500
index options for call options of different strike
prices on December 21, 1995, with twenty-eight and
fifty-six days to maturity. The implied volatilities in
the Standard and Poor’s 500 index options market
tend to decrease as the strike price increases; this
pattern is sometimes referred to as a volatility
smirk. Similarly, in some other options markets,
such as the currency options market, the implied
32
volatilities decrease initially as the strike price
increases and then increase a little—a U-shaped
pattern often referred to as a smile. Chart 4 also
makes it apparent that for options of the same strike
price, implied volatility differs depending on the
maturity of the option. For example, if the strike
price is $570, the implied volatility of the option
with twenty-eight days to maturity is 18.7 percent
whereas the implied volatility of the option with
fifty-six days to maturity is 16.7 percent. Such variations in implied volatilities across strike prices and
maturities are inconsistent with the basic premise of
the Black-Scholes-Merton model, which accommodates only one implied volatility irrespective of
strike prices and maturities. Before examining the
hedging implications of this bias, it is important to
understand what could possibly be causing such a
phenomenon for index options.
One possibility for the existence of the smirk pattern in implied volatilities is that the options market
expects the Standard and Poor’s 500 index to go
down with a higher probability than that suggested
by the statistical distribution postulated for the
returns of the index in the Black-Scholes-Merton
model. As a result, the market would put a higher
price on an out-of-the-money put than would the
Black-Scholes-Merton model. Since option prices
(both puts and calls) under Black-Scholes-Merton
increase as volatility increases, the implied volatility
using the Black-Scholes-Merton model would be
higher than it would otherwise be. In fact, if the distribution of the returns of the underlying asset is
seen as embedded in a cross section of option prices
with different strike prices (see Jackwerth and
Rubinstein 1996), the distribution appears to be one
in which, given today’s index level, the probability of
negative returns in the future is higher than the
probability of positive returns of equal magnitude.
Such distributions are said to be skewed to the left.16
In contrast, the statistical distribution that drives the
returns of an underlying asset under the BlackScholes-Merton model is Gaussian/normal, which
does not involve skewness. In other words, given
today’s index level, the probability of positive returns
is the same as the probability of negative returns of
equal magnitude.
Is it possible to get such negatively skewed distributions under alternative assumptions of the statistical process that generates returns? It turns out
that allowing for future changes in volatility to be
random and allowing volatility to be negatively correlated with the returns of the underlying asset can
generate negatively skewed distributions of the
returns of the underlying asset.17 Indeed, option
pricing models have been developed in which the
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volatility of the underlying asset varies randomly
through time and is correlated with the returns of
the underlying asset. One class of such models,
known as implied binomial tree/deterministic
volatility models, was first proposed by Dupire
(1994), Derman and Kani (1994), and Rubinstein
(1994). In these models the current volatility
(sometimes known as local volatility) is a function of
the current asset price and time, unlike in the BlackScholes-Merton model, in which volatility is constant through time.18 These models are also known
as path-independent time-varying volatility models
in that the current volatility does not depend on the
history or path of the asset price. In another class of
models, sometimes known as path-dependent timevarying volatility models, the current volatility is the
function of the entire history of asset prices and not
just the current asset price.19
Testing the hedging efficacy of an option valuation model often involves measuring the errors
incurred in replicating the option with the prescribed replicating portfolio of the model. In other
words, the replicating portfolio is formed today, and
at a future time the value of the replicating portfolio is compared with the option price observed in
the market as of that time. In empirical tests of
path-independent time-varying volatility models,
Dumas, Fleming, and Whaley (1998) show that in
the Standard and Poor’s 500 index options market
the replication errors of delta-neutral portfolios of
path-independent volatility models are greater than
those of the very simple Black-Scholes-Merton
model. In fact, in terms of replication errors of
delta-neutral portfolios, a very simple implementation of the model also appears to dominate an ad
hoc variation of the model that uses a separate
implied volatility for each option to fit to the
smile/smirk curve. The Black-Scholes-Merton
model proves more useful for hedging despite the
fact that in terms of predicting option prices (that
is, computing option prices out-of-sample) it is
dominated by the ad hoc rule and the time-varying
path-independent volatility model.
Why is it more useful? As discussed above, the
hedge ratio, or the delta, which measures the rate
of the change in option price with respect to the
change in the price of the underlying asset, is an
important consideration. If a replicating/hedge
portfolio (from an option pricing model) is formed
to replicate the value of the option at the next
period, it can be shown that to a large extent the
hedging/replication error reflects the difference in
the pricing or valuation error between the two
periods (see Dumas, Fleming, and Whaley 1998).
Though one model, model A for example, may result
in a lower pricing error (even out-of-sample) than
another model, in order for model A to result in
lower hedging errors than model B, it could also
often be necessary that the change (across two time
periods) in valuation error under model A be less
than that under model B. More often than not, however, the differences in the valuation errors (across
two time periods) between models turn out not to
be very significant for most classes of options (that
is, options of different strike prices and maturities).
In other words, although the Black-Scholes-Merton
model exhibits pricing biases, as long as these
biases remain relatively stable through time, its
hedging performance can be better than the performance of a more complex model that can account for
many of the biases, especially if the more complex
model does not adequately characterize the way
asset prices evolve over time.
Hedging with Ad Hoc Models. How do traders
or investors who routinely use the Black-ScholesMerton model to arrive at hedge ratios/deltas use
the model, despite the fact that patterns in implied
volatilities across options of different strike prices
15. An example of a barrier option is a down-and-out call option in which a regular call option gets knocked out; that is, it ceases
to exist if the asset price hits a certain preset level.
16. The distribution that is skewed is the risk-neutral distribution of asset returns (see Nandi 1998 for risk-neutral probabilities/distributions) and not necessarily the actual distribution of asset returns.
17. Negative correlation implies that lower returns are associated with higher volatility. As a result, the lower or left tail of the
distribution spreads out when returns go down, generating negative skewness. This negative correlation is often referred to
as the leverage effect (Black 1976; Christie 1982) in equities. One possible explanation for this effect is that as the stock
price goes down, the amount of leverage (ratio of debt to equity) goes up, thus making the stock more risky and thereby
increasing volatility. An argument against this explanation is that the negative correlation can be observed for stocks of corporations that do not have any debt in their capital structure.
18. Since the future level of the asset price is unknown, the future local volatility is also not known, and, strictly speaking, unlike
in the Black-Scholes-Merton model, volatility is not deterministic in these models.
19. See Heston (1993) and Heston and Nandi (forthcoming) for option pricing models with path-dependent volatility models in
continuous and discrete time, respectively. These models are sometimes known as continuous time stochastic volatility and
discrete-time GARCH models, respectively. Continuous time models are very difficult to implement due to the fact that
volatility is unobservable given the history of asset prices.
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CHART 4
Implied Volatilities of Call Options
Implied Volatility (Black-Scholes)
Twenty-Eight Days to Maturity
0.2
0.1
550
600
Strike Price
Implied Volatility (Black-Scholes)
Fifty-Six Days to Maturity
0.2
0.1
550
600
650
Strike Price
Note: The chart shows the implied volatilities from Standard and Poor’s 500 call options of different strike prices on December 21, 1995.
The Standard and Poor’s 500 index level was at approximately 610.
34
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B O X
2
Parameter Estimation
he Black-Scholes-Merton-2 version of the model
uses a procedure called nonlinear least squares
(NLS) to estimate a single implied volatility across
all options each Wednesday. The NLS procedure
minimizes the squared errors between the market
option prices and model option prices. The difference between the model price (given an implied
volatility, σ ) and the observed market price of the
option is denoted by ei(σ ). As mentioned in Box 1,
the midpoint of the bid-ask quote is used for the
T
and maturities are inconsistent with the model? As
it turns out, such traders or market makers often
use certain theoretically ad hoc variations of the
basic Black-Scholes-Merton model to circumvent its
biases. Such ad hoc variations allow the implied
volatilities input to the Black-Scholes-Merton model
to differ across strike prices and maturities. Using a
separate implied volatility for each option is inconsistent with the basic theoretical underpinning of
the Black-Scholes-Merton model, but it is a common
practice among traders and market makers in certain options exchanges (Dumas, Fleming, and
Whaley 1998). In the course of implementing such
ad hoc variations, options traders or investors can
be thought of as using the Black-Scholes-Merton
model as a translation device to express their opinion on a more complicated distribution of asset
returns than the Gaussian distribution that underlies the Black-Scholes-Merton model.
Ad hoc variations of the basic Black-ScholesMerton model, depending on the way they are
designed, may result in prices that better match
observed market prices. But do they necessarily
result in better hedging performance? Four versions
of the Black-Scholes-Merton model that differ from
one another in terms of fitting a cross section of
option prices (in-sample errors) and also in predicting option prices (out-of-sample errors) will be
presented; these examples illustrate that the differences between the models in terms of hedging/replication errors are not as significant as the differences
in valuation errors for most options. In fact, if the
models are ranked in terms of the replication errors
of the delta-neutral portfolios, the ranking could
prove different than when the models are ranked in
terms of valuation errors.
observed market price of the option. Thus the criterion function minimized at each t (over σ ) is
Nt
∑ ei ( σ )2 ,
i =1
where Nt is the number of sampled bid-ask quotes on
day t. In essence, this procedure attempts to find a
single implied volatility that minimizes the squared
pricing errors of the model.
There are many different ways in which a trader or
investor can input a value for volatility in the BlackScholes-Merton formula for computing the delta of
an option. The Black-Scholes-Merton model assumes
that the volatility of an asset’s returns is constant
through time. However, an investor trying to use the
model in the real world is not constrained to hold the
volatility constant and can periodically estimate
volatility from past observations of asset prices. As
an alternative to using the historical data, a single
implied volatility for all options (of different strikes
and maturities every day) can be estimated that
minimizes a criterion function involving the
squared price differentials between model prices
and the observed prices in the market (see Box 2
for details).
This approach results in a single implied volatility
for all options every day. On the other hand, implied
volatility can be based on observation of a particular option so that a different implied volatility exists
for each option. As an alternative to using the exact
implied volatility for each option, a procedure that
“merely smoothes Black/Scholes implied volatilities
across exercise prices and times to expiration” is
used by some options market makers at the
Chicago Board Options Exchange (CBOE) (Dumas,
Fleming, and Whaley 1998). For example, given
that the shape of the smirk in implied volatilities
resembles a parabola, one can choose the implied
volatility to be a function of the strike price and the
square of the strike price. However, implied volatilities differ across maturities even for the same
strike price. Thus the time to maturity—and possibly the square of the time to maturity—can also be
included in the function. The equation below is
used in Dumas, Fleming, and Whaley (1998).
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σ (K, τ) = a0+ a1K + a2 K 2
(14)
+ a3τ + a4τ2 + a5 Kτ,
where K is the strike price and τ is the time to maturity of the option. Since the implied volatility σ (K,τ)
is observable for each K and τ, one can use the
above equation as an ordinary least squares (OLS)
regression of the implied volatilities on the various
right-hand variables to get the coefficients a0, a1, a2,
and so on. These coefficients provide an estimated
implied volatility for each option.20 To summarize,
one can use the Black-Scholes-Merton model to
arrive at the delta in four different ways: (a) compute the delta with volatility estimated from historical prices, (b) compute the delta using a single
implied volatility that is common across all options,
(c) compute the delta using the exact implied
volatility for each option, and (d) compute the delta
using an estimated implied volatility for each option
that fits to the shape of the smirk across strike
prices and time to maturities.
Of the four different versions of the BlackScholes-Merton discussed above, the two that allow
implied volatilities to differ across options of different strike prices and maturities are indeed ad hoc.
The other two versions that result in a single implied
volatility across all strikes and maturities are much
less ad hoc. Implementing the four different versions of the Black-Scholes-Merton model in the
Standard and Poor’s 500 index options makes it possible to explore the differences in hedging errors
produced by these approaches.
The market for Standard and Poor’s 500 index
options is the second most active index options market in the United States, and in terms of open interest in options it is the largest. It is also one of the
most liquid options markets.21 These models test
data for the time period from January 5, 1994, to
October 19, 1994.22 Box 3 gives a detailed description
of the options data used for the empirical tests. The
replicating/hedge portfolios are formed on day t
from the first bid-ask quote in that option after
2:30 P.M. (central standard time). The portfolio is liquidated on one of the following days—t + 1, t + 3, or
t + 5.23 The hedging error for each version of the
Black-Scholes-Merton model is the difference
between the value of the replicating portfolio and
the option price (measured as the midpoint of the
bid-ask prices) at the time of the liquidation.
The first panel of Table 4 shows the mean absolute
hedging errors (for the whole sample and across all
options) of the four versions of the Black-ScholesMerton (BSM) model.24 Black-Scholes-Merton-1 is
the version of the model in which volatility is computed from the last sixty days of closing Standard
36
and Poor’s 500 index levels. Black-Scholes-Merton-2
is the version of the model in which a single implied
volatility is estimated for all options each day. Ad
hoc-1 is the ad hoc version of the Black-ScholesMerton model in which each option has its own
implied volatility each day, and ad hoc-2 is the other
ad hoc version, in which the implied volatility (on
each day) for each option is estimated via the OLS
procedure discussed previously.
The first panel clearly shows that judging models
on the basis of hedging/replication errors could be
somewhat different from judging them on the basis
of valuation errors, as discussed previously; valuation errors could include either in-sample errors
that show how well the model values fit market
prices or out-of-sample/predictive error.25 For
example, ad hoc-2 yields substantially lower prediction errors than the Black-Scholes-Merton-2 version
(Heston and Nandi forthcoming) but is the least
competitive in terms of hedging errors. On the other
hand, the magnitude of hedging errors of ad hoc-1, in
which the in-sample valuation errors is essentially
zero (as each option is priced exactly), is not very
different from that of Black-Scholes-Merton-1. In
fact, Black-Scholes-Merton-1, which has the highest
in-sample valuation errors (as volatility is not
T A B L E 4 Mean Absolute Hedging Errors
BSM-1
BSM-2
Ad Hoc-1
Ad Hoc-2
Whole Sample, All Options
One-day
$0.46
$0.45
$0.43
$0.52
Three-day
$0.66
$0.65
$0.62
$0.78
Five-day
$0.98
$0.94
$0.87
$1.07
Far-out-of-the-Money Puts under Forty Days to Maturity
One-day
$0.22
$0.16
$0.10
$0.19
Three-day
$0.23
$0.19
$0.20
$0.26
Five-day
$0.63
$0.50
$0.40
$0.64
Near-the-Money Calls under Forty Days to Maturity
One-day
$0.25
$0.33
$0.24
$0.34
Three-day
$0.49
$0.52
$0.44
$0.60
Five-day
$0.98
$1.08
$0.90
$0.83
Near-the-Money Puts Forty to Seventy Days to Maturity
One-day
$0.52
$0.56
$0.53
$0.62
Three-day
$0.74
$0.76
$0.77
$0.91
Five-day
$1.20
$1.34
$1.15
$1.17
Source: Calculated by the Federal Reserve Bank of Atlanta using
data from Standard and Poor’s 500 index options market
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B O X
3
Data Description
T
he data set used for hedging is a subset of the tickby-tick data on the Standard and Poor’s 500
options that includes both the bid-ask quotes and the
transaction prices; the raw data set is obtained directly from the exchange. The market for Standard and
Poor’s 500 index options is the second most active
index options market in the United States, and in terms
of open interest in options it is the largest. It is also
easier to hedge Standard and Poor’s 500 index options
because there is a very active market for the Standard
and Poor’s 500 futures that are traded on the Chicago
Mercantile Exchange.
Since many of the stocks in the Standard and Poor’s
500 index pay dividends, a time series of dividends for
the index is necessary. The daily cash dividends for
the index collected from the Standard and Poor’s 500
information bulletin for the years 1992–94 can be
used.1 The present value of the dividends (until the
option expires) is computed and subtracted from the
current index level. For the risk-free rate, the continuously compounded Treasury bill rates (from the average of the bid and ask discounts reported in the Wall
Street Journal) are interpolated to match the maturity of the option.
The raw intraday data set is sampled every
Wednesday (or the next trading day if Wednesday is a
holiday) between 2:30 P.M. and 3:15 P.M. central standard time to create the data set.2 In particular, given a
particular Wednesday, an option must be traded on
the following five trading days to be included in the
sample. The study follows Dumas, Fleming, and
Whaley (1998) in filtering the intraday data to create
weekly data sets and use the midpoint of the bid-ask
as the option price. As in Dumas, Fleming, and
Whaley (1998), options with moneyness, | K/F – 1 | (K
is the strike price and F is the forward price), less
than or equal to 10 percent are included. In terms of
maturity, options with time to maturity less than six
days or greater than one hundred days are excluded.3
An option of a particular moneyness and maturity is
represented only once in the sample on any particular
day. In other words, although the same option may be
quoted again in our time window (with the same or different index levels) on a given day, only the first record
of that option is included in our sample for that day.
A transaction must satisfy the no-arbitrage relationship (Merton 1973) in that the call price must
be greater than or equal to the spot price minus the
present value of the remaining dividends and the discounted strike price. Similarly, the put price has to be
greater than or equal to the present value of the
remaining dividends plus the discounted strike price
minus the spot price.
The entire data set consists of 7,404 records and
observations spanning each trading day from January 5,
1994, to October 19, 1994.
1. Thanks to Jeff Fleming of Rice University for making the dividend series available.
2. Wednesdays are used as fewer holidays fall on Wednesdays.
3. See Dumas, Fleming, and Whaley (1998) for justification of the exclusionary criteria about moneyness and maturity.
20. If the number of options on a given day is too few, then there is a potential problem of overfitting in that more independent
variables exist in the right-hand side but only a limited number of observations. However, such a problem can be partially
mitigated by using a subset of the above regression (see Dumas, Fleming, and Whaley 1998).
21. One would want to test any options model in a very liquid options market so that prices are more reliable and do not reflect
any liquidity premium.
22. The 1994 data were the latest full-year data available at the time of this writing.
23. The day t is usually a Wednesday. If Wednesday is a holiday, then the next trading day is chosen.
24. The mean absolute hedging error is the mean of the absolute values of the hedging errors. The conclusions do not change if
a slightly different criterion is used, like root mean squared hedging error.
25. Prediction or out-of-sample valuation errors measure how well a given model values options based on the model parameters
that were estimated in a previous time period.
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G L O S S A R Y
Call option: Gives the owner of the option the right
(but not the obligation) to buy the underlying asset at
a fixed price (called the strike or exercise price). This
right can be exercised at some fixed date in the future
(European option) or at any time until the option
matures (American option).
Put option: Gives the owner of the option the right
(but not the obligation) to sell the underlying asset at
a fixed price (called the strike or exercise price). This
right can be exercised at some fixed date in the future
(European option) or at any time until the option
matures (American option).
Long position: In a security, implies that one has
bought the security and currently owns it.
Short position: In a security, implies that one has
sold a security that one does not own, but has only borrowed, with the hope of buying it back at a lower price
in the future.
Implied volatility: The value of the volatility in the
Black-Scholes-Merton formula that equates the model
value of the option to its market price.
implied but is computed from history of returns of
the S&P 500 index), is quite competitive in terms of
hedging across the entire sample of options.
Given the hedging results in the previous paragraph, which model would one choose among the
four for constructing a hedge portfolio? The answer
may very well depend on which option is to be
hedged. The other panels of Table 4 show the mean
absolute hedging errors of the four versions for
three different classes of options: near-the-money
call and put options and some relatively far-out-ofthe-money put options. Most of these options are
heavily traded in the Standard and Poor’s 500 index
options market.26
The table shows that the differences in hedging
errors among most of the versions are more clearly
manifested in far-out-of-the-money put options. The
ad hoc-1 version, in which the delta of an option is
computed from its exact implied volatility, clearly
dominates in terms of hedging out-of-the-money
puts, irrespective of the maturity. For near-themoney options, the differences between the various
versions are not that significant, especially if the
portfolio is rebalanced on the next day. In fact, the
least complex of all the versions, Black-ScholesMerton-1, is quite competitive in terms of hedging
near-the-money options.
38
In-sample errors: Errors in fitting a model to data
under a particular criterion function. For example,
an options valuation model may have a few parameters or variables, the values of which are not observed
directly. In such a case these parameters are estimated by minimizing a criterion function, such as the
sum of squared differences between the model values
and the market prices; this procedure is often called
in-sample estimation. The differences between the
model option values, evaluated at the estimates of the
parameters, and the market option prices are called
in-sample errors.
Out-of-sample errors: Measure the difference between the model option values and the market option
prices on a sample of option prices that were observed
at a later date than the sample on which the parameters of the model were estimated. In computing outof-sample option values, the model parameters are
fixed at the estimates obtained from the in-sample
estimation.
Conclusion
lthough the classic Black-Scholes-Merton
paradigm of dynamic hedging is elegant from
a theoretical perspective, it is often fraught
with problems when it is implemented in the real
world. Even if the Black-Scholes-Merton model
were free of its known biases, the replicating/hedge
portfolio of the model, which requires continuous
trading, would rarely be able to match its target
because trading can occur only at discrete intervals
of time. Nevertheless, because of its simplicity and
tractability, the model is widely used by options
traders and investors. The basic delta-neutral hedge
portfolio of the Black-Scholes-Merton model is also
sometimes supplemented with other options to
hedge a time-varying volatility (vega hedging).
Although hedging a time-varying volatility is inconsistent with the Black-Scholes-Merton model, it can
often prove useful in practice.
One would expect the presence of biases
observed in the Black-Scholes-Merton model, such
as the smile or smirk in implied volatilities, to result
in further deterioration of the model’s hedging performance. More advanced option pricing models
(for example, random volatility models) that can
account for some of the biases turn out to be useful
mostly for deep out-of-the-money options but not
A
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necessarily for near-the-money options. Ad hoc variations of the Black-Scholes-Merton model sometimes employed by options traders or investors to
overcome the biases may also generate higher hedging errors than the very basic model despite the fact
that ad hoc models often dominate the simple model
in terms of matching observed option prices and
predicting them. Although the simple BlackScholes-Merton model can exhibit pricing biases, it
is often competitive in terms of hedging because the
pricing biases that it exhibits remain relatively stable through time.
Static hedging, an alternative to dynamic hedging,
may seem promising because it is independent of
any particular option pricing model. In particular,
static hedging could prove useful for certain kinds of
exotic options. However, static hedging requires
hedging an option via other options so that the efficacy of static hedging depends on the liquidity of
the options market, which often is not as liquid as
the market on the underlying asset.
26. Far-out-of-the-money puts are those for which K/F < 0.95 where K is the strike price and F is the forward price for maturity τ—that is, F(t) = S(t)exp[r(τ)τ], where τ is the time to maturity of the option. Near-the-money options are those for
which | K/F – 1 | ≤ 0.01.
REFERENCES
BATES, DAVID. 1996. “Jumps and Stochastic Volatility:
Exchange Rate Processes Implicit in Deutschemark
Options.” Review of Financial Studies 9 (Spring):
69–107.
HESTON, STEVEN L. 1993. “A Closed-Form Solution for
Options with Stochastic Volatility with Applications to
Bond and Currency Options.” Review of Financial
Studies 6, no. 2:327–43.
BLACK, FISCHER. 1976. “Studies of Stock Price Volatility
Changes.” In Proceedings of the 1976 Meetings of the
American Statistical Association, Business and
Economic Statistics Section, 177–81. Alexandria, VA:
American Statistical Association.
HESTON, STEVEN L., AND SAIKAT NANDI. Forthcoming.
“A Closed-Form GARCH Option Valuation Model.”
Review of Financial Studies.
BLACK, FISCHER, AND MYRON S. SCHOLES. 1973. “The Pricing
of Options and Corporate Liabilities.” Journal of Political Economy 81 (May/June): 637–54.
CARR, PETER. 1998. “Randomization and the American
Put.” Review of Financial Studies 11 (Fall): 327–43.
CHRISTIE, ANDREW A. 1982. “The Stochastic Behavior of
Common Stock Variances: Value, Leverage, and Interest
Rate Effects.” Journal of Financial Economics 10
(December): 407–32.
HUANG, JING-ZHI, MARTI G. SUBRAHMANYAM, AND G. GEORGE
YU. 1996. “Pricing and Hedging American Options: A
Recursive Integration Method.” Review of Financial
Studies 9 (Spring): 277–300.
HULL, JOHN C. 1997. Options, Futures, and Other Derivatives. 3d ed. Upper Saddle River, N.J.: Prentice-Hall.
JACKWERTH, JENS, AND MARK RUBINSTEIN. 1996. “Recovering
Probability Distributions from Option Prices.” Journal of
Finance 51 (December): 1611–52.
COX, JOHN C., AND MARK RUBINSTEIN. 1985. Options
Markets. Englewood Cliffs, N.J.: Prentice-Hall.
JU, NENGJIU. 1998. “Pricing an American Option by
Approximating Its Early Exercise Boundary as a Multipiece Exponential Function.” Review of Financial
Studies 11 (Fall): 627–46.
DERMAN, EMANUEL, AND IRAJ KANI. 1994. “Riding on the
Smile.” Risk 7 (February): 32–39.
MERTON, ROBERT C. 1973. “The Theory of Rational Option
Pricing.” Bell Journal of Economics 4 (Spring): 141–83.
DUMAS, BERNARD, JEFF FLEMING, AND ROBERT WHALEY.
1998. “Implied Volatility Functions: Empirical Tests.”
Journal of Finance 53 (December): 2059–2106.
NANDI, SAIKAT. 1998. “How Important Is the Correlation
between Returns and Volatility in a Stochastic Volatility
Model? Empirical Evidence from Pricing and Hedging in
the S&P 500 Index Options Market.” Journal of
Banking and Finance 22 (May): 589–610.
DUPIRE, BRUNO. 1994. “Pricing with a Smile.” Risk 7
(February): 18–20.
HARRISON, J. MICHAEL, AND STANLEY R. PLISKA. 1981.
“Martingales and Stochastic Integrals in the Theory of
Continuous Trading.” Stochastic Processes and Their
Applications 11:215–60.
RUBINSTEIN, MARK. 1994. “Implied Binomial Trees.”
Journal of Finance 49 (July): 771–818.
VARIAN, HAL R. 1984. Microeconomic Analysis. New
York: W.W. Norton and Company.
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Evidence on the
Efficiency of Index
Options Markets
L U C Y F. A C K E R T
Y I S O N G S . T I A N
A N D
Ackert is a senior economist in the financial section
of the Atlanta Fed’s research department. Tian is an
associate professor of finance at the Schulich School
of Business, York University. The authors thank Jerry
Dwyer, Mark Fisher, Saikat Nandi, and Steve Smith
for helpful comments.
S
INCE THE
CHICAGO BOARD OPTIONS EXCHANGE INTRODUCED THE FIRST INDEX OPTION CON-
TRACT IN
1983, INDEX OPTIONS MARKETS HAVE HAD A SIGNIFICANT ROLE IN FINANCIAL MAR-
KETS. INDEX OPTIONS HAVE BEEN ONE OF THE MOST SUCCESSFUL OF THE MANY INNOVATIVE
FINANCIAL INSTRUMENTS INTRODUCED OVER THE LAST FEW DECADES, AS THEIR HIGH TRADING
volume indicates. Index options give market participants the ability to participate in anticipated market movements without having to buy or sell a large
number of securities, and they permit portfolio
managers to limit downside risk. Given their prominence and functions, the pricing efficiency of these
markets is of great importance to academics, practitioners, and regulators.
Well-functioning financial markets are vital to a
thriving economy because these markets facilitate
price discovery, risk hedging, and allocating capital
to its most productive uses. Inefficient pricing of
index options indicates that their market (and, possibly, other financial markets) is not doing the best
possible job at these important functions. To detect
inefficient pricing (often called mispricing) requires
computing a theoretically efficient price or price
range and comparing it with prices of options traded
in financial markets. But valuing an index option in
theory is complicated and challenging.
40
One popular approach to deriving option pricing
relationships is based on a principal called noarbitrage. This approach is a very powerful tool in
the valuation of financial assets because it does not
make strong assumptions about traders’ behavior or
market price dynamics. The principle simply
assumes that arbitrageurs enter the market and
quickly eliminate mispricing if a riskless profit
opportunity exists. An arbitrageur is an individual
who takes advantage of a situation in which securities are mispriced relative to each other. The arbitrageur buys the underpriced asset and sells the
overpriced asset, locking in a riskless profit. In
doing so, the arbitrageur drives the price of the
underpriced asset up and the price of the overpriced asset down, thus eliminating mispricing.
However, in a well-functioning economy—where
there are no free lunches—there is no portfolio of
assets that has zero cost today and a certain, positive payoff in the future. Similarly, there is no port-
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folio of assets that pays a positive amount with certainty today and requires no payment in the future.
Arbitrage is critical for ensuring market efficiency
because it forces asset prices to return to their
implied, no-arbitrage values.
Many earlier studies report evidence of mispricing
of index options, though arbitrage might have been
limited. In some situations, market frictions restrict
arbitrage so that investors simply cannot take
advantage of available profit opportunities. For
example, if arbitrageurs are subject to capital constraints and they cannot raise the capital necessary
to form the riskless hedge, they will be unable to
undertake trades that would move the market toward an efficient state (Shleifer and Vishny 1997).
Similarly, the activity of arbitrageurs may be limited
because the stock index underlying the option is
often relatively difficult and costly to reproduce.
To arbitrage based on a mispriced index option,
investors may need to replicate the index by buying
or selling a large basket or set of stocks that is
perfectly correlated with the index. Doing so may be
relatively difficult and costly, even for large investors (Ackert and Tian 1998b, 1999).1
The evidence of index option mispricing has been
taken to indicate that options markets are inefficient and casts doubt on their contributions to price
discovery, hedging, and efficient capital allocation.
This article is a discussion of index option pricing
aimed at analyzing earlier evidence of mispricing
and presenting new evidence on index option pricing and its evolution. It first presents theoretical
pricing relationships implied by no-arbitrage conditions. These conditions place bounds on possible
efficient call and put option prices and imply relative pricing relationships between call and put
option prices. A call (put) option is the option to
buy (sell) an asset. Empirical tests of the conditions
presented provide powerful insight into how options market efficiency has evolved over time. In
contrast to many previous studies of options market
efficiency, the arbitrage strategies examined here
do not involve trading a stock index, and the relationships hold for any given value of the underlying
asset. This approach avoids some of the difficulties
that arise from impediments to arbitrage when, for
example, an investor might have to short sell a large
stock basket—that is, sell shares he or she does not
own by borrowing them from another investor.
The article also reviews earlier studies of the pricing efficiency of index options markets and provides
an empirical examination of the efficiency of the
market for the popular Standard and Poor’s (S&P)
500 index options. The results indicate some substantial deviations of market prices from theoretical
pricing relationships. Importantly, S&P 500 index
options are frequently mispriced, and the mispricing
does not appear to have abated over time. The mispricing may not, however, indicate market inefficiency because there
are limits to arbitrage.
Arbitrage Pricing
Relationships
Index options give market
participants the ability to
participate in anticipated
market movements without
having to buy or sell a
large number of securities,
and they permit portfolio
managers to limit downside risk.
n evaluating the
efficiency of option pricing, a theoretical optimal price
derived from a model
frequently provides
the basis for comparison. Such theoretical
models often assume
specific dynamics for
the underlying asset
in order to derive
more well-defined restrictions on the efficient price.
In contrast, tests of pricing efficiency based solely
on no-arbitrage arguments may be more informative
if the relationships are independent of the models,
though restrictions they place on price may not be
very demanding.
Arbitrage pricing relationships are based on the
simple assumption that investors prefer more to less.
If these pricing relationships are violated by actual
prices after adjustment for the bid-ask spread and
transaction costs, arbitrage profits may be possible by
buying the underpriced asset(s) and short-selling the
overpriced asset(s). As discussed previously, rational
pricing of options imposes explicit restrictions on
the relative prices of call and put options. If these
restrictions are violated, arbitrage opportunities
exist. Some arbitrage pricing relationships jointly
test options and stock market efficiency and allow
examination of the information exchange between
these markets whereas others test options market
efficiency alone and allow examination of how pricing has evolved over time.2 The relationships and
I
1. Note that traders can use the very liquid S&P 500 futures contract to replicate the index.
2. Billingsley and Chance (1985) and Ronn and Ronn (1989) note that some tests are joint tests of options and stock market
efficiency while others consider only options market efficiency. See Ackert and Tian (1998a, 1999) for examples of both types
of relationships.
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empirical tests reported in this article are of the latter type. Because stock market transactions are not
involved, examining these relationships may provide
a superior test of pricing across index options.
Another advantage of these types of relationships is
that they are unaffected by the different closing
times in stock and options markets.
The arbitrage pricing relationships presented
below allow examining whether options market efficiency improved over the sample period. Options on
the S&P 500 index are European, and the discussion
below applies to European options only. A European
option may be exercised only at maturity whereas
an American option may be exercised prior to maturity. All relationships account for the bid-ask spread
because bid-ask spreads result in significant transaction costs for participants in options markets
(Phillips and Smith 1980; Baesel, Shows, and Thorp
1983). Define:
Cb:
Ca:
Pb:
Pa:
S:
X:
T:
r:
ti:
bid price of European call option;
ask price of European call option;
bid price of a European put option;
ask price of a European put option;
price of underlying asset;
strike price;
maturity of the option;
risk-free rate of interest or Treasury bill
rate;3
transaction costs (other than those arising from the bid-ask spread) of buying or
selling calls, puts, or Treasury bills, i = c,
p, or r.
Three sets of arbitrage pricing relationships are
presented: the box spread, call and put spreads, and
call and put convexity. The box spread is a combination of call and put spreads that matches two
pairs of call and put options.4 This strategy requires
that an investor purchase and sell calls (bullish call
spread) with strike prices X1 and X2, respectively,
while simultaneously selling and purchasing puts
(bearish put spread) with strike prices X1 and X2,
respectively. The box spread is a riskless strategy
because the future payoff is always positive: the difference between two strike prices, X2 – X1, where X1
< X2. The payoff is illustrated in Chart 1. If bid-ask
spreads and transaction costs are taken into
account, the box spread is expressed by the following two inequalities:
(C1a – C2b) – (P1b – P 2a) + (X1 – X2)e–rT + t1 ≥ 0 (1a)
and
(C 2a – C1b) – (P 2b – P1a) + (X2 – X1)e–rT + t1 ≥ 0, (1b)
42
where t1 = 2tc + 2tp + tr. In the absence of arbitrage,
inequalities (1a) and (1b) hold.
In contrast to the box spread, the call (put)
spread combines two call (put) options with identical maturity. The call spread strategy requires purchase of call option 1 and sale of call option 2, where
X1 < X2, as illustrated in Chart 2. The call spread is
expressed as
(C 2a – C 1b) + (X2 – X1)e–rT + t2a ≥ 0,
(2a)
where t2a = 2tc + tr. Similarly, the put spread involves
the sale of put option 1 and purchase of put option 2
and is expressed as
(P 1a – P 2b) + (X2 – X1)e–rT + t2b ≥ 0,
(2b)
where t2b = 2tp + tr.
Finally, call (put) convexity creates a riskless position by combining three call (put) options where
X1 < X2< X3. The call (put) convexity strategy
requires purchase of call (put) options 1 and 3 and
sale of call (put) option 2. Call convexity is
expressed as
wC1a + (1 – w)C 3a – C 2b + 2tc ≥ 0
(3a)
and put convexity is
wP 1a + (1 – w)P 3a – P 2b + 2tp ≥ 0,
(3b)
where w = (X3 – X2)/(X3 – X1). So, for example, if
w = 1⁄2, call convexity involves the purchase of one
call option 1 and one call option 3 for every two call
option 2s sold. The payoff from this strategy, commonly referred to as a butterfly spread, is illustrated
in Chart 3.
If the box spread, call spread, put spread, call convexity, or put convexity is violated, arbitrage profits
are possible by taking appropriate option positions.
For example, if the call spread (2a) is violated,
index call option 1 is overvalued relative to call
option 2. The arbitrageur would sell call 1 and buy
call 2, investing the balance in a Treasury bill earning the risk-free rate. In the case of exercise of both
call options at maturity, the arbitrageur closes the
index position and earns a risk-free profit at maturity (time T) of (C 1b – C 2a) + (X1 – X2)e–rT – t2a ≥ 0,
where t2a = 2tc + tr (see the box for the details of this
arbitrage).
Although a violation of any of the inequalities
above indicates the presence of an arbitrage
opportunity, the box spread inequalities (1a) and
(1b) place more demanding restrictions on the
pricing of options. In the absence of transaction
costs, the box spread requires an equality among
four option prices. In contrast, even ignoring trans-
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C H A R T 1 Box Spread Payoff
C H A R T 2 Call Spread Payoff
Box Spread
X2 – X1
Bullish Call Spread
(Buy 1, Sell 2)
Payoff
Payoff
Buy Call 1
X2 – X1
Bearish Put Spread
(Sell 1, Buy 2)
X1
X2
Call Spread
X1
Sell Call 2
ST
C H A R T 3 Call Convexity ( w =
action costs, call and put spreads and convexity are
minimum-maximum (inequality) restrictions, and a
wide range of prices is consistent with the boundaries they place on option prices, as is apparent in
Charts 1–3.
Payoff
any empirical studies have tested pricing
relations between put and call options, particularly for options on individual stocks.
See, for example, Stoll (1969), Gould and Galai
(1974), and Klemkosky and Resnick (1979). Some
of these tests are based on theoretical option pricing
models, such as the Black-Scholes (1973) or the
Cox, Ross, and Rubinstein (1979) binomial option
pricing models. Other tests are based simply on
arbitrage arguments and are model-independent,
including, for example, the box spread.
Although the empirical evidence generally supports some pricing relationships like put-call parity
for individual stock options, significant mispricing
has been reported in stock index options markets.
For example, Evnine and Rudd (1985) use intraday
data for a two-month period in 1984 and find frequent violations of boundary conditions and put-call
parity for S&P 100 and Major Market index options,
1
⁄2
)
Buy Call 1
Buy Call 3
X3 – X2 = X2 – X1
Efficiency of Index Options Markets
M
ST
X2
X1
X2
X3
ST
Call Convexity
Sell 2, Call 2
both of which are American options.5 Evnine and
Rudd further conclude that these options are significantly mispriced relative to theoretical values
based on the binomial option pricing model. Chance
(1987) also finds that put-call parity and the box
spread are violated frequently for S&P 100 index
options and that the violations are significant in
size.6 However, these results may not indicate market inefficiency for several reasons.
3. The assumption is that borrowing and lending rates are equal. Regarding the impact of this assumption on the results, see
note 14.
4. The box spread is also a simple algebraic combination of the put-call parity relationship for each option. Put-call parity relates
the put price, call price, exercise price, risk-free interest rate, and underlying asset price for options on the same asset with
identical exercise price and expiration date. According to put-call parity, a pair of call and put options with identical maturity and strike price should be priced such that C + Xe–rT = P + S, ignoring transaction costs and the bid-ask spread.
5. A boundary condition specifies a maximum or minimum price for an option. For example, an upper bound on the price of a
call option is the value of the underlying asset because, no matter what happens, the option can never be worth more than
the asset, that is, C ≤ S. See note 4 above on put-call parity. For derivations of the various pricing relationships, see Merton
(1973), Cox and Rubinstein (1985), Chance (1987), and Hull (1997).
6. In another study, Chance (1986) examines whether S&P 100 option prices are consistent with the Black-Scholes model and
concludes that the model cannot be used to generate abnormal returns.
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B O X
Arbitrage Opportunity When the
Call Spread Does Not Hold
gnoring the bid-ask spread and transaction costs, the
call spread is expressed as
I
(C2 – C1) + (X2 – X1)e–rT ≥ 0.
Without loss of generality, it is always possible to
rearrange the pair of calls so that X1 < X2. Given this
arrangement, the price of the first call should always be
greater than that of the second call; that is, C1 > C2. If
the call spread is violated, that is, if
(C2 – C1) + (X2 – X1)e–rT < 0,
then risk-free profit opportunities are present.
Arbitrage profits are possible by taking appropriate
positions in the options market. In this case, index call
option 1 is overvalued relative to call option 2. The
arbitrageur would sell call 1, buy call 2, and invest the
balance in a Treasury bill earning the risk-free rate. At
maturity, a call option is exercised if the stock price
exceeds its exercise price. The cash flows from the
strategy are as shown in the table.
At the inception of this strategy there is no initial
investment, and at maturity there are three possible
cash flows, all of which are positive. When S < X1, nei-
ther option is exercised upon maturity and the arbitrageur accrues the entire amount invested in the
Treasury bill as profit; that is, (C1 – C2)erT > 0. When X1
< S ≤ X2, option 1 is exercised but not option 2 and the
investment in the Treasury bill more than offsets the
loss on the first option so that
(C1 – C2)erT – (S – X1) > (C1 – C2)erT – (X2 – X1) > 0.
The first inequality holds because (X1 – S) > (X1 –
X2) when X1 < S ≤ X2, and the second, because the call
spread is violated. Finally, if both options are exercised
at maturity, positive profit also accrues, again because
the call spread is violated. Therefore, profit is made by
the arbitrageur in all three possible outcomes. After
commission fees and the bid-ask spread are recognized, the violation of (2a) (see Tables 4 and 5) must
be large enough to compensate for transaction costs;
that is,
(C2a – C1b) + (X2 – X1)e–rT + t2a < 0.
Such an opportunity cannot persist as arbitrageurs will
take advantage of the mispricing until (2a) holds.
Arbitrage When the Call Spread Is Violated
Cash Flow
At Options’ Maturity
Today
S < X1
X1 < S ≤ X2
S > X2
Sell call 1
+C1
—
–(S – X1)
– (S – X1)
Sell call 2
–C2
—
—
S – X2
–(C1 – C2)
(C1 – C2)erT
(C1 – C2)erT
(C1 – C2)erT
0
(C1 – C2)erT
(C1 – C2)erT – (S – X1)
(C1 – C2)erT – (X2 – X1)
Strategy
Buy Treasury bill
Total
These tests of market efficiency may be misleading
because they use American options and the arbitrage
conditions are for European options. Kamara and
Miller (1995) point out that prior to their examination all tests of put-call parity used American options.
In addition, tests of other arbitrage pricing relationships such as the box spread used data for American
44
options (for example, Billingsley and Chance 1985).
Because of the possibility of early exercise, these
relationships may not be expected to hold for
American options, and similar conditions for
American options are frequently intractable. In their
tests using S&P 500 index options that are European,
Kamara and Miller find fewer and smaller violations.
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Tests of put-call parity may also fail to indicate
market inefficiency if arbitrage at low cost is not possible. Introducing index options helps to reduce arbitrage costs. In Canada, a stock basket, Toronto Index
Participation Units (TIPS 35), has been traded since
1990. Ackert and Tian (1998a) examine the efficiency of Canadian index and options markets by
comparing the number and size of violations in theoretical pricing relationships before and after the
introduction of TIPs. They conclude that, although
options market efficiency improved over their test
period, the connection between options markets
and stock markets did not. Ackert and Tian (1999)
also examine the impact of a traded stock basket,
Standard and Poor’s Depositary Receipts (SPDRs),
on the link between U.S. index options markets.
They conclude that the introduction of a stock
basket can enhance market efficiency because it
removes one limit to arbitrage.
In summary, the results reported in earlier studies
suggest that put-call parity is frequently violated in
index options markets and that these options are
often mispriced relative to prices predicted by theoretical models. To overcome problems in earlier
studies, this study tests theoretical pricing relationships based on no-arbitrage conditions for European
stock index options. It focuses on tests of options
market efficiency independent of the stock market
and includes the effects of transaction costs and
bid-ask spreads. It also examines whether deviations from pricing relations declined over the
1986–96 sample period.
Empirical Results
ll three arbitrage pricing relationships presented earlier are investigated for S&P 500
index options on each trading day in the
sample, as described subsequently. The number and
size of violations are recorded and analyzed. This
approach allows examining the evolution of the
index options market and provides insight into
whether market efficiency increased over the sam-
A
ple period. Arbitrage based on violations of the relationships considered does not require a position in
the underlying asset. In addition, all of the pricing
relationships are independent of an option pricing
model so that no assumption concerning the process
underlying the stock price is required. Thus, the
empirical tests are true tests of market efficiency
instead of joint tests of market efficiency and model
specification.7 Finally, the analysis recognizes the
limits that transaction costs and bid-ask spreads
place on arbitrage.
The empirical investigation analyzes the efficiency
of the S&P 500 index options market using daily data
for the S&P 500 index and index options from January 1, 1986, through December 31, 1996. Daily closing
prices, trading volume, and open interest for S&P 500
index call and put options are from the Chicago Board
Options Exchange.8 The three-month Treasury bill
rate (a proxy for the risk-free interest rate) is from
the Federal Reserve Bulletin. Bid-ask spreads and
commissions are included so that the analysis recognizes the effect of transaction costs on pricing efficiency. The approach is conservative in that it uses
closing bid and ask prices, rather than closing prices,
in testing the pricing relationships.9 Following
Harris, Sofianos, and Shapiro (1994) and Kamara
and Miller (1995), this research constructs bid and
ask prices, based on the usual spread in option
prices, from closing prices. The option bid-ask
spread is estimated by adding or subtracting 1⁄32 (1⁄16)
of a point if the price is less than (greater than or
equal to) $3.10 Following Kamara and Miller (1995),
commission costs (ti) are $30 for Treasury bills and
$2 ($4) per option contract for 100 shares if the price
is less than (greater than or equal to) $1.
On each trading day during the test period, the
three pricing relationships discussed above are
tested: the box spread (1a) and (1b), call and put
spreads (2a) and (2b), and call and put convexity
(3a) and (3b). For each maturity month, two pairs
of put and call options are used to examine the box
spread. The put and call within each pair are matched
7. So, for example, there is no test of whether prices are consistent with those predicted by a particular model such as the
Black-Scholes option pricing model.
8. All relationships tested require synchronous option prices. Inferences are limited by the fact that closing prices may be nonsynchronous. However, Evnine and Rudd (1985) and Kamara and Miller (1995) find very similar results using intraday and
closing price data for S&P 100 and S&P 500 index options, respectively.
9. See Ronn and Ronn (1989), who demonstrate that the use of bid-ask prices is conservative. They note that the market maker
commits to transacting at least one contract at the bid-ask quotes, but the effective spread may be narrower. Traders are sometimes able to bargain to obtain better prices so that trades occur inside the quoted spread.
10. Some traders may have access to better price quotes. The assumption in this article concerning the constructed spread
appears to be reasonable based on the results reported by others, though the results may be affected by the assumption to
the extent that the spread is over- or underestimated.
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with an identical strike price, but two different
strike prices are used for the two pairs. In contrast,
the call (put) spread combines two call (put)
options with identical maturity and different strike
prices. Finally, call (put) convexity combines three
call (put) options with identical maturity and different strike prices.
The frequency and severity of violations are tabulated for the full sample period as well as for each
year in the sample.11 Examining violations in the
pricing relationships for each sample year provides
insight into how the efficiency of the options market
has changed as the market has developed over time.
Tables 1 through 9 report the percentage of violations as well as the mean violation in dollars.
Significant dollar violations are tested for by testing
the null hypothesis that the mean dollar violation is
zero. All reported t statistics use standard errors
corrected for autocorrelation using a maximum likelihood procedure estimated by a Gauss-Marquardt
algorithm (Judge and others 1985).12
To further investigate the persistence of violations
in pricing relationships, the study examines whether
arbitrage opportunities are evident the day following
observed violations. Doing so provides an ex ante
test, which, as Galai (1977) argues, a true test of
market efficiency must be. Ex ante tests are executed from the trader’s point of view and reflect the
trader’s ability to actually form the required, profitable portfolio. In an ex ante approach, current
prices reveal arbitrage opportunities but execution
is at prices that are yet to be revealed. Conducting
ex ante tests involves identifying each day on which
a particular violation occurs and tracking whether
the violation persisted on the following trading day.
Existence on the following day implies that traders
did not fully eliminate arbitrage opportunities.
Table 1 reports the frequency and severity of violations and ex ante violations of the box spread,
inequalities (1a) and (1b). For the two inequalities,
the percentage and dollar amount of violations are
similar (21.02 percent and $1.07 versus 23.78 percent and $1.08). For each relationship, the percentage of violations is substantial and the mean dollar
violation is significantly different from zero.13 The ex
ante tests indicate that significant abnormal profit
opportunities existed even on the day following the
violation of a pricing relationship. For example,
28,292 violations of (1a) occurred, and of these violations 2,785 or 9.84 percent persisted on the following
day with a significant mean violation of $1.02.14
Tables 2 and 3 report the percentage and dollar size
of violations and ex ante violations of (1a) and (1b),
respectively, for each year in the 1986–96 sample
period. All mean dollar violations are significantly
different from zero at the 1 percent significance
level. Although some variation is observed in the
extent to which the pricing relationships are violated
across years, the results provide no evidence that
options market efficiency improved over the sample
T A B L E 1 Violations and Ex Ante Violations of the Box Spread (1a) and (1b)
Box Spread (1a)
Total
Violations
Ex Ante
Violations
Box Spread (1b)
Total
Violations
Ex Ante
Violations
Frequency of Violations
Number of Observations
Number of Violations
Percentage of Violations
134,606
28,292
134,606
32,014
28,292
2,785
32,014
3,210
21.02
9.84
23.78
10.03
Violations, in Dollars
Mean
1.07
1.02
1.08
1.11
Standard Deviation
1.05
1.00
1.07
1.09
t statistic for nonzero mean
170.00***
54.19***
180.36***
57.74***
Note: This table reports the frequency and dollar size of violations of the box spread (1a) and (1b) using daily data for the S&P 500 index
and index options from January 1, 1986, through December 31, 1996. An ex ante violation occurs when a particular violation persists into
the following trading day. Asterisks *,**, or *** denote significance at the 10 percent, 5 percent, and 1 percent levels, respectively, in a
two-tailed test.
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T A B L E 2 Violations and Ex Ante Violations of the Box Spread (1a) by Year
Total Violations
Ex Ante Violations
Sample Year
Percentage of
Violations
Mean Dollar
Violation
Percentage of
Violations
Mean Dollar
Violation
1986
18.72
0.83
4.35
0.76
1987
22.30
1.24
8.03
1.13
1988
20.34
0.86
5.83
0.85
1989
13.94
0.91
7.54
0.81
1990
24.06
1.06
12.18
1.01
1991
21.62
0.99
9.11
0.90
1992
17.03
0.78
9.21
0.92
1993
18.29
0.86
9.42
0.76
1994
17.69
0.92
8.99
0.87
1995
20.76
1.02
10.75
1.05
1996
25.96
1.35
11.14
1.23
Overall
21.02
1.07
9.84
1.02
Note: This table reports the percentage and dollar size of violations of the box spread (1a) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular
violation persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
period. The frequency of violations remains high
at approximately 20 percent of observations, even
after taking into account trading costs, including the
bid-ask spread and commission fees.
Next, violations of call and put spreads (2a) and
(2b) and call and put convexity (3a) and (3b) are
examined. As reported in Tables 4 and 7, significant
mean dollar violations and ex ante dollar
violations were observed for all four relationships.
However, for all four the frequency of violations is
quite low. The maximum percentage of violations
(ex ante violations) across the four inequalities for
the full sample is only 3.08 percent (8.04 percent).
When the percentage and dollar violations by year
reported in Tables 5 and 6 (8 and 9) for call and
put spreads (convexity) are considered, there is no
apparent trend. Although market efficiency does
not appear to have improved over the sample
period, the results suggest that options market valuations were generally consistent with these theoretical predictions.
A numerical example for the call spread provides
perspective on the size of the violations reported in
this article. On January 4, 1996, call options expiring on March 16, 1996, with strike prices 610 (X1)
and 615 (X2) were priced at $23.25 (C1) and $15.50
(C2). The maturity date translates into a time to
maturity of 0.1973 years (T), and the continuously
compounded Treasury bill rate is 5.29 percent (r).
Using inequality (2a) and ignoring transaction costs
results in 15.50 – 23.25 + (615 – 610)e(–0.0529 × 0.1973)
= –2.8019 so that the size of the violation is $2.80.
Transaction costs are the sum of commission fees
and the bid-ask spread and are (4 + 4 + 30)/100
+ 1/8 = 0.505, which gives a net violation of $2.30
(–2.8019 + 0.505).
11. In some cases, a few extreme outliers were detected. After checking and rechecking the original data sources, these outliers
remained. However, removing these outliers does not change statistical inferences.
12. Autocorrelation in the dollar violations might be expected because the time to maturity for sample options may overlap.
Diagnostic tests confirm the presence of significant positive autocorrelation. However, inferences are unchanged if ordinary
least squares standard errors are used.
13. Note that inequality (1a) involves lending whereas inequality (1b) requires borrowing. Because similar frequency and magnitude of violations are observed across the two inequalities, the results suggest that the assumption of equal borrowing and
lending rates does not explain the extent of profit opportunities.
14. Abnormal profit opportunities are not expected to persist and, thus, the mean ex post violation is expected to be zero.
However, no directional relationship in the percentage of violations over the two-day time period is posited.
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T A B L E 3 Violations and Ex Ante Violations of the Box Spread (1b) by Year
Total Violations
Percentage of
Violations
Sample Year
Ex Ante Violations
Mean Dollar
Violation
Percentage of
Violations
Mean Dollar
Violation
1986
21.32
0.83
7.07
0.60
1987
26.23
1.27
11.16
1.26
1988
21.56
0.95
5.65
0.67
1989
15.90
0.90
6.36
0.86
1990
25.11
1.03
8.99
1.12
1991
24.16
0.99
9.28
1.03
1992
20.25
0.88
8.45
0.84
1993
19.19
0.81
6.77
0.67
1994
19.73
0.91
8.89
1.04
1995
23.87
0.95
10.61
0.83
1996
30.54
1.41
13.17
1.42
Overall
23.78
1.08
10.03
1.11
Note: This table reports the percentage and dollar size of violations of the box spread (1b) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular
violation persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
TABLE 4
Violations and Ex Ante Violations of the Call Spread (2a) and Put Spread (2b)
Call Spread (2a)
Total
Violations
Put Spread (2b)
Ex Ante
Violations
Total
Violations
Ex Ante
Violations
Frequency of Violations
Number of Observations
Number of Violations
Percentage of Violations
283,345
5,806
537,701
2,159
5,806
467
2,159
145
2.05
8.04
0.40
6.72
Violations, in Dollars
Mean
1.05
1.09
1.30
1.08
Standard Deviation
1.04
1.06
1.22
1.06
t statistic for nonzero mean
77.24***
22.13***
49.47***
12.27***
Note: This table reports the frequency and dollar size of violations of the call spread (2a) and put spread (2b) using daily data for the S&P
500 index and index options from January 1, 1986, through December 31, 1996. An ex ante violation occurs when a particular violation
persists into the following trading day. Asterisks *,**, or *** denote significance at the 10 percent, 5 percent, and 1 percent levels,
respectively, in a two-tailed test.
48
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TABLE 5
Page 49
Violations and Ex Ante Violations of the Call Spread (2a) by Year
Total Violations
Ex Ante Violations
Percentage of
Violations
Mean Dollar
Violation
Percentage of
Violations
1986
1.08
0.81
7.69
1.72
1987
2.10
0.92
8.70
0.70
1988
1.14
0.85
2.69
1.95
1989
1.71
0.91
6.54
0.86
1990
0.80
0.85
7.07
1.11
1991
3.14
1.17
8.36
0.90
1992
1.64
0.98
7.51
1.21
1993
1.48
0.73
6.65
0.52
1994
0.70
0.88
2.46
0.91
1995
3.92
0.99
11.16
1.14
1996
2.21
1.36
6.18
1.47
Overall
2.05
1.05
8.04
1.09
Sample Year
Mean Dollar
Violation
Note: This table reports the percentage and dollar size of violations of the call spread (2a) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular violation
persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
TABLE 6
Violations and Ex Ante Violations of the Put Spread (2b) by Year
Total Violations
Ex Ante Violations
Sample Year
Percentage of
Violations
Mean Dollar
Violation
Percentage of
Violations
Mean Dollar
Violation
1986
0.31
0.97
0
0
1987
1.83
1.75
4.75
1.66
1988
0.48
0.81
3.91
0.37
1989
0.19
0.90
3.70
0.25
1990
0.87
1.13
8.00
0.96
1991
0.25
0.97
4.42
0.43
1992
0.24
0.79
7.29
0.74
1993
0.17
0.96
2.78
0.19
1994
0.45
1.05
9.51
0.88
1995
0.12
1.00
9.80
1.06
1996
0.29
1.65
8.06
1.34
Overall
0.40
1.30
6.72
1.08
Note: This table reports the percentage and dollar size of violations of the put spread (2b) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular violation
persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
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T A B L E 7 Violations and Ex Ante Violations of Call Convexity (3a) and Put Convexity (3b)
Call Convexity (3a)
Total
Violations
Ex Ante
Violations
Put Convexity (3b)
Total
Violations
Ex Ante
Violations
Frequency of Violations
Number of Observations
882,954
27,206
2,244,467
20,439
Number of Violations
27,206
1,659
20,439
844
3.08
6.10
0.91
4.13
Percentage of Violations
Violations, in Dollars
Mean
0.91
1.13
0.95
1.21
Standard Deviation
0.94
1.07
1.04
1.14
t statistic for
nonzero mean
159.98***
43.01***
131.12***
30.81***
Note: This table reports the frequency and dollar size of violations of call convexity (3a) and put convexity (3b) using daily data for the
S&P 500 index and index options from January 1, 1986, through December 31, 1996. An ex ante violation occurs when a particular
violation persists into the following trading day. Asterisks *,**, or *** denote significance at the 10 percent, 5 percent, and 1 percent
levels, respectively, in a two-tailed test.
T A B L E 8 Violations and Ex Ante Violations of Call Convexity (3a) by Year
Total Violations
Ex Ante Violations
Percentage of
Violations
Mean Dollar
Violation
1986
2.11
0.72
0.61
0.02
1987
3.83
0.99
9.82
1.29
1988
1.37
0.79
1.12
0.20
1989
2.23
0.76
2.29
0.80
1990
1.76
0.64
4.07
0.72
1991
4.32
1.00
8.41
1.09
1992
1.82
0.75
3.17
1.02
1993
1.66
0.54
4.00
0.36
1994
0.85
0.55
4.09
0.82
1995
4.98
0.85
6.65
1.05
1996
3.45
1.07
5.12
1.33
Overall
3.08
0.91
6.10
1.13
Sample Year
Percentage of
Violations
Mean Dollar
Violation
Note: This table reports the percentage and dollar size of violations of call convexity (3a) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular violation
persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
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T A B L E 9 Violations and Ex Ante Violations of Put Convexity (3b) by Year
Total Violations
Ex Ante Violations
Percentage of
Violations
Mean Dollar
Violation
1986
1.06
0.58
0
0
1987
4.21
1.53
6.31
1.76
1988
0.77
0.71
2.06
0.49
1989
0.40
0.64
1.03
0.13
1990
1.82
0.82
5.59
0.93
1991
0.56
0.76
2.00
0.66
1992
0.45
0.70
2.91
0.70
1993
0.20
0.56
0.36
0.33
1994
0.76
0.78
3.94
1.13
1995
0.29
0.52
2.05
0.36
1996
1.01
0.89
3.51
1.06
Overall
0.91
0.95
4.13
1.21
Sample Year
Percentage of
Violations
Mean Dollar
Violation
Note: This table reports the percentage and dollar size of violations of put convexity (3b) using daily data for the S&P 500 index and index
options for each year in the January 1, 1986, through December 31, 1996, sample period. An ex ante violation occurs when a particular violation
persists into the following trading day. All mean dollar violations are significantly different from zero at the 1 percent significance level.
Taken together, significant violations of arbitrage
pricing relationships are observed, even using ex
ante tests, particularly for the box spread relationship. The differing results across the relationships
tested are not surprising because the box spread is
a more demanding test of market efficiency as compared with call and put spreads or convexity. The
overall finding is that S&P 500 index options are frequently mispriced to a significant extent and that
options market efficiency has not changed markedly
over time.
Conclusion
his article examines the efficiency of the S&P
500 index options market using theoretical
pricing relationships derived from stock
index option no-arbitrage principles. It reports frequent and substantial violations of the box spread
T
relationship in particular, even though the analysis
reflects transaction costs. The results do not provide support for the argument that options market
efficiency improved over time. However, at the same
time, there were few violations of call and put
spreads and convexity, which are less demanding
tests of pricing efficiency than the box spread.
Market frictions appear to have a significant effect
on arbitrageurs’ abilities to take advantage of violations of no-arbitrage pricing relationships. Although
the analysis reflects the market frictions imposed
by the bid-ask spread and commission costs, other
frictions may be significant. One such friction may
be insufficient liquidity, which increases option
traders’ risk and may prevent them from eliminating
arbitrage opportunities. In a liquid market a transaction can be quickly completed with little impact
on prices.
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