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Federal Reserve Bank of Chicago
Explaining Asset Pricing Puzzles
Associated with the 1987 Market Crash
Luca Benzoni, Pierre Collin-Dufresne, and
Robert S. Goldstein
November 8, 2010
WP 2010-10
Explaining Asset Pricing Puzzles Associated
with the 1987 Market Crash∗
Luca Benzoni†
Pierre Collin-Dufresne‡
Robert S. Goldstein§
November 8, 2010
Abstract
The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility
curve for equity index options, despite minimal changes in aggregate consumption. We explain these events
within a general equilibrium framework in which expected endowment growth and economic uncertainty are
subject to rare jumps. The arrival of a jump triggers the updating of agents’ beliefs about the likelihood
of future jumps, which produces a market crash and a permanent shift in option prices. Consumption
and dividends remain smooth, and the model is consistent with salient features of individual stock options,
equity returns, and interest rates.
Key words: Volatility Smile; Volatility Smirk; Implied Volatility; Option Pricing; Portfolio Insurance; Market
Risk.
JEL Classification: G12, G13.
∗
We thank Raj Aggarwal, Gordon Alexander, George Constantinides, Mariacristina De Nardi, Darrell Duffie, Bernard
Dumas, Nicolae Gârleanu, Jun Liu, Massimo Massa, Jun Pan, Monika Piazzesi, Mark Rubinstein, Bill Schwert (the Editor),
Costis Skiadas, Tan Wang, Stanley Zin, an anonymous referee, and seminar participants at the March 2006 NBER Asset Pricing
Meeting, Chicago, the 2006 Western Finance Association conference, the 2007 Econometric Society conference, and the 2008
Chicago/London Conference on Financial Markets for helpful comments and suggestions. Olena Chyruk and Andrea Ajello
provided excellent research assistance. All errors remain our sole responsibility. The views expressed herein are those of the
authors and not necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve System. The most recent
version of this paper can be downloaded from http://ssrn.com/abstract=1543467.
†
Benzoni is at the Federal Reserve Bank of Chicago, 230 S. LaSalle Street, Chicago, IL 60604, 312-322-8499,
lbenzoni@frbchi.org.
‡
Collin-Dufresne is at the Columbia Business School, 3022 Broadway, Uris Hall 404, New York, NY 10027, 212-854-6471,
pc2415@columbia.edu, and NBER.
§
Goldstein is at the Carlson School of Management, University of Minnesota, 321 19th Ave S., Minneapolis, MN 55455,
phone 612-624-8581, fax 612-626-1335, golds144@umn.edu, and NBER.
1.
Introduction
The 1987 stock market crash has generated many puzzles for financial economists. In spite of little
change in observable macroeconomic fundamentals, market prices fell 20–25% and interest rates dropped
about 1–2%. Moreover, the crash triggered a permanent shift in index option prices: Prior to the crash,
implied ‘volatility smiles’ for index options were relatively flat. Since the crash, however, the Black-Scholes
formula has been significantly underpricing short-maturity, deep out-of-the-money Standard and Poor’s
(S&P) 500 put options (Rubinstein, 1994; Bates, 2000). This feature, often referred to as the ‘volatility
smirk,’ is demonstrated in Fig. 1, which shows the spread of both in-the-money (ITM) and out-of-the-money
(OTM) implied volatilities relative to at-the-money (ATM) implied volatilities from 1985–2006. This figure
clearly shows that on October 19, 1987, the volatility smirk spiked upward, and that this shift has remained
ever since.
Not only is this volatility smirk puzzling in its own right, but it is also difficult to explain relative to
the shape of implied volatility functions (IVF) for individual stock options, which are much flatter and
more symmetric (see, e.g., Bollen and Whaley, 2004; Bakshi, Kapadia, and Madan, 2003; and Dennis
and Mayhew, 2002). Indeed, Bollen and Whaley (2004) argue that the difference in the implied volatility
functions for options on individual firms and on the S&P 500 index cannot be explained by the differences
in their underlying asset return distributions.
In this paper, we attempt to explain these puzzles while simultaneously capturing other salient features
of asset prices. In particular, we examine a representative-agent general equilibrium endowment economy
that can simultaneously explain:
• The prices of deep OTM put options for both individual stocks and the S&P 500 index;
• Why the slope of the implied volatility curve changed so dramatically after the crash;
• Why the regime shift in the volatility smirk has persisted for more than 20 years;
• How the market can crash with little change in observable macroeconomic variables.
We build on the long-run risk model of Bansal and Yaron (2004, BY), who show that if agents have a
preference for early resolution of uncertainty, e.g., have Kreps and Porteus (1978) / Epstein and Zin (1989),
or KPEZ, preferences with elasticity of intertemporal substitution EIS > 1, then persistent shocks to the
expected growth rate and volatility of aggregate consumption will be associated with large risk premiums
in equilibrium. Their model is able to explain a high equity premium, low interest rates, and low interest
rate volatility while matching important features of aggregate consumption and dividend time series. We
extend their model in two dimensions. First, we add a jump component to the shocks driving the expected
consumption growth rate and consumption volatility. These jumps (typically downward for expected growth
rates and upward for volatility) are bad news for the agent with KPEZ preferences, who will seek to reduce
her position in risky assets. In equilibrium, this reduction in demand leads to asset prices exhibiting a
downward jump, even though aggregate consumption and dividends are smooth. That is, in our model,
the level of consumption and dividends follows a continuous process; it is their expected growth rates and
volatilities that jump. Since shocks to expected consumption growth rate and consumption volatility are
associated with large risk premiums, jumps in asset prices can be substantial, akin to market ‘crashes.’
1
Our second contribution relative to BY (2004) is to allow for parameter uncertainty and learning. Specifically, we assume the jump frequency is governed by a hidden two-state continuous Markov chain, which
needs to be filtered in equilibrium. This adds another source of risk to the economy, namely the posterior
probability of the hidden state. We show that the risk premium associated with revisions in posterior beliefs
about the hidden state can be large, as they are a source of ‘long-run risk.’ In fact, we show that it can
explain the dramatic shift in the shape of the implied volatility skew observed in 1987. If, prior to 1987,
agents’ beliefs attribute a very low probability to high jump intensities then, prior to 1987, prices mostly
correspond to a no-jump Black-Scholes type economy. However, after a jump in prices occurs as a result
of the jump in expected growth rates and volatility of fundamentals, agents update their beliefs about the
likelihood (i.e., intensity) of future jumps occurring, which contributes to the severity of the market crash
and leads to the steep skew in implied volatilities observed in the data henceforth. Because these beliefs are
very persistent, the skew is long-lived after the crash.
Although the two new features, jumps and learning, dramatically impact the prices of options, we
show that our model still matches salient features of U.S. economic fundamentals. Because jumps impact
the expected consumption growth rate and consumption volatility, but not the level of consumption, the
consumption process remains smooth in our model, consistent with the data. Further, as noted by BY
(2004) and Shephard and Harvey (1990), it is very difficult to distinguish between a purely independent
identically distributed (i.i.d.) process and one which incorporates a small persistent component. Indeed, we
show that the dividend and consumption processes implied by the model fit the properties of the data well,
in that we cannot reject the hypothesis that the observed data were generated from our model.
Nonetheless, and as in BY (2004), the asset pricing implications of our model differ significantly from
those of an economy in which dividends are i.i.d. Specifically, the calibrated model matches the typical level
of the price-dividend ratio and produces reasonable levels for the equity premium, the risk-free rate, and
their standard deviation. In the same calibration we show that the pre-crash implied volatility function
for short-maturity index options is nearly flat, while it becomes a steep smirk immediately after the crash.
Moreover, the model predicts a downward jump in the risk-free rate during the crash event, consistent with
observation.
Finally, the model reproduces the stylized properties of the implied volatility functions for individual
stock option prices. We specify individual firm stock dynamics by first taking our model for the S&P 500
index and then adding idiosyncratic shocks, both of the diffusive and the jump types. We then calibrate
the coefficients of the idiosyncratic components to match the distribution of returns for the ‘typical’ stock.
In particular, we match the cross-sectional average of the high-order moments (variance, skewness, and
kurtosis) for the stocks in the Bollen and Whaley (2004) sample. We simulate option prices from this model
and compute Black-Scholes implied volatilities across different moneyness. Consistent with the evidence in
Bollen and Whaley (2004), Bakshi, Kapadia, and Madan (2003), and Dennis and Mayhew (2002), we find
an implied volatility function that is considerably flatter than that for S&P 500 options. Bakshi, Kapadia,
and Madan (2003) conclude that the differential pricing of individual stock options is driven by the degree
of skewness/kurtosis in the underlying return distribution in combination with the agent’s high level of risk
aversion. Here, we propose a plausible endowment economy that, in combination with recursive utility,
2
yields predictions consistent with their empirical findings.
Related literature. Motivated by the empirical failures of the Black-Scholes model in post-crash S&P 500
option data, prior studies have examined more general option pricing models (see, e.g., Bates, 1996; Duffie et
al., 2000; and Heston, 1993). A vast literature explores these extensions empirically,1 reaching the conclusion
that a model with stochastic volatility and jumps significantly reduces the pricing and hedging errors of the
Black-Scholes formula.2 These previous studies, however, focus on post-1987 S&P 500 option data. Further,
they follow a partial equilibrium approach and let statistical evidence guide the exogenous specification of
the underlying return dynamics.
Reconciling the findings of this literature in a rational expectations general equilibrium setting has proven
difficult. For instance, Pan (2002) notes that the compensation demanded for the ‘diffusive’ return risk is
very different from that for jump risk. Consistent with Pan’s finding, Jackwerth (2000) shows that the risk
aversion function implied by S&P 500 index options and returns in the post-1987 crash period is partially
negative and increasing in wealth; similar results are presented in Aı̈t-Sahalia and Lo (2000) and Rosenberg
and Engle (2002). This evidence is difficult to reconcile in the standard general equilibrium model with
constant relative risk-aversion utility and suggests that there may be a lack of integration between the
option market and the market for the underlying stocks.
Several papers have investigated the ability of equilibrium models to explain post-1987 S&P 500 option
prices. Liu, Pan, and Wang (2005, LPW) consider an economy in which the endowment is an i.i.d. process
that is subject to jumps. They show that, in this setting, neither constant relative risk aversion nor Epstein
and Zin (1989) preferences can generate a volatility smirk consistent with post-1987 evidence on S&P 500
options. They argue that in order to reconcile the prices of options and the underlying index, agents must
exhibit ‘uncertainty aversion’ towards rare events that is different from the standard ‘risk-aversion’ they
exhibit towards diffusive risk. This insight provides a decision-theoretic basis to the idea of crash aversion
advocated by Bates (2008), who considers an extension of the standard power utility that allows for a special
risk-adjustment parameter for jump risk distinct from that for diffusive risk. These prior studies assume
that the dividend level is subject to jumps, while the expected dividend growth rate is constant. Thus, in
these models a crash like that observed in 1987 is due to a 20–25% downward jump in the dividend level.3
Moreover, their model predicts no change in the risk-free rate during the crash event. In our setting, it is the
expected endowment growth rate that is subject to jumps. Thus, in our model, dividends and consumption
are smooth and the market can crash with minimal change in observable macroeconomic fundamentals.
Further, the risk-free rate drops around crash events, consistent with empirical evidence.
Other studies explore the option pricing implications of models with state dependence in preferences
1
Among recent contributions, Bakshi et al. (1997, 2000), Bates (2000), and Huang and Wu (2004) focus on derivatives prices
alone. Pan (2002), Broadie et al. (2007), Chernov and Ghysels (2000), Jones (2003), Eraker (2004), and Benzoni (2002) use
data on both underlying and derivatives prices to fit the model.
2
A related literature investigates the profits of option trading strategies (e.g., Coval and Shumway, 2001; and Santa-Clara
and Saretto, 2009) and the economic benefits of giving investors access to derivatives when they solve the portfolio choice
problem (e.g., Constantinides et al., 2009; Driessen and Maenhout, 2007; Liu and Pan, 2003).
3
Barro (2006) makes a similar assumption about output dynamics. His model captures the contractions associated with the
Great Depression and the two World Wars, but it does not match the evidence around the 1987 crash, when the output level
remained smooth.
3
and/or fundamentals; see, e.g., Bansal, Gallant, and Tauchen (2007), Bondarenko (2003), Brown and Jackwerth (2004), Buraschi and Jiltsov (2006), Chabi-Yo, Garcia, and Renault (2008), David and Veronesi (2002,
2009), and Garcia, Luger, and Renault (2001, 2003). These papers do not study the determinants of stock
market crashes, the permanent shift in the implied volatility smirk that followed the 1987 events, and the
difference between implied volatility functions for individual and index stock options. To our knowledge,
our paper is the first to focus on these issues.
Also related is a growing literature that investigates the effect of changes in investors’ sentiment (e.g.,
Han, 2008), market structure, and net buying pressure (e.g., Bollen and Whaley, 2004; Dennis and Mayhew,
2002; and Gârleanu et al., 2009) on the shape of the implied volatility smile. This literature argues that
due to the existence of limits to arbitrage, market makers cannot always fully hedge their positions (see,
e.g., Green and Figlewski, 1999; Figlewski, 1989; Hugonnier et al., 2005; Liu and Longstaff, 2004; Longstaff,
1995; and Shleifer and Vishny, 1997). As a result, they are likely to charge higher prices when asked to
absorb large positions in certain option contracts. These papers, however, do not address why end users buy
these options at high prices relative to the Black-Scholes value or why the 1987 crash changed the shape of
the volatility smile so dramatically and permanently. Our paper offers one possible explanation.
Finally, the large impact that learning can have on asset price dynamics has been shown previously
(e.g., David, 1997; Veronesi, 1999, 2000). One important difference between these papers and ours is that
our agent learns from jumps rather than diffusions, as in Benzoni, Collin-Dufresne, Goldstein, and Helwege
(2010), leading to different updating dynamics.
The main contribution of our paper is to explain pre- and post-1987 crash asset prices in a rationalexpectation framework that is consistent with underlying fundamentals. However, to our knowledge this is
also the first article to examine the effect of jumps in the Bansal and Yaron (2004) economy. This has proven
to be a fruitful extension of the long-run risk framework and has been further explored by, e.g., Drechsler
and Yaron (2008), Eraker (2008), and Eraker and Shaliastovich (2008).
The rest of the paper proceeds as follows. In Section 2, we present the model and discuss our solution
approach. Section 3 shows that the model matches the relevant asset pricing facts while being consistent
with underlying fundamentals. Section 4 concludes the paper.
2.
The model
We specify the dynamics for log-consumption (c ≡ log C) and log-dividend (δ ≡ log D) as
µ
¶
√
1
dc =
µC + x − Ω dt + Ω dzC
2
¶
µ
q
´
√ ³
1 2
dδ =
µD + φ x − σD Ω dt + σD Ω ρDC dzC + 1 − ρ2DC dzD
2
√
p
dx = −κx x dt + σxc Ω dzc + σx0 + σxΩ Ω dzx + νe dN
q
´
√ ³
¢
¡
f dN .
dΩ = κΩ Ω − Ω dt + σΩ Ω ρΩC dzC + 1 − ρ2ΩC dzΩ + M
(1)
(2)
(3)
(4)
These are continuous-time versions of the dividend and consumption dynamics considered in BY (2004),
except that the predictable dividend component x and the measure of economic uncertainty Ω are subject
4
to jumps. The diffusive shocks {dzC , dzD , dzx , dzΩ } are uncorrelated Brownian motions, while jumps are
governed by a Poisson process dN with Prob(dNt = 1|Ft ) = λt dt. The jump intensity λt can take two
possible values, λG and λB , and it transitions from one state to the other via the dynamics:
¡
¢
π λt+dt = λB |λt = λG = φGB dt
¡
¢
π λt+dt = λG |λt = λB = φBG dt.
f are drawn from the distribution
Following Eraker (2000), the jump size variables νe and M
õ
¶·
µ
¶¶¸2 !
µ
³
´
1
1
1
f = M = ξ e−ξM 1
p
π νe = ν, M
exp
− 2
ν− ν−α M −
.
{M >0}
2σν
ξ
2πσν2
(5)
(6)
As in BY (2004), changes in economic fundamentals are driven by the continuous trickling down of new
information, modeled through the diffusive shocks {dzC , dzD , dzx , dzΩ }. Jumps add an additional source of
risk to this framework and capture the notion of sudden unexpected changes in economic fundamentals. In
Eq. (6), ν denotes the average value of νe. When ν is negative, the typical jump in the x state variable lowers
the expected growth rate of the agent’s endowment. Moreover, a jump increases economic uncertainty Ω
f is higher than 1 , then the average jump in x is
f, whose average value is 1 . If the realization of M
by M
ξ
h
³
´i ξ
reduced to ν − α M − 1ξ
(assuming α > 0). Hence, α controls the level of correlation between the jump
in volatility and the jump in expected consumption growth.
We model jumps as rare events, i.e., both λB and λG are small. Yet, we specify λB = 0.035 to be
considerably larger than λG = 0.0005. Thus, the agent views the economy with λt = λB as the ‘bad’
economy and λt = λG as the ‘good’ economy.
The agent does not observe the state of the economy directly, i.e., she does not know whether λt = λB
or λt = λG . Instead, she observes only the process {dNt }. We define the state variable p(t) as her date-t
estimate that the economy is in the good state (i.e., her prior):
p(t) ≡ π(λt = λG | dNt ) .
(7)
Over each interval dt, the agent updates her prior. The solution to the filtering problem with Markov
switching is studied in Liptser and Shiryaev (2001). Applying the results in their Theorem 19.6, p. 332, and
Example 1, p. 333, we obtain the Bayesian dynamics for the probability of being in a good state:4
¶
µ G
£
¡
¢
¤
λ − λ(p)
dN + − p λG − λ(p) + (1 − p)φBG − pφGB dt ,
(8)
dp = p
λ(p)
£
¤
where λ(p(t)) = p(t)λG + (1 − p(t)) λB . Thus, at time t, the agent perceives the probability of a future
jump to be:
Prob(dNt = 1|Ft ) = λ(p(t)) dt.
(9)
Over each interval dt, the agent updates her prior according to Eq. (8). The first term captures updating
due to an observation of a jump (dN = 1). Since jumps are rare events, this term is zero most of the time.
4
For reasons of parsimony, we assume the agent learns about the state of the economy only by observing market crashes.
It is possible to allow the agent to also learn about the state of the economy by observing external signals (see, e.g., Veronesi,
2000). Such a generalization would allow us to capture higher-frequency fluctuations in option prices, which is not our focus.
5
However, if a jump occurs during the interval dt, the agent’s prior that the economy is in the good state
³ G
´
−λ(p)
shifts by dpt = pt λ λ(p)
. This change is negative for interior values of 0 < pt < 1, but is zero when
either pt = 0 or pt = 1. Intuitively, when pt is either zero or one, the agent knows the state for certain,
and thus does not update her priors even if she observes a jump. The second term in Eq. (8) captures
deterministic fluctuations in the agent’s prior, and is controlled by the Markov chain transition coefficients
φGB and φBG . In our calibration, these coefficients are small, which implies that the agent’s beliefs are very
persistent. This feature helps explain why the shift in the volatility smirk observed around the 1987 crash
has persisted since then.
For future reference, it is convenient to define the state vector as Xt = (xt , Ωt , pt ). Further, we write the
state vector dynamics as:
dct = µC (Xt ) dt + σC (Xt ) dz(t)
dδt = µD (Xt ) dt + σD (Xt ) dz(t) + σDD (Xt ) dzD (t)
e dN (t) ,
dXt = µX (Xt ) dt + σX (Xt ) dz(t) + Γ
(10)
e Note
with the vector of independent Brownian motions z = (zc , zx , zΩ ) and the vector of jump variables Γ.
that (ct , Xt ) and (δt , Xt ) are both Markov systems.
2.1.
Recursive utility
Following Epstein and Zin (1989), we assume that the representative agent’s preferences over a consumption process {Ct } are represented by a utility index U (t) that satisfies the following recursive equation:
½
¾ 1
¡
¢ 1−ρ 1−ρ
1−ρ
1−γ 1−γ
−βdt
−βdt
.
U (t) = (1 − e
)Ct + e
Et U (t + dt)
(11)
With dt = 1, this is the discrete time formulation of KPEZ preferences, in which Ψ ≡ 1/ρ is the EIS and γ
is the risk-aversion coefficient.
The properties of the stochastic differential utility in (11) and the related implications for asset pricing
have been previously studied by, e.g., Duffie and Epstein (1992a,b), Duffie and Skiadas (1994), Schroder
and Skiadas (1999, 2003), and Skiadas (2003). In Appendix A, we extend their results to the case in which
the aggregate output has jump-diffusion dynamics. The solution to our model follows immediately from
Propositions 1 and 2 in Appendix A. Specifically, let us define
¶
µ
1
U (t)(1−γ) .
J(t) =
1−γ
Then, it is well-known that J(t) has the following representation:
·Z ∞
¸
J(t) = Et
f (Cs , J(s)) ds ,
(12)
t
where f (C, J) is the normalized aggregator defined in Duffie and Epstein (1992), which we reproduce in
Eq. (A.7) of Appendix A.
6
For the cases ρ, γ 6= 1, Proposition 1 gives the agent’s value function as:5
J=
where we have defined θ ≡
1−γ
1−ρ .
ec(1−γ) θ
β I(X)θ ,
1−γ
(13)
Further, we show in Appendix A that I is the price-consumption ratio, and
satisfies the relation:
¶
µ
2
DI(X)θ
2 ||σC (X)||
− βθ +
−θ = I(X) (1 − γ)µC (X) + (1 − γ)
2
I(X)(θ−1)
+(1 − γ)θσC (X)σX (X)> IX (X) + I(X)λ(X)J I(X)θ ,
(14)
where we define the (continuous diffusion, jump, and jump-compensator) operators D, J , J in Eq. (A.13)
in Appendix A.
2.2.
Pricing kernel and risk-free rate
When ρ, γ 6= 1, Proposition 1 in Appendix A identifies the pricing kernel as
Rt
Π(t) = e
0
ds [(θ−1)I(s)−1 −βθ]
β θ e−γct I(Xt )(θ−1) .
(15)
Using Itô’s lemma, we obtain the dynamics of the pricing kernel, which identifies both the diffusion and the
jump risk premiums, as well as the risk-free rate:
dΠ(t)
= −rt dt − (γσC (Xt ) + (θ − 1)σI (Xt )) dz(t) + J I(Xt )(θ−1) dNt − λ(Xt )J I(Xt )(θ−1) dt.
Π(t)
Here, we have defined the diffusion-volatility of the price-consumption ratio as σI (X) =
(16)
1
>
I(X) IX (X) σX (X),
and the risk-free rate via:6
¶
µ
¶
µ
||σC (X)||2
1
||σC (X)||2
>
− γ(1 + ρ)
− (1 − θ)σI (X)
σC (X) + σI (X)
r(X) = β + ρ µC (X) +
2
2
2
µ
¶
θ−1 θ
+λ(X)
J I − J I (θ−1) .
(17)
θ
Note that this result reduces to the standard result for the constant relative risk aversion (CRRA) exchange
economy if ρ = γ (i.e., θ = 1). Instead, if agents have a preference for resolution of uncertainty, then riskaversion affects the interest rate via the precautionary savings motive if consumption has positive volatility.
Further, if agents display a preference for early resolution (γ > ρ) and if the EIS is greater than one (i.e,
ρ < 1), then the greater the volatility of the price-consumption ratio, the lower the equilibrium interest
rate, as agents want to divest from the risky asset because of long-run risk (this follows since under these
conditions, 1 − θ =
5
6
γ−ρ
1−ρ
> 0).
Other cases with either ρ or γ equal to one can be treated similarly, as shown in Appendix A.
Again, we only present results for the case where ρ, γ 6= 1. The other cases are treated in Appendix A.
7
2.3.
Price-dividend ratio and equity premium
The stock market portfolio is a claim to aggregate dividends D(t). Thus, its value is obtained by the
standard discounted cash-flow formula:
S(t) = Et
·Z
∞
t
¸
Π(s)
D(s) ds .
Π(t)
(18)
This equation implies that the excess return on the stock is given by:7
·
¸
·
¸
dS(t)
dΠ(t) dS(t)
1
D(t)
1
Et
+
− rt = − Et
.
dt
S(t)
S(t)
dt
Π(t) S(t)
(19)
Defining the price-dividend ratio via St = Dt L(Xt ) and substituting into Eq. (19), we obtain a partial
differential equation (PDE) for L(X) similar to that obtained for the price-consumption ratio. To save on
space, we relegate this expression to Eq. (B.7) in Appendix B. Using the definition of the pricing kernel, we
can compute the right-hand side of Eq. (19) more explicitly to obtain the following expression for the risk
premium on the dividend claim:
µS (X) +
1
− r(X) = (γσC (X) + (1 − θ)σI (X))> (σD (X) + σL (X))
L(X)
³
´
+λ(X) J {I(X)θ−1 L(X)} − J I(X)θ−1 − J L(X) ,
where we have defined σL (X) =
when (1 − θ) =
γ−ρ
1−ρ
1
>
L(X) LX (X) σX (X)
(20)
as the diffusion of the price-dividend ratio. Note that
> 0 (which holds, in particular, when agents have a preference for early resolution of
uncertainty, that is γ > ρ, and the EIS is greater than one, that is ρ < 1), then the higher the volatility of
the price-consumption ratio, the greater the equity premium.
2.4.
Pricing options on the market portfolio and individual stocks
The date-t value of a European call option on the stock market portfolio S(t), with maturity T and
strike price K, is given by
¸
· R
− T r(X(s)) ds
+
(S(T ) − K)
,
C(S(t), X(t), K, T ) = Et e t
Q
(21)
where the expectation is computed under the risk-neutral measure Q. The risk-neutral dynamics of the
stock price, St = Dt L(Xt ) are:
µ
¶
dS
1
Q
=
rt −
dt + (σD (Xt ) + σL (Xt )) dz Q (t) + σDD (Xt ) dzD
(t)
S
L(Xt )
+J L(Xt ) dNt − λQ (Xt )J L(Xt ) dt .
(22)
The risk-neutral dynamics for D, x, Ω, and p are given in Appendix B.4.
As in Bakshi, Kapadia, and Madan (2003), we specify return dynamics on an individual stock,
dSi
Si ,
as the
sum of a systematic component and an idiosyncratic component. In particular, we assume that individual
firm dynamics follow
·³
¸
´
h
i
dSi
dS
νei
νei
=
+ σi dzi + e − 1 dNi − E e − 1 λi dt ,
Si
S
7
This follows from Itô’s lemma, the dynamics of Π(t), and the fact that Π(t)S(t) +
8
Rt
0
Π(s)D(s) ds is a P-martingale.
(23)
where the market return dynamics
dS
S
are given in Eq. (22). Here, σi captures the volatility of the idiosyn-
cratic diffusive shock, while the diversifiable jump component has Poisson arrival rate Ni with constant
intensity λi and normally distributed jump size νei ; N (µνi , σνi ). The free parameters (σi , λi , µνi , σνi ) are
chosen to match historical moments of the return distribution on individual firms. By definition, the diversifiable shocks do not command a risk premium, while the risk adjustments on the systematic component
are identical to those that we have applied to price the options on the S&P 500 index. Thus, the price of
an option on an individual stock is given by a formula similar to Eq. (21).8
3.
Data and model implications
Here, we calibrate the model coefficients to match economic fundamentals, solve the model numerically,
and study its asset pricing implications.
3.1.
Baseline model coefficients
Table 1 reports the coefficients for our baseline model calibration, expressed with a yearly decimal scaling.
They are organized in seven groups, which we briefly discuss below.
1. Preferences:
We use a time discount factor coefficient β = 0.0176 and fix the coefficient of relative risk aversion
γ at 10, a value that is generally considered to be reasonable (e.g., Mehra and Prescott, 1985). The
magnitude of the EIS coefficient Ψ is more controversial. Hall (1988) argues that the EIS is below 1.
However, Guvenen (2001) and Hansen and Singleton (1982), among others, estimate the EIS to be
in excess of 1, and Attanasio and Weber (1989), Bansal, Gallant, and Tauchen (2007), and Bansal,
Tallarini, and Yaron (2006) find it to be close to 2. Accordingly, we fix Ψ = 2 in our baseline case.
2. Aggregate consumption and dividends:
We fix µC = 0.018 and µD = 0.025, consistent with the evidence that, historically, dividend growth has
exceeded consumption growth (Section 3.2 below). As in, e.g., BY (2004), we set φ > 1 to allow the
sensitivity of dividend growth to shocks in x to exceed that of consumption growth. Setting ρDC > 0
guarantees a positive correlation between consumption and dividends.
3. Predictable mean component, x:
Similar to BY (2004), in the dynamics (3) we use κx = 0.2785, which makes x a highly persistent
process (if we adjust for differences in scaling and map the BY (2004) AR(1) ρ coefficient into the κx
of our continuous-time specification, we find κx = 0.2547). We decompose the shocks to the x process
into two terms that are orthogonal and parallel to consumption shocks, with σxc > 0, σxΩ > 0, and
σx0 = 0.
8
There might be a potential concern that the dynamics (23) for the individual firms and the dynamics
P(22) for the aggregate
index are not self-consistent. That is, the terminal value of a strategy that invests an amount S(0) = N
i=1 Si (0) in the index
does not necessarily have the same terminal value of a strategy that invests an amount Si (0) in each of
P the individual stocks,
i = 1, . . . , N . However, we find in unreported simulations that the discrepancy is negligible, i.e., S(T ) ≈ N
i=1 Si (T ). Intuitively,
the idiosyncratic shocks that we specify are in fact diversifiable when the portfolio is composed of a sufficiently large number of
firms.
9
4. Economic uncertainty, Ω:
In the Ω-dynamics (4), we fix Ω and σΩ at values similar to those used in BY (2004). However, in
our calibration κΩ = 1.0484, which makes the Ω process much less persistent than the x process (the
half life of a shock is around seven to eight months). This is in contrast to the high persistence of the
volatility shocks in the BY (2004) calibration, and more akin to the calibration in Drechsler and Yaron
(2008). This feature is important in the presence of jumps to volatility, since with a highly persistent
Ω process, as in BY (2004), volatility would remain high for years after a jump. Finally, we allow for
a negative correlation between shocks to consumption and volatility, ρΩC < 0.
5. Jumps:
We set λG = 0.0005 and λB = 0.035. Thus, if the jump intensity λt equals λG , a jump occurs
about once every 2,000 years. In contrast, if λt = λB , the average jump time is approximately 30
years. Jumps in x have negative mean, ν < 0, i.e., the typical jump carries bad news for the growth
prospects of the economy.9 Similarly, jumps to Ω are positive and increase the level of economic
uncertainty. Finally, we set the transition probabilities φBG = 0.025 and φGB = 0.0025. With these
values, in steady state the economy is in the ‘good’ state λG with probability φBG /(φBG +φGB ) = 0.91.
6. Individual stock returns:
For each of the 20 stocks in the Bollen and Whaley (2004) study, we compute standard deviation,
skewness, and kurtosis by using daily return series for the sample period from January 1995 to December 2000 (the same period considered by Bollen and Whaley). For each of these statistics, we evaluate
cross-sectional averages. We find an average standard deviation of 37.6% per year and average skewness and kurtosis of 0.12 and 7.12, respectively. Four coefficients characterize the distribution of the
idiosyncratic shocks in Eq. (23): the standard deviation of the diffusive firm-specific shock, σi ; the
intensity of the diversifiable jump component, λi ; and the mean and standard deviation of the jump
size, µνi and σνi . After some experimentation, we fix the jump intensity to λi = 5, which corresponds
to an expected arrival rate of five jumps per year. We choose the remaining coefficients to match the
average standard deviation, skewness, and kurtosis reported above. This approach yields σi = 0.3137,
µνi = 0.0036, and σνi = 0.0632. To confirm that the results are robust to this approach, we solve
for σi , µνi , and σνi when λi takes different values in the 1–10 range. The results are similar to those
discussed below.
7. Initial conditions:
Before the crash, the agent is nearly sure that the economy is in the good state λt = λG . Specifically,
we set pP re = 99.85%. When the agent perceives a jump in fundamentals, she updates her prior
³ G
´
λ −λ(pP re )
P
ost
P
re
P
re
according to Eq. (8). In our calibration, this yields p
=p
+p
= 90.48%. We
λ(pP re )
fix the remaining state variables at their steady-state values, x0 = ν λ/κx and Ω0 = Ω + λ/(ξκΩ ).
9
In our calibration, jumps are extremely rare events that are typically associated with large jumps in asset prices. This
distinguishes our paper from other studies that consider higher-frequency jumps. For instance, Drechsler and Yaron (2008)
assume that jumps have mean zero and occur, on average, 0.8 times per year.
10
3.2.
Aggregate consumption and dividends
Here we demonstrate that the calibration discussed in the previous section matches the historical data
well. In Table 2, we report summary statistics for the series of yearly growth rates on aggregate consumption
(Panel A) and dividends (Panel B). We focus not only on low-order moments, like mean, standard deviation,
auto- and cross-correlations, but also on higher-order moments like skewness and kurtosis. We report
empirical results for two sample periods. The first spans 80 years of data, from 1929 to 2008, the second
spans the post-World War II period, from 1946 to 2008. In addition to point estimates for these moments,
we report standard errors robust with respect to both auto-correlation and heteroskedasticity. The measure
for aggregate consumption is the real (in chained, year 2000, dollars) yearly series of per-capita consumption
expenditures in nondurable goods and services from the National Income and Product Accounts (NIPA)
tables published by the Bureau of Economic Analysis. Following Fama and French (1988), we obtain a
monthly dividend proxy by subtracting returns without dividends from returns with dividends on the valueweighted market index as reported by the Center for Research in Security Prices (CRSP). We sum the
monthly dividends to obtain the yearly dividend, and we deflate the yearly dividend series using Consumer
Price Index (CPI) data.
To examine the model implications, we simulate 10,000 samples of monthly consumption and dividend
data, each spanning a period of 80 years (same as the length of the 1929–2008 sample period). We aggregate
the monthly series to obtain yearly dividends and consumption, and we compute the series of yearly growth
rates
∆C
C
and
∆D
D .
For each of the 10,000 samples, we compute summary statistics for these series. In the
table, we report the mean value of these statistics, as well as the 5th, 50th, and 95th percentiles. In the
simulations, we use two different initial conditions. First, we initialize the Markov chain for the λ process
at λ(t = 0) = λG . That is, we assume that at the beginning of time, the economy is in the ‘good’ state.
Second, we initialize λ(t = 0) = λB , i.e., we assume that at t = 0, the economy is in the ‘bad’ state. In the
table, we report the results for each of these two cases.
In both sets of simulations, the moments of
∆C
C
and
∆D
D
are very close to the sample moments. For
most moments, the mean and the median computed in model simulations are essentially identical to those
computed with the data. See, for instance, AC(1) for
∆C
C :
it is 0.42 in the data (1929–2008 sample),
compared to a median simulated value of 0.42 when λ(t = 0) = λG , and 0.44 when λ(t = 0) = λB .
In a few cases, the median values in model simulations do not perfectly match the estimates in the
data. However, the sample estimates are in the 90% confidence interval computed from model simulations.
See, e.g., the skewness of
∆D
D .
It is 0.38 in the 1929–2008 sample, a value that falls well within the model
confidence interval of [−0.20, 0.71] when λ(t = 0) = λG , and [−0.24, 0.71] when λ(t = 0) = λB . These
results reflect the fact that some statistics are imprecisely estimated. For instance, the skewness of
∆D
D
is much higher in the 1946–2008 sample, and the standard error associated to this estimate is very high.
Similarly, the model cannot match the extreme
∆C
C
skewness estimated over 1929–2008 (due to the drop in
consumption during the Great Depression), but it gets close to the −0.53 estimate for the 1946–2008 period.
There is one fact that the model does not capture well. The kurtosis of
∆D
D
is 5.30 in the 1946–2008
sample, and 9.06 in the 1929–2008 sample. Both values exceed the 4.16 upper bound in model simulations.
This may not be a serious shortcoming of the model, for two reasons. First, kurtosis is very imprecisely
11
estimated in the data (note the difference in point estimates across the two sample periods and the huge
standard errors). Second, it is arguably a good thing that our results are not driven by an excess of
skewness/kurtosis built in the model.
3.3.
Stock market return and risk-free rate
Before turning to option prices, we further validate our calibration by showing that the model is also
consistent with a wide range of asset pricing facts. Table 3 reports key asset pricing moments computed
with data spanning the 1929–2008 and 1946–2008 sample periods. The real annualized total market return,
( ∆S
S +
D
S ),
is the yearly return, inclusive of all distributions, on the CRSP value-weighted market index,
adjusted for inflation using the CPI. The real risk-free rate rf is the inflation-adjusted three-month rate from
the ‘Fama Risk-Free Rates’ database in CRSP. In computing the logarithmic price-dividend ratio, log(S/D),
we consider two measures of dividends. The first is the real dividend on the CRSP value-weighted index,
which we have already discussed in Section 3.2 above. The second is the real dividend on the CRSP
value-weighted market index, adjusted for share repurchases (Boudoukh et al., 2007).
We solve the model numerically (Appendix C discusses the numerical approach) and simulate 10,000
samples of monthly stock market returns, risk-free rates, and price-dividend ratios, each spanning a period
of 80 years. We aggregate the monthly series at the yearly frequency. For each of the 10,000 simulated
samples, we compute summary statistics for these series. We report the mean value of these statistics, as
well as the 5th, 50th, and 95th percentiles. We repeat the analysis with two simulation schemes. In the
first set of simulations, we initialize the probability process p at p(t = 0) = pP re , which corresponds to the
pre-crash economy. In the second set, we initialize p(t = 0) = pP ost , which corresponds to the post-crash
economy.
Panel A in Table 3 shows the moments for the real stock market return. The sample mean estimate
is very close to the mean in model simulations. The sample standard deviation for the yearly return is
a bit high relative to the model predictions when estimated over the 1929–2008 sample. However, the
estimate computed with post-World War II data falls well within the model confidence bands, and the
model matches the (annualized) monthly return standard deviation estimate well. Moreover, the model fits
the market return kurtosis accurately, both in monthly and annual data. The sample skewness is more
imprecisely estimated, but remains reasonably close to the model predictions, especially in monthly data.
Panel B shows the mean and standard deviation of the risk-free rate. While the model matches the mean
accurately, the standard deviation is somewhat higher in the data than in the model. This is a well-known
feature of the long-run risk setup. It is arguably a desirable property of the model, rather than a weakness.
For instance, Beeler and Campbell (2009, p. 8) report similar results and note that “the data record the ex
post real return on a short-term nominally riskless asset, not the ex ante (equal to ex post) real return on a
real riskless asset. Volatile inflation surprises increase the volatility of the series in the data, but not in the
model.”10
10
Other studies have attempted to filter out the predictable component in real rate fluctuations prior to computing its volatility.
For instance, Barro (2006) finds that the annual standard deviation of the residuals from an AR(1) process for realized real rates
of return on U.S. Treasury bills or short-term commercial paper from 1880 to 2004 is 0.018. In our calibration, the risk-free
rate volatility does not exceed that value.
12
The properties of the stock market return and the risk-free rate extend to the equity premium, which
the model matches quite well (Panel C). Finally, the model seems to underestimate the mean level of the
logarithmic price-dividend ratio (Panel D). However, when we account for share repurchases in the measure
of dividends, as in, e.g., Boudoukh et al. (2007), the sample estimate of the price-dividend ratio is revised
downward and is perfectly in line with the model predictions.
In sum, these results support two main conclusions. First, the model matches several important asset
pricing moments quite well. Second, the asset pricing moments predicted by the model pre- and post-1987
crash are similar. Yet, we show in the next section that option prices differ quite dramatically before and
after the crash.
3.4.
Option prices
Fig. 1 shows the spread of in-the-money and out-of-the-money implied volatilities relative to at-the-money
implied volatilities from 1985 to 2006. (Appendix D explains how we constructed the implied volatility
series.) Prior to the crash, 10% OTM puts with one month to maturity had an average implied volatility
spread of 1.83%. Similarly, the spread for 2.5% ITM put options averaged −0.12% prior to the crash. On
some dates the implied volatility function had the shape of a mild ‘smile’ and on others it was shaped
like a mild ‘smirk.’ Overall, the Black-Scholes formula priced all options relatively well prior to the crash,
underpricing deep OTM options only slightly. This all changed on October 19, 1987, when the spread for
OTM puts spiked up to a level above 10%. Since then, implied volatilities for deep OTM puts have averaged
8.21% higher than ATM implied volatilities. Moreover, since the crash, implied volatilities for ITM options
have been systematically lower than ATM implied volatilities, with an average spread of −1.33%.
We simulate 500,000 paths and compute option prices on the S&P 500 index, with one month to maturity,
across different strike prices. Fig. 2 illustrates the results for the pre-crash economy, i.e., p = pP re , and for
the post-crash economy, i.e., p = pP ost . The two plots capture the stark regime shift in index option prices.
Prior to the crash, the implied volatility function is nearly flat, with a very mild upward tilt. In the model,
the pre-crash spread between OTM and ATM implied volatilities is 1.69%, while the spread between ITM
and ATM implied volatilities is −0.06%. After the crash, the implied volatility function tilts into a steep
smirk: in the model, the spread between OTM and ATM implied volatilities is 8.39%, while the spread
between ITM and ATM implied volatilities is −0.38%. These values closely match the numbers we find in
the data.
Another important property of S&P 500 option prices, which is evident from Fig. 1, is that the regime
shift in S&P 500 options has persisted since the 1987 crash. Fig. 3 demonstrates that our model is also able
to capture this empirical observation. Indeed, it shows the volatility smirk retains a value of approximately
8% for values of p in the range of p ∈ (0, 0.95). Thus, our model predicts that as the value of p updates via
Eq. (8), the post-crash smirk will be persistent. As a particular example, if we assume that no additional
crashes are observed for 20 years after the 1987 crash, Eq. (8) implies that the probability of being in the
low jump state will drift up to approximately p ≈ 0.94. This value of p still generates a volatility smirk of
almost 8%. Moreover, there may be additional jumps in fundamentals after the 1987 crash. If that were to
happen, the agent would revise her posterior probability down, erasing any increase in p due to the effect
13
of the drift. Also in this case, Fig. 3 shows that lowering p below pP ost would have a minimal effect on the
shape of the volatility smirk.
3.4.1.
Sensitivity analysis
We illustrate the sensitivity of the pre- and post-crash volatility functions to some key underlying parameters:
Preferences coefficients. Fig. 4 shows that when the coefficient of risk aversion is lowered to 7.5, most of the
post-crash volatility smirk remains intact. Increasing the value of γ to 12.5 steepens the post-crash smirk
considerably. Most importantly, even when γ = 12.5, a value that exceeds the range that most economists
find to be ‘reasonable’ (e.g., Mehra and Prescott, 1985), the pre-crash smirk remains relatively flat.
As noted previously, researchers have obtained a wide array of estimates for the EIS parameter Ψ. In
our baseline case, we use Ψ = 2. Fig. 5 shows that even lower estimates for Ψ, such as 1.5, still produce
steep post-crash volatility smirks.
Jump coefficients. Fig. 6 illustrates the effect of a one-standard-deviation perturbation of the average jump
size coefficient. Not surprisingly, the steepness of the post-crash smirk is quite sensitive to the level of this
coefficient. Lowering ν increases the steepness of the smirk, especially in the post-crash economy.
In contrast, a change in the expected size of the jump in volatility, 1/ξ, has a limited effect on the
steepness of the smirk. This is evident from Fig. 7, which shows results for ξ = ξhigh , a value that corresponds
to an average jump size 1/ξhigh ≈ 0, and ξ = ξlow , a value that corresponds to an average jump size that
is double the baseline case. This result is due to the low persistence of the Ω processes in our calibration:
Unlike shocks to x, shocks to Ω are short-lived. Moreover, jumps are rare events in our calibration. Thus,
in sum, changing the expected volatility jump size has little impact on the volatility smirk.
3.4.2.
Options on individual stocks
We now turn to the pricing of individual stock options. We simulate option prices for a typical stock, as
discussed in Section 2.4, and extract Black-Scholes implied volatilities for different option strike prices. Fig.
8 compares this implied volatility function to the volatility smirk for S&P 500 options. Consistent with the
empirical evidence, our model predicts that the volatility smile for individual stock options is considerably
flatter than that for S&P 500 options.
Bakshi, Kapadia, and Madan (2003) conclude that the differential pricing of individual stock options
is driven by the degree of skewness/kurtosis in the underlying return distribution in combination with the
agent’s high level of risk aversion. Here, we propose a plausible endowment economy that, in combination
with recursive utility, yields predictions consistent with their empirical findings. Combined with our results
discussed above, this evidence is not inconsistent with the notion that the markets for S&P 500 and individual
stock options, as well as the market for the underlying stocks, are well integrated.
3.5.
The change in stock and bond prices around the 1987 crash
In our model, the 1987 crash is caused by a downward jump νe in expected consumption growth x
f in consumption volatility Ω. Here, we quantify the magnitude of
and a simultaneous upward jump M
14
these jumps implied by our model. Prior to the crash, the stock market price-dividend ratio is LP re =
L(xP re , ΩP re , pP re ), where pP re is the agent’s prior on the probability that λt = λG . In our calibration,
pP re ≈ 1. Immediately after the crash, the price-dividend ratio drops to LP ost = L(xP re + νe, ΩP re +
f, pP ost ), where pP ost is the agent’s posterior probability that λt = λG , i.e., according to Eq. (8), pP ost =
M
³ G
´
−λ(pP re )
pP re + pP re λ λ(p
. Similarly, the change in the risk-free rate at the time of the crash is
P re )
f, pP ost ) − r(xP re , ΩP re , pP re ) .
∆r = rP ost − rP re = r(xP re + νe, ΩP re + M
(24)
Using S&P 500 data, we find LP re = 32.64 and LP ost = 25.96, while the change in the three-month
Treasury bill yield measured over the two weeks before and after the crash is −1.39%. The model matches
these numbers when pP re = 99.85%, pP ost = 90.48%, xP re = 0.0317, νe = −0.0034, ΩP re = 0.000533, and
f = 0.000169.
M
Now, νe = −0.0034 is a very small jump in x. The expected growth rate of a persistent process is difficult
to measure. Thus, it will be difficult for the econometrician to detect this jump in ex post dividend data.
f = 0.000169, is bigger relative to the long-run mean of Ω, but in the
The jump in economic uncertainty, M
calibration the persistence of the economic uncertainty process Ω is much lower than the persistence of the
growth process x. After a jump, the process Ω is quickly pulled back towards its steady-state value by its
drift (the half-life of a shock to Ω is around seven to eight months). Moreover, prior to the crash the value
of economic uncertainty, ΩP re = 0.000533, is not far from the steady-state value of Ω, 0.0006 (the standard
deviation of a diffusive shock is 0.000068). Thus, this jump will be hardly detectable in low-frequency
dividend and consumption data as well.
These computations show that the model explains the change in stock and bond prices around the 1987
market crash with minimal change in fundamentals. In fact, the crash is mainly driven by the updating
f and focus only on the effect of the
of the agent’s beliefs. If we omit the effect of the jumps νe and M
change in the agent’s prior from pP re to pP ost , we find L(xP re , ΩP re , pP ost ) = 26.70. That is, Bayesian
updating over p alone drives an 18.20% drop in prices (26.70/32.64 − 1 = −0.1820). Similarly, we find
r(xP re , ΩP re , pP ost ) − r(xP re , ΩP re , pP re ) = −0.95%, which accounts for most of the drop in the risk-free
rate.
3.6.
Final thoughts
We conclude this section with two observations. First, Fig. 1 conveys two main points. One, as mentioned
previously, there has been a permanent shift in the shape of the implied volatility function due to the crash.
Two, there are daily fluctuations in the shape of the smirk. This second feature has been studied extensively
in the literature. Prior contributions have shown that these fluctuations can be understood in both a general
equilibrium framework (e.g., David and Veronesi (2002 and 2009)) and a partial equilibrium setting (e.g.,
Bakshi et al., 1997, 2000, Bates, 2000, Pan, 2002, and Eraker, 2004). Such high-frequency fluctuations
can be captured within the context of our model by introducing additional state variables that drive highfrequency changes in expected dividend growth and/or volatility. However, since these daily fluctuations
have already been explained, we do not investigate such variables in order to maintain parsimony.
15
Second, another aspect of S&P 500 options is that expected return volatility computed under the riskadjusted probability measure is typically higher than expected return volatility computed under the actual
probability measure. The difference between these two expected volatility measures is often termed the
‘variance risk premium,’ or VRP. Moreover, previous studies have shown that the VRP fluctuates over time
and predicts future stock market returns at the short/medium horizon (Bollerslev, Tauchen, and Zhou, 2009,
and Drechsler and Yaron, 2008). Our model does not capture this evidence, but it can be extended to include
higher-frequency jumps (similar to Drechsler and Yaron, 2008) or time-varying volatility-of-volatility (as in
Bollerslev, Tauchen, and Zhou, 2009). Since this is not our contribution, we point the interested reader to
those studies for more details.
4.
Conclusions
The 1987 stock market crash is associated with many asset pricing puzzles. Examples include: i)
Stocks fell 20-25%, interest rates fell approximately 1-2%, yet there was minimal impact on observable
economic variables (e.g., consumption), ii) the slope of the implied volatility curve on index options changed
dramatically after the crash, and this change has persisted for more than 20 years, iii) the magnitude of
this post-crash slope is difficult to explain, especially in relation to the implied volatility slope on individual
firms. We propose a general equilibrium model that can explain these puzzles while capturing many other
salient features of the U.S. economy. We accomplish this by extending the model of Bansal and Yaron
(2004) to account for jumps and learning. In particular, we specify the representative agent to be endowed
with KPEZ preferences and assume that the aggregate dividend and consumption processes are driven by a
persistent stochastic growth variable that can jump. Economic uncertainty fluctuates and is also subject to
jumps. Jumps are rare and driven by a hidden state the agent filters from past data. In such an economy,
there are three sources of long-run risk: expected consumption growth, volatility of consumption growth, and
posterior probability of the jump intensity in expected growth rates and volatility. Jumps in fundamentals,
even small, can lead to substantial jumps in prices of long-lived assets because of the updating of beliefs
about the likelihood of future such jumps. In that sense, learning acts as an amplifier of long-run risk
premiums associated with small persistent jumps in growth rates and their volatility. Indeed, we identify
a realistic calibration of the model that matches the prices of short-maturity at-the-money and deep outof-the-money S&P 500 put options, as well as the prices of individual stock options. Further, the model,
calibrated to the stock market crash of 1987, generates the steep shift in the implied volatility ‘smirk’ for
S&P 500 options observed around the 1987 crash. This ‘regime shift’ occurs in spite of a minimal change in
observable macroeconomic fundamentals.
In sum, our model points to a simple mechanism, based on learning about the riskiness of the economy,
that explains why market prices suddenly crashed with little change in fundamentals, and why buyers of
OTM put options were willing to pay a much higher price for these securities after the crash. Of course,
we acknowledge that other mechanisms probably also contributed to the crash. For example, portfolio
insurance and its implementation via dynamic hedging strategies is often cited as a major culprit. Let us
just point out that, while not directly a ‘shock to fundamentals,’ the failure of portfolio insurance could
16
well have contributed to deteriorating prospects for economic fundamentals through a ‘financial accelerator’
mechanism. It is a common belief that the growth rate of consumption and consumption volatility are tied
to the strength of the financial system. Thus, if the crash revealed that risk-sharing was not as effective
as previously thought, then this could have negatively affected investors’ expectations about the future
prospects of the economy. In this respect, further learning about economic fundamentals occurs through the
experience of a crash in prices and might result in a further drop in prices via the mechanism we describe.
Explicitly modeling this feedback mechanism between prices and economic fundamentals is outside the scope
of the present paper, but seems an interesting avenue for future research.
17
Appendix A.
Equilibrium prices in a jump-diffusion exchange economy with recursive utility
There are several formal treatments of stochastic differential utility and its implications for asset pricing
(see, e.g., Duffie and Epstein, 1992a,b, Duffie and Skiadas, 1994, Schroder and Skiadas, 1999, 2003, and
Skiadas, 2003). For completeness, in this Appendix we offer a simple formal derivation of the pricing
kernel that obtains in an exchange economy where the representative agent has a KPEZ recursive utility.
Our contribution is to characterize equilibrium prices in an exchange economy where aggregate output has
particular jump-diffusion dynamics (Propositions 1 and 2).
A.1.
Representation of preferences and pricing kernel
We assume the existence of a standard filtered probability space (Ω, F, {Ft }, P ) on which there exists a
P
vector z(t) of d independent Brownian motions and one counting process N (t) = i 1{τ ≤t} for a sequence
i
of inaccessible stopping times τi , i = 1, 2, . . ..11
Aggregate consumption in the economy is assumed to follow a continuous process, with stochastic growth
rate and volatility, which both may experience jumps:
d log Ct = µC (Xt ) dt + σC (Xt ) dz(t)
e dN (t) ,
dXt = µX (Xt ) dt + σX (Xt ) dz(t) + Γ
(A.1)
(A.2)
where Xt is an n-dimensional Markov process [we assume sufficient regularity on the coefficient of the
stochastic differential equation (SDE) for it to be well-defined, e.g., Duffie, 2001, Appendix B]. In particular,
e is an (n, 1) vector of i.i.d. random variables with joint
µ is an (n, 1) vector, σ is an (n, d) matrix, and Γ
X
X
density (conditional on a jump dN (t) = 1) of (ν). We further assume that the counting process has a
³
´
Rt
(positive integrable) intensity λ(Xt ) in the sense that N (t) − 0 λ(Xs ) ds is a (P, Ft ) martingale.
Following Epstein and Zin (1989), we assume that the representative agent’s preferences over a consumption process {Ct } are represented by a utility index U (t) that satisfies the following recursive equation:
½
¾ 1
¡
¢ 1−ρ 1−ρ
1−ρ
1−γ 1−γ
−βdt
−βdt
U (t) = (1 − e
)Ct + e
Et U (t + dt)
.
(A.3)
With dt = 1, this is the discrete time formulation of KPEZ, in which Ψ ≡ 1/ρ is the EIS and γ is the
risk-aversion coefficient.
To simplify the derivation, let us define the function
( 1−α
x
(1−α)
uα (x) =
log(x)
Further, let us define
g(x) =
uρ (u−1
γ (x))
≡
0 < α 6= 1
α = 1.
((1−γ)x)1/θ
(1−ρ)
uρ (ex )
log((1−γ)x)
(1−γ)
γ, ρ 6= 1
γ = 1, ρ 6= 1
ρ = 1, γ 6= 1 ,
11
N (t) is a pure jump process and hence is independent of z(t) by construction (in the sense that their quadratic co-variation
is zero).
18
where
θ=
1−γ
.
1−ρ
Then, defining the ‘normalized’ utility index J as the increasing transformation of the initial utility index
J(t) = uγ (U (t)), Eq. (A.3) becomes:
g(J(t)) = (1 − e−βdt )uρ (Ct ) + e−βdt g (Et [J(t + dt)]) .
(A.4)
Using the identity J(t + dt) = J(t) + dJ(t) and performing a simple Taylor expansion, we obtain:
0 = βuρ (Ct )dt − βg(J(t)) + g 0 (J(t)) Et [dJ(t)] .
(A.5)
Slightly rearranging the above equation, we obtain a backward recursive stochastic differential equation
that could be the basis for a formal definition of stochastic differential utility (see Duffie and Epstein,
1992a,b; Skiadas, 2003):
Et [dJ(t)] = −
βuρ (Ct ) − βg(J(t))
dt .
g 0 (J(t))
Indeed, let us define the so-called ‘normalized’ aggregator function:
βuρ (C)
− βθJ
((1−γ)J)1/θ−1
βuρ (C) − βg(J)
f (C, J) =
≡
(1 − γ)βJ log(C) − βJ log((1 − γ)J)
g 0 (J)
βuρ (C)
β
− 1−ρ
e(1−ρ)J
(A.6)
γ, ρ 6= 1
γ 6= 1, ρ = 1
γ = 1, ρ 6= 1 .
We obtain the following representation for the normalized utility index:
µZ T
¶
J(t) = Et
f (Cs , J(s)) + J(T ) .
(A.7)
(A.8)
t
Further, if the transversality condition limT →∞ Et (J(T )) = 0 holds, letting T tend to infinity, we obtain
the simple representation:
µZ
J(t) = Et
∞
¶
f (Cs , J(s))ds .
(A.9)
t
Fisher and Gilles (1999) discuss many alternative representations and choices of the utility index and
associated aggregator as well as their interpretations. Here, we note only the well-known fact that when
ρ = γ (i.e., θ = 1), then f (C, J) = βuρ (C) − βJ, and a simple application of Itô’s lemma shows that
µZ ∞
¶
J(t) = Et
e−β(s−t) βuρ (Cs )ds .
t
To obtain an expression for the pricing kernel, note that under the assumption (which we maintain
throughout) that an ‘interior’ solution to the optimal consumption-portfolio choice of the agent exists, a
necessary condition for optimality is that the gradient of the utility index is zero for any small deviation
of the optimal consumption process in a direction that is budget feasible. More precisely, let us define the
utility index corresponding to such a small deviation by:
µZ ∞ ³
´ ¶
δ
∗
δ
J (t) = Et
f Cs + δ C̃(s), J (s) ds .
t
19
Then we may define the gradient of the utility index evaluated at the optimal consumption process C ∗ (t)
in the direction C̃(t):
J δ (t) − J(t)
δ→0
"Z δ
#
∞
f (Cs∗ + δ C̃(s), J δ (s)) − f (Cs , J δ (s))
= lim Et
ds
δ→0
δ
t
·Z ∞
¸
∗
∗
= Et
fC (Cs , J(s))C̃s + fJ (Cs , J(s))∇J(Cs ; C̃s )ds .
∇J(Ct∗ ; C̃t ) = lim
(A.10)
t
Assuming sufficient regularity (essentially the gradient has to be a semi-martingale and the transversality
condition has to hold: limT →∞ Et [e
RT
t
fJ (Cs ,Js )ds
∇J(CT∗ ; C̃T ) = 0), a simple application of the generalized
Itô-Doeblin formula gives the following representation:
µZ ∞ R
¶
s
fJ (Cu ,Ju )du
δ
∗
∇ J(Ct ; C̃t ) = Et
et
fC (Cs , Js )C̃s ds .
(A.11)
t
This shows that
Π(t) = e
Rt
0
fJ (Cs ,Js )ds
fC (Ct , Jt )
(A.12)
is the Riesz representation of the gradient of the normalized utility index at the optimal consumption. Since
a necessary condition for optimality is that ∇J(Ct∗ ; C̃t ) = 0 for any feasible deviation C̃t from the optimal
consumption stream Ct∗ , we conclude that Π(t) is a pricing kernel for this economy; see, e.g., Chapter 10 of
Duffie (2001) for further discussion.
A.2.
Equilibrium prices
Assuming the equilibrium consumption process given in Eqs. (A.1)–(A.2) above, we obtain an explicit
characterization of the felicity index J and the corresponding pricing kernel Π.
For this we define, respectively, the continuous diffusion, jump, and jump-compensator operators for any
h(·) : Rn − R:
1
Dh(x) = hx (x)µX (x) + trace(hxx σX (x)σX (x)> )
2
h(x + ν̃)
J h(x) =
−1
h(x)
Z
Z
h(x + ν)
(ν)dν1 . . . dνn − 1,
J h(x) = E[J h(x)] = . . .
h(x)
(A.13)
where hx is the (n, 1) Jacobian vector of first derivatives and hxx denotes the (n, n) Hessian matrix of second
derivatives. With these notations, we find:
Proposition 1. Suppose I(x) : Rn → R solves the following equation:
³
´
2
2 ||σC (x)|| − βθ +
0
=
I(x)
(1
−
γ)µ
(x)
+
(1
−
γ)
C
2
DI(x)θ
>
+ (1 − γ)θσC (x)σX (x) Ix (x) + θ + I(x)λ(x)J I(x)θ
for ρ, γ 6= 1
I(x)(θ−1)
¡
¢
0 = I(x) ((1
+ 1 + I(x)λ(x) log 1 + J I(x) for γ = 1, ρ 6= 1
³ − ρ)µC (x) − β) + I(x)D log I(x)
2´
||σ
(x)||
2
0 = I(x) (1 − γ)µC (x) + (1 − γ) C 2
+ DI(x)+
for ρ = 1, γ 6= 1
(1 − γ)σC (x)σX (x)> Ix (x) − βI(x) log I(x) + I(x)λ(x)J I(x)
(A.14)
20
and satisfies the transversality condition (limT →∞ E[J(T )] = 0 for J(t) defined below), then the value function is given by:
θ
J(t) = uγ (Ct )(βI(Xt ))
t ))
J(t) = log(Ct ) + log(βI(X
1−ρ
J(t) = u (C )I(X )
γ
t
t
for ρ, γ 6= 1
for γ = 1, ρ 6= 1
for ρ = 1, γ 6= 1 .
(A.15)
The corresponding pricing kernel is:
R
(1−θ)
− 0t (βθ+ I(X ) )ds
s
Π(t)
=
e
(Ct )−γ (I(Xt ))(θ−1) for ρ, γ 6= 1
Rt β
− 0 I(X ) )ds
1
s
for γ = 1, ρ 6= 1
(Ct I(Xt ))
Π(t) = e R t
for ρ = 1, γ 6= 1 .
Π(t) = e− 0 β(1+log I(Xs ))ds (Ct )−γ I(Xt )
(A.16)
Proof. We provide the proof for the case γ, ρ 6= 1. The special cases are treated similarly.
From its definition
µZ
J(t) = Et
Thus, J(Xt , Ct ) +
Rt
0
t
∞
¶
f (Cs , J(s)) .
(A.17)
f (Cs , J(Xs , Cs ))ds is a martingale. This observation implies that:
Equivalently:
E[dJ(Xt , Ct ) + f (Ct , J(Xt , Ct ))dt] = 0 .
(A.18)
DJ(Ct , Xt )
f (Ct , J(Ct , Xt ))
+ J J(Ct , Xt ) +
= 0.
J(Ct , Xt )
J(Ct , Xt )
(A.19)
To obtain the equation of the proposition, we use our guess (J(t) = uγ (Ct )β θ I(Xt )θ ) and apply the ItôDoeblin formula using the fact that
f (C, J)
J
DJ
J
uρ (C)
θ
− βθ
− βθ =
1/θ−1
I(X)
((1 − γ)J)
J
1
DI(X)θ
= (1 − γ)µC (X) + (1 − γ)2 ||σC (X)||2 +
+ (1 − γ)θσC (X)σI (X)> ,
2
I(X)θ
=
where we have defined σI (x)> =
(A.20)
(A.21)
σX (x)> Ix (x)
.
I(x)
Now suppose that I(·) solves this equation. Then, applying the Itô-Doeblin formula to our candidate
J(t), we obtain
Z
J(T ) = J(t) +
Z
= J(t)−
t
Z
T
Z
T
T
J(s− )J I(Xs )θ dN (s)
t
t
t
Z T
Z T
T
f (Cs , Js )ds+
J(s) {(1 − γ)σC (Xs ) + θσI (Xs )} dz(s) +
dM (s),
DJ(s)ds +
(JC σC + JX σX )dz(s) +
t
t
where we have defined the pure jump martingale
Z t
Z t
−
θ
M (t) =
J(s )J I(Xs ) dN (s) −
λ(Xs− )J(s− )J I(Xs )θ ds .
0
0
21
If the stochastic integral is a martingale,12 and if the transversality condition is satisfied, then we obtain
the desired result by taking expectations and letting T tend to infinity:
¸
·Z ∞
J(t) = E
f (Cs , Js ) ds ,
(A.22)
t
which shows that our candidate J(t) solves the recursive stochastic differential equation. Uniqueness follows
(under some additional technical conditions) from the appendix in Duffie and Epstein (1992a).
The next result investigates the property of equilibrium prices.
Proposition 2. The risk-free interest rate is given by:
||σ (x)||2
||σ (x)||2
) − γ(1 + ρ) C 2 ¡ −
r(x) = β + ρ(µC (x) + C 2
¢
θ − J I (θ−1)
J
I
for ρ 6= 1
(1 − θ)σI (x)> (σC (x) + 21 σI (xt )) + λ(x) θ−1
θ
||σC (x)||2
2
r(x) = β + µC (x) +
− γ||σC (x)||
for ρ = 1 .
2
Further, the value of the claim to aggregate consumption is given by:
(
V (t) = C(t)I(Xt ) for ρ 6= 1
V (t) = C(t)
for ρ = 1 .
β
Thus,
³
´
dVt
= µV (Xt )dt + σC (Xt ) + σI (Xt )1{ρ6=1} dz(t) + J I(Xt )dN (t) .
Vt
(A.23)
(A.24)
(A.25)
The risk premium on the claim to aggregate consumption is given by
µV (X) +
1
− r(X) = (γσC (X) + (1 − θ)σI (X))> (σC (X) + σI (X))
I(X)
³
´
+λ(X) J I(X)θ − J I(X)θ−1 − J I(X) .
(A.26)
Proof. To prove the result for the interest rate, apply the Itô-Doeblin formula to the pricing kernel. It
follows from r(t) = −E[ dΠ(t)
Π(t) ]/dt that:
r(Xt ) = βθ +
(1 − θ)
1
DI(Xt )(θ−1)
+ γµc (Xt ) − γ 2 ||σc (Xt )||2 −
− λ(Xt )J I(X)θ−1 .
I(Xt )
2
I(Xt )(θ−1)
Now substitute the expression for
1
I(X)
(A.27)
from the equation in (A.14) to obtain the result.
To prove the result for the consumption claim, define V (t) = ct I(Xt ). Then using the definition of
R
−βθt− 0t
Π(t) = e
we obtain:
−βθt−
d (Π(t) V (t)) = e
12
Rt
(θ−1)
0 I(Xs ) ds
(1−θ)
ds
I(Xs )
θ−1
c−γ
,
t I(Xt )
¶ ¶
µ
µ
(1 − θ)
dt .
dJ(t) − J(t) βθ +
I(Xt )
Sufficient conditions are:
·Z
T
E
0
¸
¡
¢
J(s)2 ||(1 − γ)σC (Xs ) + θσI (Xs )||2 ds < ∞ ∀T > 0 .
22
(A.28)
Note that by definition we have:
dJ(t) = −f (ct , J)dt + dMt
µ
¶
θ
= −J(t)
− θβ dt + dMt
I(Xt )
(A.29)
for some P -martingale M . Combining this observation with (A.28), we get:
−βθt−
d (Π(t) V (t)) = e
Rt
(1−θ)
0 I(Xs ) ds
R (1−θ)
(−J(t))
−βθt− 0t I(X ) ds
s
dt + e
dMt
I(Xt )
= −Π(t) c(t) dt + e
−βθt−
Rt
(1−θ)
0 I(Xs ) ds
dMt .
(A.30)
Thus integrating we obtain
Z
Π(T )V (T ) +
t
Z
T
Π(s)cs ds = Π(t)V (t) +
T
e
R
−βθ(u−t)− tu
(1−θ)
ds
I(Xs )
t
dMu .
(A.31)
Taking expectations, letting T → ∞, and assuming the transversality condition holds
(i.e., limT →∞ E[Π(T )V (T )] = 0), we obtain the desired result:
·Z ∞
¸
Π(t)V (t) = Et
Π(s) cs ds .
(A.32)
t
To derive the excess return equation, note that the martingale condition implies:
Et [
dΠ(t)V (t)
D(t)
]+
dt = 0 .
Π(t)V (t)
V (t)
(A.33)
Further, Itô’s lemma implies:
1
dΠ(t)V (t)
Et [
] =
dt
Π(t)V (t)
·
¸
1
dΠ(t) dV (t) dΠ(t) dV (t)
Et
+
+
dt
Π(t)
V (t)
Π(t) V (t)
= µV (X) − r(X) + (γσC (X) + (1 − θ)σI (X))> (σC (X) + σI (X)) +
1
Et [J I(X)θ−1 J I(X)]
dt
= µV (X) − r(X) + (γσC (X) + (1 − θ)σI (X))> (σC (X) + σI (X))
³
´
+λ(X) J I(X)θ − J I(X)θ−1 − J I(X) .
(A.34)
Combining Eqs. (A.33) and (A.34), we get the expression for the excess return on the consumption claim
given in Eq. (A.26).
Appendix B.
Application to the three-dimensional model
Here we apply the general equations given in Appendix A to our three-state variable model, where the
state vector is Xt = (xt , Ωt , pt ), whose dynamics are given in Eqs. (3), (4), and (8).
23
B.1.
Price-consumption ratio
The equation for the price-consumption ratio follows immediately from the dynamics of (xt , Ωt , pt ) given
in Eqs. (3), (4), and (8) and the PDE (14):
h
i
γ
0 = I (1 − γ)µC + (1 − γ)x − (1 − γ)Ω − βθ − θIx κx x
2
#
µ
¶"
µ ¶2
¡
¢
θ
Ix
2
+
σx0 + (σxc + σxΩ )Ω (θ − 1)
I + Ixx + θIΩ κΩ Ω − Ω
2
I
Ã
!
µ ¶2
·
µ ¶µ ¶
¸
σΩ2
IΩ
Ix
IΩ
+ Ωθ IΩΩ + (θ − 1)
I + θ (θ − 1)
I + IxΩ σxc σΩ ρΩC Ω
2
I
I
I
£
¡
¢
¤
+θ − p λG − λ(p) + (1 − p)φBG − pφGB Ip + (1 − γ)θIx σxc Ω
h i
+(1 − γ)θIΩ σΩ ρΩC Ω + λ(p)I J I θ + θ .
B.2.
(B.1)
Pricing kernel and risk-free rate
When ρ, γ 6= 1, the pricing kernel in our three factor economy is
Rt
Π(t) = e
0
ds [(θ−1)I(s)−1 −βθ]
β θ e−γc I(t)(θ−1) .
(B.2)
Ito’s lemma gives
dΠ
Π
µ
µ ¶
µ ¶
¶
√
√
√
I
I
= −r dt + dzC −γ Ω + (θ − 1) x σxc Ω + (θ − 1) Ω σΩ ΩρΩC
I
I
¸
·
µ ¶
¸
·
µ ¶
√ q
IΩ
Ix p
2
σx0 + σxΩ Ω + dzΩ (θ − 1)
σΩ Ω 1 − ρΩC
+dzx (θ − 1)
I
I
"
#
"
#
f, p + ∆p)
f, p + ∆p)
I θ−1 (x + νe, Ω + M
I θ−1 (x + νe, Ω + M
+dN
− 1 − λ(p) dt E
−1 ,
I θ−1 (x, Ω, p)
I θ−1 (x, Ω, p)
(B.3)
where the risk-free rate r equals:
¶¸
µ ¶
·µ
£
¤
1
I
1
(θ − 1)I −1 − βθ − γ µC + x − Ω + γ 2 Ω + (θ − 1) x [−κx x]
2
2
I
"
µ ¶2 µ
¶# µ
¶
µ ¶
£ ¡
¢¤
1
I
I
Ixx
2
+ (θ − 1) (θ − 2) x
σx0 + (σxc
+ σxΩ )Ω + (θ − 1) Ω
κΩ Ω − Ω
+
2
I
I
I
"
"
#
µ ¶2 µ
¶#
θ−1 (x + ν
f, p + ∆p)
IΩ
I
e
,
Ω
+
M
1
IΩΩ
σΩ2 Ω + λ(p) E
−1
+ (θ − 1) (θ − 2)
+
2
I
I
I θ−1 (x, Ω, p)
µ ¶
¢
¡
¤
I £
I
+(θ − 1) p − p λG − λ(p) + (1 − p)φBG − pφGB − γ(θ − 1) x σxc Ω
I
I
µ ¶
·
µ ¶µ ¶ µ
¶¸
I
I
IΩ
IxΩ
−γ(θ − 1) Ω σΩ ρΩC Ω + (θ − 1) (θ − 2) x
+
σxc σΩ ρΩC Ω.
(B.4)
I
I
I
I
−r(x, Ω, p) =
B.3.
Price-dividend ratio and equity premium
The price of the stock market portfolio, S(t), satisfies the well-known formula
0 =
=
1
E [d (Π(t) S(t))] + Π(t)D(t)
dt t
1
E [S(t) dΠ(t) + Π(t) dS(t) + dΠ(t) dS(t)] + Π(t)D(t).
dt t
24
(B.5)
Dividing by Π(t) and using E
£ dΠ ¤
Π
= −r dt yields
0 = −rS(t) +
·
¸
1
1
dΠ
Et [dS(t)] + Et
dS(t) + D(t).
dt
dt
Π
(B.6)
We define the price-dividend ratio L(x, Ω, p) via S(x, Ω, p, D) = DL(x, Ω, p) and substitute in Eq. (B.6).
Then, dividing by D we find
·
¸
·
µ
¶¸
dL dD dD dL
dΠ dL dD dD dL
L
L
+
+
+ Et
+
+
+ 1.
0 = −rL + Et
dt
L
D
D L
dt
Π
L
D
D L
(B.7)
To solve this equation, we need the dynamics of L(x, Ω, p), which we obtain from Ito’s lemma and the
dynamics of the state vector in Eqs. (3), (4) and (8). Substituting in Eq. (B.7), we find
£
¤
¡
¢ 1
1
2
)Ω + LΩ κΩ Ω − Ω + LΩΩ σΩ2 Ω
0 = 1 − rL + (µD + φx) L − κx xLx + Lxx σx0 + (σxΩ + σxc
2¢
2
£
¡
¤
+LxΩ σxc σΩ ρΩC Ω + Lp − p λG − λ(p) + (1 − p)φBG − pφGB + σD ρDC [Lx σxc + LΩ σΩ ρΩC ] Ω
·
¸·
µ ¶
µ ¶
¸
I
I
+Ω σD ρDC L + Lx σxc + LΩ σΩ ρΩC −γ + (θ − 1) x σxc + (θ − 1) Ω σΩ ρΩC
I
I
µ ¶
µ ¶
¡
¢
IΩ
Ix
(σx0 + σxΩ Ω) + (θ − 1)LΩ
σΩ2 Ω 1 − ρ2ΩC
+(θ − 1)Lx
I
I
"Ã
!µ
¶#
I (θ−1) (x + ν, Ω + M, p + ∆p)
+λ(p) E
L(x + ν, Ω + M, p + ∆p) − L(x, Ω, p) .
(B.8)
I (θ−1) (x, Ω, p)
1
Next, we derive the equity risk premium, (µ − r), via the relation (µ − r) = − dt
E
£ dΠ dS ¤
Π S
, which yields
¶µ ¶µ
¶
µ ¶µ ¶
¡
¢
Lx
I
LΩ
Ix
σx0 + σxΩ Ω − (θ − 1) Ω
σΩ2 Ω 1 − ρ2ΩC
(µ − r) = −(θ − 1)
I
L
I
L
·
µ ¶
µ ¶
¸·
µ ¶
µ ¶
¸
Ix
IΩ
Lx
LΩ
−Ω −γ + (θ − 1)
σxc + (θ − 1)
σΩ ρΩC σD ρDC +
σxc +
σΩ ρΩC
I
I
L
L
"Ã
!µ
¶#
I (θ−1) (x + ν, Ω + M, p + ∆p)
L(x + ν, Ω + M, p + ∆p)
−λ(p)E
−1
−1
.
(B.9)
L(x, Ω, p)
I (θ−1) (x, Ω, p)
µ
B.4.
Risk-neutral dynamics
From the pricing kernel, we identify the following risk-neutral dynamics:
·
µ
µ ¶
µ ¶
¶¸
√
1
I
I
dc = µC + x − Ω + Ω −γ + (θ − 1) x σxc + (θ − 1) Ω σΩ ρΩC
dt + Ω dzCQ
2
I
I
·
µ
µ ¶
µ ¶
¶¸
Ix
IΩ
dD
= µD + φ x + ρDC σD Ω −γ + (θ − 1)
σxc + (θ − 1)
σΩ ρΩC
dt
D
I
I
q
´
√ ³
Q
+σD Ω ρDC dzCQ + 1 − ρ2DC dzD
µ ¶
¶¸
·
µ
µ ¶
IΩ
Ix
σxc + (θ − 1)
σΩ ρΩC
dt
dx = −κx x + σxc Ω −γ + (θ − 1)
I
I
·
µ ¶¸
√
p
I
+ (σx0 + σxΩ Ω) (θ − 1) x
dt + σxc Ω dzcQ + σx0 + σxΩ Ω dzxQ + νe dN
I
·
µ
µ ¶
µ ¶
¶¸
¢
¡
Ix
IΩ
dΩ = κΩ Ω − Ω + σΩ ρΩC Ω −γ + (θ − 1)
σxc + (θ − 1)
σΩ ρΩC
dt
I
I
25
·
µ
¶¸
q
´
√ ³
f dN
+ σΩ (1 − ρΩC )Ω(θ − 1)
dt + σΩ Ω ρΩC dzCQ + 1 − ρ2ΩC dzΩQ + M
"
#
f, p + ∆p)
I (θ−1) (x + νe, Ω + M
Q
λ (p) = λ(p) E
,
(B.10)
I (θ−1) (x, Ω, p)
2
2
IΩ
I
where
f = M ) = π(e
f = M) ∗
π Q (e
ν = ν, M
ν = ν, M
Appendix C.
I (θ−1) (x + ν, Ω + M, p + ∆p)
h
i.
f, p + ∆p)
E I (θ−1) (x + νe, Ω + M
(B.11)
Affine approximation to the model
To solve the model, we approximate the price-consumption ratio with an exponential affine function,
I(x, Ω, p) = eA(p)+B(p)x+F (p)Ω .
(C.1)
Plugging the approximation of I into Eq. (B.1) and dividing by I, we find
¡ 2
¢ ¤
γ
θ2 £
0 = (1 − γ)µc + (1 − γ)x − (1 − γ)Ω − βθ − θBκx x +
σx0 + σxc
+ σxΩ Ω B 2
2
à 2!
¡
¢
σΩ2
+θF κΩ Ω − Ω +
θ2 F 2 Ω + θ2 BF σxc σΩ ρΩC Ω
2
¤
£ ¡
¢
¤£
+θ −p λG − λ(p) − pφGB + (1 − p)φBG Ap + xBp + ΩFp
θ
+(1 − γ)θBσxc Ω + (1 − γ)θF σΩ ρΩC Ω + − λ(p)
I
θ[A(pλG /λ(p))−A(p)] θx(B(pλG /λ(p))−B(p)) θΩ(F (pλG /λ(p))−F (p))
e
+λ(p) e
e
µ
¶
2 θ2
σ
α
ξ
θB(pλG /λ(p))(ν+ ξ )+ ν2 B(pλG /λ(p))2
×e
.
ξ + αθB(pλG /λ(p)) − θF (pλG /λ(p))
(C.2)
To solve Eq. (C.2), we apply a continuous-time analog of the Campbell and Shiller (1988) log-linear approximation; see, e.g., Campbell and Viceira (2002) and Chacko and Viceira (2005). We use Taylor’s formula to
³
´
³ ´
expand the exponential terms in x and Ω around the points x0 ≡ κν λ(p) and Ω0 ≡ Ω + ξκ1 λ(p). We
x
Ω
then collect terms linear in x, linear in Ω, and independent of x and Ω to obtain a system of three equations
that define the functions A(p), B(p), and F (p):
θ2
0 = (1 − γ)µc − βθ + σx0 B 2 + θF κΩ Ω − λ(p) + θe−(A+x0 B+Ω0 F ) (1 + x0 B + Ω0 F )
£
¡ 2
¢
¡
¢¤
+ζ1 (p) 1 − θx0 B(pλG /λ(p)) − B(p) − θΩ0 F (pλG /λ(p)) − F (p)
£
¤
+θ −p(λG − λ(p)) − pφGB + (1 − p)φBG Ap
¤
£
0 = (1 − γ) − θBκx − θBe−(A+Bx0 +F Ω0 ) + ζ1 (p)θ B(pλG /λ(p)) − B(p)
£
¤
+θ −p(λG − λ(p)) − pφGB + (1 − p)φBG Bp
à !
³γ ´
¢ 2
σΩ2
θ2 ¡ 2
(1 − γ) +
θ2 F 2 + θ2 BF σxc σΩ ρΩC
σxc + σxΩ B − θF κΩ +
0 = −
2
2
2
£
¤
+(1 − γ)θBσxc + (1 − γ)θF σΩ ρΩC − θF e−(A+Bx0 +F Ω0 ) + ζ1 (p)θ F (pλG /λ(p)) − F (p)
£
¤
+θ −p(λG − λ(p)) − pφGB + (1 − p)φBG Fp ,
(C.3)
26
where we have defined the function:
ζ1 (p) ≡ λ(p) eθ[A(pλ
G /λ(p))−A(p)
×eθΩ0 [F (pλ
G /λ(p))−F (p)
]
] eθ(ν+ αξ ) B(pλG /λ(p))+
µ
2 θ2
σν
B 2 (pλG /λ(p))
2
eθx0 [B(pλ
¶
ξ
.
ξ + αθB(pλG /λ(p)) − θF (pλG /λ(p))
]
G /λ(p))−B(p)
(C.4)
We approximate the functions A(p), B(p), and F (p) with a linear combination of general Chebyshev polynomials, and determine the coefficients of the approximation via least-squares minimization of the approximation error (e.g., Judd, 1998). We extend the approximation to include Chebyshev polynomials up to order 20
(adding higher-order polynomials does not change the solution). This approach gives us a semi-closed-form
solution to the model, which facilitates the analysis greatly. To check the accuracy of this approach, we also
solve the model via fixed-point iterations over the price-consumption ratio I. Albeit considerably slower,
this alternative method converges to a nearly identical solution.
We continue our approximation by looking for a price-dividend ratio of the form
L(x, Ω, p ) = eA
L (p)+B L (p)x+F L (p)Ω
.
(C.5)
We plug this expression into the price-dividend ratio Eq. (B.8), divide by L, and Taylor expand the exponential terms to be linear in x and Ω around the points x0 and Ω0 . We then collect terms to obtain a system
of three equations that define the functions AL (p), B L (p), and F L (p):
¢
¤
£
¡
0 = AL
− p λG − λ(p) + (1 − p)φBG − pφGB
p
L (p)−B L (p)x
+e−A
0 −F
L (p)Ω
0
(1 + B L (p)x0 + F L (p)Ω0 )
1
+(θ − 1) e−A−Bx0 −F Ω0 (1 + Bx0 + F Ω0 ) − βθ − γµC + (θ − 1)2 B 2 σx0
2
+(θ − 1)F κΩ Ω − λ(p)
£
¡
¢
¤
+(θ − 1)Ap (p) − p λG − λ(p) + (1 − p)φBG − pφGB
(B L )2
σx0 + F L κΩ Ω + (θ − 1)B L Bσx0
2
¤
¤
£
£
+ζ2 (p) 1 − [(θ − 1) B(pλG /λ(p)) − B(p) + B L (pλG /λ(p)) − B L (p) x0
£
£
¤
¤
− (θ − 1) F (pλG /λ(p)) − F (p) + F L (pλG /λ(p)) − F L (p) Ω0 ]
£
¡
¢
¤
0 = BpL − p λG − λ(p) + (1 − p)φBG − pφGB
+µD +
L (p)−B L (p)x
−e−A
0 −F
L (p)Ω
0
B L (p) − (θ − 1) e−A−Bx0 −F Ω0 B − γ − (θ − 1)Bκx
£
¡
¢
¤
+(θ − 1)Bp (p) − p λG − λ(p) + (1 − p)φBG − pφGB
+φ − κx B L
¤
¤
£
£
+ζ2 (p) (θ − 1) B(pλG /λ(p)) − B(p) + B L (pλG /λ(p)) − B L (p)
£
¡
¢
¤
0 = FpL − p λG − λ(p) + (1 − p)φBG − pφGB
L
L
L
−e−A (p)−B (p)x0 −F (p)Ω0 F L (p) − (θ − 1) e−A−Bx0 −F Ω0 F
γ(1 + γ) 1
1
2
+
+ (θ − 1)2 B 2 (σxc
+ σxΩ ) − (θ − 1)F κΩ + (θ − 1)2 F 2 σΩ2
2
2£
2
¡
¢
¤
+(θ − 1)Fp (p) − p λG − λ(p) + (1 − p)φBG − pφGB
−γ(θ − 1)Bσxc − γ(θ − 1)F σΩ ρΩC
27
(B L )2
(F L )2 2
2
(σxΩ + σxc
) − F L κΩ +
σΩ
2
£2 L
¤
+ σD ρDC B σxc + F L σΩ ρΩC
+(θ − 1)2 BF σxc σΩ ρΩC +
+B L F L σxc σΩ ρΩC
+[σD ρDC + B L σxc + F L σΩ ρΩC ] [−γ + (θ − 1)Bσxc + (θ − 1)F σΩ ρΩC ]
¡
¢
+(θ − 1)B L BσxΩ + (θ − 1)F L F σΩ2 1 − ρ2ΩC
£
£
¤
¤
+ζ2 (p) (θ − 1) F (pλG /λ(p)) − F (p) + F L (pλG /λ(p)) − F L (p) ,
(C.6)
where we have defined the function ζ2 :
ζ2 (p) = λ(p)e(θ−1)[A(pλ
∗ eA
[B(pλG /λ(p))−B(p)]x0 +[F (pλG /λ(p))−F (p)]Ω0 ]
G /λ(p))−A(p)+
L (pλG /λ(p))−AL (p)+
[B L (pλG /λ(p))−B L (p)]x0 +[F L (pλG /λ(p))−F L (p)]Ω0
∗χ(B L (pλG /λ(p)) + (θ − 1) B(pλG /λ(p)), F L (pλG /λ(p)) + (θ − 1) F (pλG /λ(p))) ,
h
i
σ2 B 2
f
B(ν+ α
)+ ν2
ξ
with χ(B, F ) ≡ E eBeν +F M = e
ξ
ξ+αB−F .
(C.7)
Similar to A(p), B(p), and F (p), we approximate
AL (p), B L (p), and F L (p) with a linear combination of general Chebyshev polynomials of order 20, and
determine the coefficients of this approximation via least-squares minimization of the approximation error.
Appendix D.
Pre- and post-crash implied volatility patterns
Fig. 1 shows the permanent regime shift in pre- and post-1987 market crash implied volatilities for
S&P 500 options. The plot in Panel A depicts the spread between implied volatilities for S&P 500 options
that have a strike-to-price ratio X = K/S − 1 = −10% and at-the-money implied volatilities. The plot
in Panel B depicts the spread between implied volatilities for options that have a strike-to-price ratio
X = K/S − 1 = 2.5% and at-the-money implied volatilities.
D.1.
American options on the S&P 500 futures
We construct implied volatility functions from 1985 to 1995 by using transaction data on American
options on S&P 500 futures. As in Bakshi et al. (1997), prior to analysis we eliminate observations that
have a price lower than $(3/8) to mitigate the impact of price discreteness on option valuation. Since nearmaturity options are typically illiquid, we also discard observations with time-to-maturity shorter than ten
calendar days. For the same reason, we do not use call and put contracts that are more than 3% in-themoney. Finally, we disregard observations on options that allow for arbitrage opportunities, e.g., calls with
a premium lower than the early exercise value.
We consider call and put transaction prices with the three closest available maturities. For each contract,
we select the transaction price nearest to the time of the market close and pair it with the nearest transaction
price on the underlying S&P 500 futures. This approach typically results in finding a futures price that is
time-stamped within six seconds from the time of the option trade. We approximate the risk-free rate with
the three-month Treasury yield and compute implied volatilities using the Barone-Adesi and Whaley (1987)
pricing formula for American options.
At each date and for each of the three closest maturities, we interpolate the cross-section of implied
volatilities with a parabola. This approach is similar to the one used in Shimko (1993). In doing so, we require
28
that we have at least three implied volatility observations, one with a strike-to-price ratio X = K/S − 1
no higher than -9%, one with X no lower than 1.5%, and one in between these two extremes. We record
the interpolated implied volatility at X = 0 and the implied volatility computed at the available X-values
closest to -10% and 2.5%.
Then, at each date and for each of the three X choices, we interpolate the implied volatility values across
the three closest maturities using a parabola. We use the fitted parabola to obtain the value of implied
volatility at 30 days to maturity. If only two maturities are available, we replace the parabola with a linear
interpolation. If only one maturity is available, we retain the value of implied volatility observed at that
maturity, provided that such maturity is within 20 to 40 days.
Trading in American options on the S&P 500 futures contracts began on January 28, 1983. Prior to
1987, only quarterly options maturing in March, June, September, and October were available. Additional
serial options written on the next quarterly futures contracts and maturing in the nearest two months were
introduced in 1987 (e.g., Bates, 2000). This data limitation, combined with the relatively scarce size and
liquidity of the options market in the early years, renders it difficult to obtain smirk observations at the
30-day maturity with -10% moneyness. Therefore, we start the plot in December 1985. After this date,
we find implied volatility values with the desired parameters for most trading days. Relaxing the time-tomaturity and moneyness requirements results in longer implied volatility series going back to January 1983.
Qualitatively, the plot during the period from January 1983 to December 1985 remains similar to that for
the period from December 1983 to October 1987 (see, e.g., Bates, 2000).
D.2.
European options on the S&P 500 index
After April 1996, we use data on S&P 500 index European options. We obtain daily SPX implied
volatilities from April 1996 to April 2006 from the OptionMetrics database. Similar to what we discussed in
Section D.1, we exclude options with a price lower than $(3/8), a time-to-maturity shorter than ten calendar
days, and contracts that are more than 3% in-the-money.
At each date and for each of the three closest maturities, we interpolate the cross-section of implied
volatilities using a parabola. We have also considered a spline interpolation, which has produced similar
results. We use the fitted parabola to compute the value of implied volatilities for strike-to-index-price ratios
X = K/S − 1 = −10%, zero, and 2.5%. Finally, we interpolate implied volatilities at each of these three
levels of moneyness across the three closest maturities. We use the fitted parabola to compute the value of
implied volatility at the 30-day maturity.
29
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34
Table 1
Model coefficients
The table reports the value of the coefficients for the baseline calibration of the model given in Eqs. (1)–
(8) and (11). The coefficients are expressed with yearly decimal scaling.
Preferences
γ = 10
Ψ=2
β = 0.0176
Consumption and dividends
µC = 0.018
µD = 0.025
φ = 2.6050
σD = 5.3229
σx0 = 0
σxΩ = 0.1301
Ω = 0.0006
σΩ = 0.004
ρΩC = −0.6502
σν = 0.0216
ξ =2100
α=3
σνi = 0.0632
λi = 5
ρDC = 0.2523
Predictable mean component, x
κx = 0.2785
σxc = 0.1217
Economic uncertainty, Ω
κΩ = 1.0484
Jumps
ν = −0.035
Transition probabilities
φGB = 0.0025
φBG = 0.025
Idiosyncratic return shocks
σi = 0.3137
µνi = 0.0036
35
λG = 0.0005
λB = 0.035
36
Mean
Std. dev.
Skewness
Kurtosis
AC(1)
AC(2)
AC(5)
AC(10)
0.0189 (0.0031)
0.0221 (0.0052)
-1.2447 (0.7983)
7.7297 (1.8802)
0.4218 (0.1153)
0.1268 (0.1569)
-0.0123 (0.1285)
0.0405 (0.0950)
Panel A: Consumption growth,
1929–2008
1946–2008
0.0190 (0.0021)
0.0129 (0.0014)
-0.5378 (0.5905)
3.3963 (0.7489)
0.3631 (0.0917)
0.0735 (0.1448)
-0.0573 (0.0656)
0.0626 (0.1017)
∆C
C
Data
0.0166
0.0245
-0.0505
2.9146
0.4185
0.1720
0.0470
-0.0268
Mean
0.0077
0.0206
-0.5225
2.2735
0.2335
-0.0627
-0.1752
-0.2500
5%
0.0166
0.0243
-0.0470
2.8256
0.4223
0.1725
0.0453
-0.0269
50%
0.0253
0.0288
0.4074
3.8531
0.5905
0.4012
0.2759
0.1974
95%
Model, w/ initial cond. λ(t = 0) = λG
0.0147
0.0252
-0.0981
2.9735
0.4385
0.1932
0.0552
-0.0260
Mean
0.0048
0.0209
-0.6251
2.2853
0.2466
-0.0443
-0.1714
-0.2541
5%
0.0148
0.0250
-0.0838
2.8589
0.4412
0.1930
0.0525
-0.0254
50%
0.0242
0.0303
0.3828
4.0348
0.6181
0.4291
0.2872
0.2060
95%
Model, w/ initial cond. λ(t = 0) = λB
Table 2
How well does the model match underlying fundamentals?
We compare sample moments of yearly growth rates in aggregate consumption and dividends to the values predicted by the baseline
calibration of the model. The measure for aggregate consumption is the real yearly series of per-capita consumption expenditures in nondurable
goods and services (source: NIPA tables, Bureau of Economic Analysis). We obtain monthly dividends from returns, with and without dividends,
on the CRSP value-weighted market index (e.g., Fama and French, 1988). We sum the monthly dividends to obtain the yearly dividend series,
and we deflate that series using CPI data. We report empirical results for two sample periods. The first spans 80 years of data, 1929–2008, the
second spans the post-World War II period, 1946–2008. Standard errors estimates, in brackets, are robust with respect to both autocorrelation
and heteroskedasticity. Next, we simulate 10,000 samples of monthly consumption and dividend data from the model, each spanning a period of
80 years (same as the length of the 1929–2008 sample period). We aggregate the monthly series to obtain series of yearly growth rates ∆C
C and
∆D
D . For each of the 10,000 simulated samples, we compute summary statistics for these series. We report the mean value of these statistics,
as well as the 5th, 50th, and 95th percentiles. We repeat the analysis with two simulation schemes. In the first set of simulations, we initialize
the Markov chain for the λ process at λ(t = 0) = λG . In the second set, we initialize λ(t = 0) = λB .
37
Mean
Std. dev.
Skewness
Kurtosis
AC(1)
corr(∆C/C, ∆D/D)
0.0173
0.1105
0.3822
9.0564
0.1877
0.5923
∆D
D
Data
(0.0114)
(0.0244)
(0.7873)
(2.4074)
(0.1398)
(0.1905)
1929–2008
Panel B: Dividend growth,
Table 2, continued
0.0224
0.0659
1.0729
5.3017
0.2482
0.1809
(0.0094)
(0.0096)
(0.4959)
(0.8784)
(0.1141)
(0.1238)
1946–2008
0.0205
0.1125
0.2388
3.0179
0.2793
0.3647
Mean
-0.0101
0.0958
-0.2034
2.2896
0.0969
0.1746
5%
0.0201
0.1122
0.2272
2.8937
0.2818
0.3690
50%
0.0508
0.1298
0.7149
4.1583
0.4514
0.5378
95%
Model, w/ initial cond. λ(t = 0) = λG
0.0157
0.1135
0.2251
3.0289
0.2887
0.3807
Mean
-0.0168
0.0965
-0.2351
2.3014
0.1028
0.1880
5%
0.0156
0.1131
0.2133
2.9045
0.2917
0.3851
50%
0.0476
0.1314
0.7110
4.1497
0.4650
0.5556
95%
Model, w/ initial cond. λ(t = 0) = λB
38
Mean
Std. dev.
∆S
S
+
D
S
0.0737 (0.0199)
0.2032 (0.0177)
0.1903 (0.0179)
-0.2086 (0.3233)
2.7813 (0.2419)
0.3309 (0.7081)
11.3264 (2.3856)
0.0064 (0.0073)
0.0386 (0.0072)
Panel B: Real risk-free rate, rf
Mean
Std. dev.
Std. dev. (monthly obs)
Skewness
Kurtosis
Skewness (monthly obs)
Kurtosis (monthly obs)
Panel A: Total real market return,
1929–2008
1946–2008
0.0069 (0.0063)
0.0316 (0.0082)
0.0760 (0.0211)
0.1809 (0.0143)
0.1477 (0.0070)
-0.3199 (0.4835)
2.9173 (0.3479)
-0.6008 (0.2583)
4.9747 (0.8870)
Data
0.0107
0.0062
0.0685
0.1618
0.1518
0.3293
3.1954
-0.0843
4.0033
Mean
0.0072
0.0046
0.0380
0.1395
0.1448
-0.1561
2.3282
-0.9511
2.7948
5%
0.0108
0.0061
0.0683
0.1614
0.1513
0.3156
3.0154
0.0090
3.0485
50%
Model (pre-crash)
0.0139
0.0082
0.0994
0.1861
0.1611
0.8580
4.6450
0.1514
11.3500
95%
0.0087
0.0061
0.0728
0.1628
0.1522
0.3178
3.2097
-0.1205
4.2790
Mean
0.0051
0.0045
0.0423
0.1402
0.1447
-0.1834
2.3381
-1.0525
2.7986
5%
0.0089
0.0060
0.0726
0.1623
0.1514
0.3074
3.0288
0.0019
3.0620
50%
Model (post-crash)
0.0120
0.0081
0.1040
0.1874
0.1628
0.8487
4.6942
0.1495
12.5785
95%
Table 3
How well does the model match basic asset pricing moments?
We compare sample asset pricing moments to the values predicted by the baseline calibration of the model. The total real yearly market
D
return, ∆S
S + S , is the yearly return, inclusive of all distributions, on the CRSP value-weighted market index, adjusted for inflation using the
CPI. The real risk-free rate rf is the inflation-adjusted three-month rate from the ‘Fama Risk-Free Rates’ database in CRSP. In computing the
logarithmic price-dividend ratio, log(S/D), we consider two measures of dividends. The first is the real dividend on the CRSP value-weighted
index. The second is the real dividend on the CRSP value-weighted market index, adjusted to include share repurchases (Boudoukh et al.,
2007). Standard errors estimates, in brackets, are robust with respect to both autocorrelation and heteroskedasticity. Next, we simulate 10,000
samples of monthly stock market returns, risk-free rates, dividends, and stock market portfolio prices, each spanning a period of 80 years. We
aggregate the monthly series at the yearly frequency. For each of the 10,000 simulated samples, we compute summary statistics for these series.
We report the mean value of these statistics, as well as the 5th, 50th, and 95th percentiles. We repeat the analysis with two simulation schemes.
In the first set of simulations, we initialize the probability process p at p(t = 0) = pP re , which corresponds to the pre-crash economy. In the
second set, we initialize p(t = 0) = pP ost , which corresponds to the post-crash economy.
39
+
D
S
− rf
0.0673 (0.0214)
0.2052 (0.0222)
-0.2692 (0.3584)
3.0852 (0.3140)
0.2203 (0.6756)
10.9809 (2.1599)
∆S
S
1929–2008
Mean
Mean (w/ shares rep.)
Std. dev.
Std. dev. (w/ shares rep.)
3.3450
3.1519
0.4239
0.2915
(0.1069)
(0.0698)
(0.0699)
(0.0248)
1946–2008
3.4346
3.2027
0.4249
0.2930
(0.1207)
(0.0814)
(0.0733)
(0.0287)
0.0691 (0.0195)
0.1742 (0.0143)
-0.2705 (0.4980)
3.0060 (0.4180)
-0.5836 (0.2622)
4.9824 (0.8981)
¡S¢
D
Data
Panel D: Logarithmic price-dividend ratio, log
Mean
Std. dev.
Skewness
Kurtosis
Skewness (monthly obs)
Kurtosis (monthly obs)
Panel C: Equity premium,
Table 3, continued
3.0625
0.0594
0.0879
0.0289
0.1384
-0.1465
2.3307
-0.9357
2.7948
5%
3.1630
0.0578
0.1606
0.3360
3.1953
-0.0817
3.9883
Mean
0.0799
3.1720
0.0575
0.1602
0.3212
3.0128
0.0103
3.0486
50%
Model (pre-crash)
0.1600
3.2127
0.0871
0.1845
0.8610
4.6461
0.1528
11.2290
95%
0.0863
3.1149
0.0641
0.1615
0.3253
3.2083
-0.1174
4.2599
Mean
0.0587
2.9791
0.0350
0.1391
-0.1693
2.3397
-1.0376
2.8008
5%
0.0791
3.1306
0.0638
0.1610
0.3117
3.0277
0.0031
3.0621
50%
Model (post-crash)
0.1458
3.1726
0.0934
0.1858
0.8517
4.6654
0.1509
12.4307
95%
0.15
0.1
0.05
06
20
05
20
03
20
01
20
99
19
97
19
95
19
93
19
91
19
19
89
0
O
ct 19
19 87
,1
98
7
10%OTM IV − ATM IV
Panel A
Panel B
2.5%ITM IV − ATM IV
0.02
0.01
0
−0.01
−0.02
−0.03
6
20
0
5
20
0
3
20
0
1
20
0
99
19
97
19
95
19
93
19
91
19
89
19
98
7
,1
O
ct
19
19
87
−0.04
Fig. 1. Pre- and post-crash implied volatility smirk for S&P 500 options with one month to maturity. The
plot in Panel A depicts the spread between implied volatilities for S&P 500 options with a strike-to-price
ratio X = K/S − 1 = −10% and at-the-money implied volatilities. The plot in Panel B depicts the spread
between implied volatilities for options with a strike-to-price ratio X = K/S − 1 = 2.5% and at-the-money
implied volatilities. Appendix D explains how we constructed the implied volatility series.
40
Pre− and post−crash implied volatilities
26
Pre crash
Post crash
Implied volatility (annual percentage)
24
22
20
18
16
14
−0.1
−0.05
0
Moneyness = K/S0−1
0.05
0.1
Fig. 2. The plots depict the model-implied volatility smirk pre- and post-1987 market crash for S&P 500
options with one month to maturity. The model coefficients are set equal to the baseline values given in
Table 1.
Implied volatility (annual percentage)
26
24
22
20
18
16
14
12
−0.1
−0.05
0
0.05
Moneyness= K/S0−1
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
p
Fig. 3. The plot depicts the model-implied volatility smirk as a function of the probability p the agent
assigns to be in the low-crash-intensity economy. Implied volatilities are for S&P 500 options with one
month to maturity. The model coefficients are set equal to the baseline values given in Table 1.
41
Pre−crash implied volatilities
Post−crash implied volatilities
30
30
γ=12.5
γ=10 (baseline)
γ=7.5
Implied volatility (annual percentage)
28
28
26
26
24
24
22
22
20
20
18
18
16
16
14
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
γ=12.5
γ=10 (baseline)
γ=7.5
14
0.1
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
0.1
Fig. 4. The plots illustrate the sensitivity of the model-implied volatility smirk to the elasticity of relative
risk-aversion coefficient γ. Implied volatilities are for S&P 500 options with one month to maturity.
Pre−crash implied volatilities
Post−crash implied volatilities
26
26
ψ=2.5
ψ=2 (baseline)
ψ=1.5
Implied volatility (annual percentage)
ψ=2.5
ψ=2 (baseline)
ψ=1.5
24
24
22
22
20
20
18
18
16
16
14
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
14
0.1
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
0.1
Fig. 5. The plots illustrate the sensitivity of the model-implied volatility smirk to the elasticity of intertemporal substitution coefficient Ψ = ρ1 . Implied volatilities are for S&P 500 options with one month to
maturity.
42
Pre−crash implied volatilities
Post−crash implied volatilities
32
32
ν
=−0.035− σ
ν
=−0.035− σ
ν
=−0.035 (baseline)
ν
=−0.035 (baseline)
ν
bar
bar
Implied volatility (annual percentage)
30
νbar=−0.035+ σν
28
26
26
24
24
22
22
20
20
18
18
16
16
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
bar
30
28
14
ν
bar
14
0.1
νbar=−0.035+ σν
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
0.1
Fig. 6. The plots illustrate the sensitivity of the model-implied volatility smirk to the jump coefficient ν̄.
Implied volatilities are for S&P 500 options with one month to maturity.
Pre−crash implied volatilities
Post−crash implied volatilities
Implied volatility (annual percentage)
26
26
ξ=ξlow
ξ=ξlow
ξ baseline
ξ=ξhigh
ξ baseline
ξ=ξhigh
24
24
22
22
20
20
18
18
16
16
14
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
14
0.1
−0.1
−0.05
0
0.05
Moneyness = K/S0−1
0.1
Fig. 7. The plots illustrate the sensitivity of the model-implied volatility smirk to the jump coefficient ξ.
Implied volatilities are for S&P 500 options with one month to maturity.
43
Implied volatilities for index and individual stock options
Index options
Individual stock options
40
Implied volatility (annual percentage)
35
30
25
20
15
−0.1
−0.05
0
Moneyness = K/S0−1
0.05
0.1
Fig. 8. The plots contrast the model-implied volatility function for individual stock options to the volatility
smirk for S&P 500 index options with one month to maturity. The model coefficients are set equal to the
baseline values given in Table 1.
44
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
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The Age of Reason: Financial Decisions Over the Lifecycle
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Information Acquisition in Financial Markets: a Correction
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Estate Taxation, Entrepreneurship, and Wealth
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Conflict of Interest and Certification in the U.S. IPO Market
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The Reaction of Consumer Spending and Debt to Tax Rebates –
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Nonparametric Analysis of Intergenerational Income Mobility
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How the Credit Channel Works: Differentiating the Bank Lending Channel
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1
Working Paper Series (continued)
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A Conversation with 590 Nascent Entrepreneurs
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Cyclical Dumping and US Antidumping Protection: 1980-2001
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WP-07-21
Health Capital and the Prenatal Environment:
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WP-07-22
The Spending and Debt Response to Minimum Wage Hikes
Daniel Aaronson, Sumit Agarwal, and Eric French
WP-07-23
The Impact of Mexican Immigrants on U.S. Wage Structure
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A Leverage-based Model of Speculative Bubbles
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Bank Lending, Financing Constraints and SME Investment
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Global Inflation
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2
Working Paper Series (continued)
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WP-08-08
WP-08-09
The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
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Consumer Choice and Merchant Acceptance of Payment Media
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Investment Shocks and Business Cycles
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New Vehicle Characteristics and the Cost of the
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Realized Volatility
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Revenue Bubbles and Structural Deficits: What’s a state to do?
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The role of lenders in the home price boom
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WP-08-16
Bank Crises and Investor Confidence
Una Okonkwo Osili and Anna Paulson
WP-08-17
Life Expectancy and Old Age Savings
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-08-18
Remittance Behavior among New U.S. Immigrants
Katherine Meckel
WP-08-19
Birth Cohort and the Black-White Achievement Gap:
The Roles of Access and Health Soon After Birth
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WP-08-20
Public Investment and Budget Rules for State vs. Local Governments
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Why Has Home Ownership Fallen Among the Young?
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Why do the Elderly Save? The Role of Medical Expenses
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-09-02
3
Working Paper Series (continued)
Using Stock Returns to Identify Government Spending Shocks
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The Effect of Disability Insurance Receipt on Labor Supply
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CEO Overconfidence and Dividend Policy
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WP-09-06
Do Financial Counseling Mandates Improve Mortgage Choice and Performance?
Evidence from a Legislative Experiment
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and Douglas D. Evanoff
WP-09-07
Perverse Incentives at the Banks? Evidence from a Natural Experiment
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WP-09-08
Pay for Percentile
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The Life and Times of Nicolas Dutot
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The Case of the Undying Debt
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WP-09-12
Paying for Performance: The Education Impacts of a Community College Scholarship
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The Price of Gasoline and the Demand for Fuel Economy:
Evidence from Monthly New Vehicles Sales Data
Thomas Klier and Joshua Linn
WP-09-15
Estimation of a Transformation Model with Truncation,
Interval Observation and Time-Varying Covariates
Bo E. Honoré and Luojia Hu
WP-09-16
Self-Enforcing Trade Agreements: Evidence from Antidumping Policy
Chad P. Bown and Meredith A. Crowley
WP-09-17
Too much right can make a wrong: Setting the stage for the financial crisis
Richard J. Rosen
WP-09-18
4
Working Paper Series (continued)
Can Structural Small Open Economy Models Account
for the Influence of Foreign Disturbances?
Alejandro Justiniano and Bruce Preston
WP-09-19
Liquidity Constraints of the Middle Class
Jeffrey R. Campbell and Zvi Hercowitz
WP-09-20
Monetary Policy and Uncertainty in an Empirical Small Open Economy Model
Alejandro Justiniano and Bruce Preston
WP-09-21
Firm boundaries and buyer-supplier match in market transaction:
IT system procurement of U.S. credit unions
Yukako Ono and Junichi Suzuki
Health and the Savings of Insured Versus Uninsured, Working-Age Households in the U.S.
Maude Toussaint-Comeau and Jonathan Hartley
WP-09-22
WP-09-23
The Economics of “Radiator Springs:” Industry Dynamics, Sunk Costs, and
Spatial Demand Shifts
Jeffrey R. Campbell and Thomas N. Hubbard
WP-09-24
On the Relationship between Mobility, Population Growth, and
Capital Spending in the United States
Marco Bassetto and Leslie McGranahan
WP-09-25
The Impact of Rosenwald Schools on Black Achievement
Daniel Aaronson and Bhashkar Mazumder
WP-09-26
Comment on “Letting Different Views about Business Cycles Compete”
Jonas D.M. Fisher
WP-10-01
Macroeconomic Implications of Agglomeration
Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited
WP-10-02
Accounting for non-annuitization
Svetlana Pashchenko
WP-10-03
Robustness and Macroeconomic Policy
Gadi Barlevy
WP-10-04
Benefits of Relationship Banking: Evidence from Consumer Credit Markets
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles
WP-10-05
The Effect of Sales Tax Holidays on Household Consumption Patterns
Nathan Marwell and Leslie McGranahan
WP-10-06
Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions
Scott Brave and R. Andrew Butters
WP-10-07
Identification of Models of the Labor Market
Eric French and Christopher Taber
WP-10-08
5
Working Paper Series (continued)
Public Pensions and Labor Supply Over the Life Cycle
Eric French and John Jones
WP-10-09
Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-10-10
6
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