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Federal Reserve Bank of Chicago

Stochastic Volatility
Torben G. Andersen and Luca Benzoni

WP 2009-04

1

Stochastic Volatility1
Torben G. Andersena and Luca Benzonib
a

Kellogg School of Management, Northwestern University, Evanston, IL; NBER, Cambridge, MA;
and CREATES, Aarhus, Denmark.
b Federal Reserve Bank of Chicago, Chicago, Illinois, USA.

Article Outline
Glossary
I

Deﬁnition of the Subject and its Importance

II

Introduction

III

Model Speciﬁcation

IV

Realized Volatility

V

Applications

VI

Estimation Methods

VII

Future Directions

VIII References

Glossary
Implied volatility The value of asset return volatility which equates a model-implied derivative
price to the observed market price. Most notably, the term is used to identify the volatility
implied by the Black and Scholes [63] option pricing formula.
Quadratic return variation The ex-post sample-path return variation over a ﬁxed time interval.
Realized volatility The sum of ﬁnely sampled squared asset return realizations over a ﬁxed time
interval. It is an estimate of the quadratic return variation over such time interval.
Stochastic volatility A process in which the return variation dynamics include an unobservable
shock which cannot be predicted using current available information.
1

This version: June 15, 2008. Chapter prepared for the Encyclopedia of Complexity and System Science, Springer.
We are grateful to Olena Chyruk, Bruce Mizrach (the Editor), and Neil Shephard for helpful comments and suggestions.
Of course, 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 work of Andersen is supported by a
grant from the NSF to the NBER and support from CREATES funded by the Danish National Research Foundation.

2

I. Deﬁnition of the Subject and its Importance
Given the importance of return volatility on a number of practical ﬁnancial management decisions,
the eﬀorts to provide good real-time estimates and forecasts of current and future volatility have
been extensive. The main framework used in this context involves stochastic volatility models. In a
broad sense, this model class includes GARCH, but we focus on a narrower set of speciﬁcations in
which volatility follows its own random process, as is common in models originating within ﬁnancial
economics. The distinguishing feature of these speciﬁcations is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable
information. In what follows, we refer to these models as genuine stochastic volatility models.
Much modern asset pricing theory is built on continuous-time models. The natural concept of
volatility within this setting is that of genuine stochastic volatility. For example, stochastic-volatility
(jump-)diﬀusions have provided a useful tool for a wide range of applications, including the pricing of
options and other derivatives, the modeling of the term structure of risk-free interest rates, and the
pricing of foreign currencies and defaultable bonds. The increased use of intraday transaction data
for construction of so-called realized volatility measures provides additional impetus for considering
genuine stochastic volatility models. As we demonstrate below, the realized volatility approach is
closely associated with the continuous-time stochastic volatility framework of ﬁnancial economics.
There are some unique challenges in dealing with genuine stochastic volatility models. For example,
volatility is truly latent and this feature complicates estimation and inference. Further, the presence of
an additional state variable—volatility—renders the model less tractable from an analytic perspective.
We review how such challenges have been addressed through development of new estimation methods
and imposition of model restrictions allowing for closed-form solutions while remaining consistent with
the dominant empirical features of the data.

II. Introduction
The label Stochastic Volatility is applied in two distinct ways in the literature. For one, it is used
to signify that the (absolute) size of the innovations of a time series displays random ﬂuctuations
over time. Descriptive models of ﬁnancial time series almost invariably embed this feature nowadays
as asset return series tend to display alternating quiet and turbulent periods of varying length and
intensity. To distinguish this feature from models that operate with an a priori known or deterministic path for the volatility process, the random evolution of the conditional return variance is termed
stochastic volatility. The simplest case of deterministic volatility is the constant variance assumption
invoked in, e.g., the Black and Scholes [63] framework. Another example is modeling the variance
purely as a given function of calendar time, allowing only for eﬀects such as time-of-year (seasonals),
day-of-week (institutional and announcement driven) or time-of-day (diurnal eﬀects due to, e.g., market microstructure features). Any model not falling within this class is then a stochastic volatility
model. For example, in the one-factor continuous-time Cox, Ingersoll, and Ross [113] (CIR) model
the (stochastic) level of the short term interest rate governs the dynamics of the (instantaneous) drift
and diﬀusion term of all zero-coupon yields. Likewise, in GARCH models the past return innovations
govern the one-period ahead conditional mean and variance. In both models, the volatility is known,

3
or deterministic, at a given point in time, but the random evolution of the processes renders volatility
stochastic for any horizon beyond the present period.
The second notion of stochastic volatility, which we adopt henceforth, refers to models in which
the return variation dynamics is subject to an unobserved random shock so that the volatility is
inherently latent. That is, the current volatility state is not known for sure, conditional on the true
data generating process and the past history of all available discretely sampled data. Since the CIR
and GARCH models described above do render the current (conditional) volatility known, they are
not stochastic volatility models in this sense. In order to make the distinction clear cut, we follow
Andersen [10] and label this second, more restrictive, set genuine stochastic volatility (SV) models.
There are two main advantages to focusing on SV models. First, much asset pricing theory is
built on continuous-time models. Within this class, SV models tend to ﬁt more naturally with a wide
array of applications, including the pricing of currencies, options, and other derivatives, as well as
the modeling of the term structure of interest rates. Second, the increasing use of high-frequency
intraday data for construction of so-called realized volatility measures is also starting to push the
GARCH models out of the limelight as the realized volatility approach is naturally linked to the
continuous-time SV framework of ﬁnancial economics.
One drawback is that volatility is not measurable with respect to observable (past) information in
the SV setting. As such, an estimate of the current volatility state must be ﬁltered out from a noisy
environment and the estimate will change as future observations become available. Hence, in-sample
estimation typically involves smoothing techniques, not just ﬁltering. In contrast, the conditional
variance in GARCH is observable given past information, which renders (quasi-)maximum likelihood
techniques for inference quite straightforward while smoothing techniques have no role. As such,
GARCH models are easier to estimate and practitioners often rely on them for time-series forecasts of
volatility. However, the development of powerful method of simulated moments, Markov Chain Monte
Carlo (MCMC) and other simulation based procedures for estimation and forecasting of SV models
may well render them competitive with ARCH over time on that dimension.
Direct indications of the relations between SV and GARCH models are evident in the sequence of
papers by Dan Nelson and Dean Foster exploring the SV diﬀusion limits of ARCH models as the case of
continuous sampling is approached, see, e.g., Nelson and Foster [219]. Moreover, as explained in further
detail in the estimation section below, it can be useful to summarize the dynamic features of asset
returns by tractable pseudo-likelihood scores obtained from GARCH-style models when performing
simulation based inference for SV models. As such, the SV and GARCH frameworks are closely related
and should be viewed as complements. Despite these connections we focus, for the sake of brevity,
almost exclusively on SV models and refer the interested reader to the GARCH chapter for further
information.
The literature on SV models is vast and rapidly growing, and excellent surveys are available, e.g.,
Ghysels et al. [158] and Shephard [239, 240]. Consequently, we focus on providing an overview of the
main approaches with illustrations of the scope for applications of these models to practical ﬁnance
problems.

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III. Model Speciﬁcation
The original econometric studies of SV models were invariably cast in discrete time and they were quite
similar in structure to ARCH models, although endowed with a more explicit structural interpretation.
Recent work in the area has been mostly directly towards a continuous time setting and motivated
by the typical speciﬁcations in ﬁnancial economics. This section brieﬂy reviews the two alternative
approaches to speciﬁcation of SV models.

III.1. Discrete-Time SV Models and the Mixture-of-Distributions Hypothesis
Asset pricing theory contends that ﬁnancial asset prices reﬂect the discounted value of future expected
cash ﬂows, implying that all news relevant for either discount rates or cash ﬂows should induce a shift
in market prices. Since economic news items appear almost continuously in real time, this perspective
rationalizes the ever-changing nature of prices observed in ﬁnancial markets. The process linking
news arrivals to price changes may be complex, but if it is stationary in the statistical sense it will
nonetheless produce a robust theoretical association between news arrivals, market activity and return
volatility. In fact, if the number of news arrival is very large, standard central limit theory will tend to
imply that asset returns are approximately normally distributed conditional on the news count. More
generally, variables such as the trading volume, the number of transactions or the number of price
quotes are also naturally related to the intensity of the information ﬂow. This line of reasoning has
motivated speciﬁcations such as,
yt |st ; N (µy st , σ 2 st )
(1)
y
where yt is an “activity” variable related to the information ﬂow, st is a positive intensity process
reﬂecting the rate of news arrivals, µy represents the mean response to an information event, and σ y
is a pure scaling parameter.
This is a normal mixture model, where the st process governs or “mixes” the scale of the distribution
across the periods. If st is constant, this is simply an i.i.d. Gaussian process for returns and possible
other related variables. However, this is clearly at odds with the empirical evidence for, e.g., return
volatility and trading volume. Therefore, st is typically stipulated to follow a separate stochastic
process with random innovations. Hence, each period the return series is subject to two separate
shocks, namely the usual idiosyncratic error term associated with the (normal) return distribution,
but also a shock to the variance or volatility process, st . This endows the return process with genuine
stochastic volatility, reﬂecting the random intensity of news arrivals. Moreover, it is typically assumed
that only returns, transactions and quotes are observable, but not the actual value of st itself, implying
that σ y cannot be separately identiﬁed. Hence, we simply ﬁx this parameter at unity.
The time variation in the information ﬂow series induces a fat-tailed unconditional distribution,
consistent with stylized facts for ﬁnancial return and, e.g., trading volume series. Intuitively, days
with a lot of news display more rapid price ﬂuctuations and trading activity than days with a low
news count. In addition, if the st process is positively correlated, then shocks to the conditional mean
and variance processes for yt will be persistent. This is consistent with the observed clustering in
ﬁnancial markets, where return volatility and trading activity are contemporaneously correlated and
each display pronounced positive serial dependence.

5
The inherent randomness and unobserved nature of the news arrival process, even during period
t, renders the true mean and variance series latent. This property is the major diﬀerence with the
GARCH model class, in which the one-step-ahead conditional mean and variance are a known function
of observed variables at time t − 1. As such, for genuine SV models, we must distinguish the full, but
infeasible, information set (st ∈ Ft ) and the observable information set (st ∈ It ). This basic latency of
/
the mixing variable (state vector) of the SV model complicates inference and forecasting procedures
as discussed below.
For short horizon returns, µy is nearly negligible and can reasonably be ignored or simply ﬁxed
at a small constant value, and the series can then be demeaned. This simpliﬁcation produces the
following return (innovation) model,
√
rt = st zt ,
(2)
Where zt is an i.i.d. standard normal variable, implying a simple normal-mixture representation,
rt |st ; N (0, st ) .

(3)

Univariate return models of the form (3) as well as multivariate systems including a return variable
along with other related market activity variables, such as the transactions count, the quote intensity
or the aggregate trading volume, stem from the Mixture-of-Distributions Hypothesis (MDH).
Actual implementation of the MDH hinges on a particular representation of the informationarrival process st . Clark [102] uses trading volume as a proxy for the activity variable, a choice
motivated by the high contemporaneous correlation between return volatility and volume. Tauchen
and Pitts [247] follow a structural approach to characterize the joint distribution of the daily return
and volume relation governed by the underlying latent information ﬂow st . However, both these
models assume temporal independence of the information ﬂow, thus failing to capture the clustering
in these series. Partly in response, Gallant et al. [153] examine the joint conditional return-volume
distribution without imposing any structural MDH restrictions. Nonetheless, many of the original
discrete-time SV speciﬁcations are compatible with the MDH framework, including Taylor [249],2 who
proposes an autoregressive parameterization of the latent log-volatility (or information ﬂow) variable
log(st+1 ) = η 0 + η 1 log(st ) + ut , ut ; i.i.d(0, σ 2 ) ,
u

(4)

where the error term, ut , may be correlated with the disturbance term, zt , in the return equation (2)
so that ρ = corr(ut , zt ) = 0. If ρ < 0, downward movements in asset prices result in higher future
volatility as also predicted by the so-called ‘leverage eﬀect’ in the exponential GARCH, or EGARCH,
form of Nelson [218] and the asymmetric GARCH model of Glosten et al. [160].
Early tests of the MDH include Lamoureux and Lastrapes [194] and Richardson and Smith [231].
Subsequently, Andersen [11] studies a modiﬁed version of the MDH that provides a much improved
ﬁt to the data. Further reﬁnements of the MDH speciﬁcation have been pursued by, e.g., Liesenfeld
[198, 199] and Bollerslev and Jubinsky [67]. Among the ﬁrst empirical studies of the related approach
of stochastic time changes are An´ and Geman [29], who focus on stock returns, and Conley et al.
e
[109], who focus on the short-term risk-free interest rate.
2

Discrete-time SV models go farther back in time, at least to the early paper by Rosenberg [232] recently reprinted
in Shephard [240].

6

III.2. Continuous-Time Stochastic Volatility Models
Asset returns typically contain a predictable component, which compensates the investor for the risk
of holding the security, and an unobservable shock term, which cannot be predicted using current
available information. The conditional asset return variance pertains to the variability of the unobservable shock term. As such, over a non-inﬁnitesimal horizon it is necessary to ﬁrst specify the
conditional mean return (e.g., through an asset pricing model) in order to identify the conditional
return variation. In contrast, over an inﬁnitesimal time interval this is not necessary because the
requirement that market prices do not admit arbitrage opportunities implies that return innovations
are an order of magnitude larger than the mean return. This result has important implications for
the approach we use to model and measure volatility in continuous time.
Consider an asset with log-price process {p(t) , t ∈ [0 , T ]} deﬁned on a probability space (Ω, F, P ).
Following Andersen et al. [19] we deﬁne the continuously compounded asset return over a time interval
from t − h to t, 0 ≤ h ≤ t ≤ T , to be
r(t, h) = p(t) − p(t − h) .

(5)

A special case of (5) is the cumulative return up to time t, which we denote r(t) ≡ r(t, t) = p(t) − p(0),
0 ≤ t ≤ T . Assume the asset trades in a frictionless market void of arbitrage opportunities and the
number of potential discontinuities (jumps) in the price process per unit time is ﬁnite. Then the logprice process p is a semi-martingale (e.g., Back [33]) and therefore the cumulative return r(t) admits
the decomposition (e.g., Protter [229])
r(t) = µ(t) + M C (t) + M J (t) ,

(6)

where µ(t) is a predictable and ﬁnite variation process, M c (t) a continuous-path inﬁnite-variation
martingale, and M J (t) is a compensated ﬁnite activity jump martingale. Over a discrete time interval
the decomposition (6) becomes
r(t, h) = µ(t, h) + M C (t, h) + M J (t, h) ,

(7)

where µ(t, h) = µ(t) − µ(t − h), M C (t, h) = M C (t) − M C (t − h), and M J (t, h) = M J (t) − M J (t − h).
Denote now with [r, r] the quadratic variation of the semi-martingale process r, where (Protter
[229])
[r, r]t = r(t)2 − 2

r(s−)dr(s) ,

(8)

and r(t−) = lims↑t r(s). If the ﬁnite variation process µ is continuous, then its quadratic variation is
identically zero and the predictable component µ in decomposition (7) does not aﬀect the quadratic
variation of the return r. Thus, we obtain an expression for the quadratic return variation over the
time interval from t − h to t, 0 ≤ h ≤ t ≤ T (e.g., Andersen et al. [21] and Barndorﬀ-Nielsen and
Shephard [51, 52]):
QV (t, h) = [r, r]t − [r, r]t−h = [M C , M C ]t − [M C , M C ]t−h +

∆M 2 (s)
t−h<s≤t

C

C

C

C

∆r2 (s) .

= [M , M ]t − [M , M ]t−h +
t−h<s≤t

(9)

7
Most continuous-time models for asset returns can be cast within the general setting of equation
(7), and equation (9) provides a framework to study the model-implied return variance. For instance,
the Black and Scholes [63] model is a special case of the setting described by equation (7) in which
the conditional mean process µ is constant, the continuous martingale M C is a standard Brownian
motion process, and the jump martingale M J is identically zero:
dp(t) = µdt + σdW (t) .

(10)

In this case, the quadratic return variation over the time interval from t − h to t, 0 ≤ h ≤ t ≤ T ,
simpliﬁes to
t

σ 2 ds = σ 2 h ,

QV (t, h) =

(11)

t−h

that is, return volatility is constant over any time interval of length h.
A second notable example is the jump-diﬀusion model of Merton [214],
dp(t) = (µ − λξ )dt + σdW (t) + ξ(t) dqt ,

(12)

where q is a Poisson process uncorrelated with W and governed by the constant jump intensity λ, i.e.,
Prob(dqt = 1) = λ dt. The scaling factor ξ(t) denotes the magnitude of the jump in the return process
if a jump occurs at time t. It is assumed to be normally distributed,
ξ(t) ; N ( ξ, σ 2 ) .
ξ

(13)

In this case, the quadratic return variation process over the time interval from t−h to t, 0 ≤ h ≤ t ≤ T
becomes
t

QV (t, h) =
t−h

J(s)2 ,

J(s)2 = σ 2 h +

σ 2 ds +

(14)

t−h≤s≤t

t−h≤s≤t

where J(t) ≡ ξ(t)dq(t) is non-zero only if a jump actually occurs.
Finally, a broad class of stochastic volatility models is deﬁned by
dp(t) = µ(t)dt + σ(t)dW (t) + ξ(t) dqt ,

(15)

where q is a constant-intensity Poisson process with log-normal jump amplitude (13). Equation (15)
is also a special case of (7) and the associated quadratic return variation over the time interval from
t − h to t, 0 ≤ h ≤ t ≤ T , is
t

QV (t, h) =

σ(s)2 ds +

t−h

J(s)2
t−h≤s≤t

J(s)2 .

≡ IV (t, h) +

(16)

t−h≤s≤t

As in the general case of equation (9), equation (16) identiﬁes the contribution of diﬀusive volatility,
termed ‘integrated variance’ (IV), and cumulative squared jumps to the total quadratic variation.
Early applications typically ignored jumps and focused exclusively on the integrated variance
component. For instance, IV plays a key role in Hull and White’s [174] SV option pricing model,

8
which we discuss in Section V.1 below along with other option pricing applications. For illustration,
we focus here on the SV model speciﬁcation by Wiggins [256]:
dp(t) = µdt + σ(t) dWp (t)

(17)

dσ(t) = f (σ(t))dt + η σ(t) dWσ (t) ,

(18)

where the innovations to the return dp and volatility σ, Wp and Wσ , are standard Brownian motions.
If we deﬁne y = log(σ) and apply Itˆ’s formula we obtain
o
1
f (σ(t))
dt + η dWσ (t) .
dy(t) = d log(σ(t)) = − η 2 +
2
σ(t)

(19)

Wiggins approximates the drift term f (σ(t)) ≈ {α + κ[ log(σ) − log(σ(t)) ]}σ(t). Substitution in
equation (19) yields
d log(σ(t)) = [ α − κ log(σ(t)) ] dt + η dWσ (t) ,
(20)
where α = α + κ log(σ) − 1 η 2 . As such, the logarithmic standard deviation process in Wiggins has
2
diﬀusion dynamics similar in spirit to Taylor’s discrete time AR(1) model for the logarithmic information process, equation (4). As in Taylor’s model, negative correlation between return and volatility
innovations, ρ = corr(Wp , Wσ ) < 0, generates an asymmetric response of volatility to return shocks
similar to the leverage eﬀect in discrete-time EGARCH models.
More recently, several authors have imposed restrictions on the continuous-time SV jump-diﬀusion
(15) that render the model more tractable while remaining consistent with the empirical features of
the data. We return to these models in Section V.1 below.

IV. Realized Volatility
Model-free measures of return variation constructed only from concurrent return realizations have
been considered at least since Merton [215]. French et al. [148] construct monthly historical volatility
estimates from daily return observations. More recently, the increased availability of transaction
data has made it possible to reﬁne early measures of historical volatility into the notion of ‘realized
volatility,’ which is endowed with a formal theoretical justiﬁcation as an estimator of the quadratic
return variation as ﬁrst noted in Andersen and Bollerslev [18]. The realized volatility of an asset
return r over the time interval from t − h to t is
n

RV (t, h; n) =

r t−h+
i=1

ih h
,
n n

2

.

(21)

Semi-martingale theory ensures that the realized volatility measure RV converges to the return quadratic
variation QV, previously deﬁned in equation (9), when the sampling frequency n increases. We point
the interested reader to, e.g., Andersen et al. [19] to ﬁnd formal arguments in support of this claim.
Here we convey intuition for this result by considering the special case in which the asset return follows
a continuous-time diﬀusion without jumps,
dp(t) = µ(t)dt + σ(t) dW (t) .

(22)

9
As in equation (21), consider a partition of the [t − h, t] interval with mesh h/n. A discretization
of the diﬀusion (22) over a sub-interval from t − h + (i−1)h to t − h + ih , i = 1, . . . , n, yields
n
n
r t−h+

ih h
,
n n

where ∆W t − h +

≈µ t−h+
ih
n

(i − 1)h
n

=W t−h+

ih
n

h
(i − 1)h
+σ t−h+
n
n

−W t−h+

Suppressing time indices, the squared return
2

2

r =µ

h
n

r2

(i−1)h
n

∆W

t−h+

ih
n

,

(23)

.

over the time interval of length h/n is therefore:

2

+ 2 µ σ ∆W

h
n

+ σ 2 (∆W )2 .

(24)

As n → ∞ the ﬁrst two terms vanish at a rate higher than the last one. In particular, to a ﬁrst order
approximation the squared return equals the squared return innovation and therefore the squared
return conditional mean and variance are
E r2 |Ft
Var r2 |Ft

≈ σ2

h
n

≈ 2 σ4

(25)
2

h
n

.

(26)

The no-arbitrage condition implies that return innovations are serially uncorrelated. Thus, summing over i = 1, . . . , n we obtain
n

E RV (t, h, n)| Ft =

E r t−h+
i=1

ih h
,
n n

n

2

| Ft

≈

σ t−h+
i=1
t

n

Var r t − h +
i=1

ih h
,
n n

| Ft

≈

2σ t − h +
i=1

≈ 2

h
n
(27)

t−h
n

2

2

σ(s)2 ds

≈
Var RV (t, h, n)| Ft =

(i − 1)h
n

h
n

t

(i − 1)h
n

σ(s)4 ds .

4

h
n

2

(28)

t−h

Equation (27) illustrates that realized volatility is an unbiased estimator of the return quadratic
variation, while equation (28) shows that the estimator is consistent as its variance shrinks to zero
when we increase the sampling frequency n and keep the time interval h ﬁxed. Taken together,
these results suggest that RV is a powerful and model-free measure of the return quadratic variation.
Eﬀectively, RV gives practical empirical content to the latent volatility state variable underlying the
models previously discussed in Section III.2.
Two issues complicate the practical application of the convergence results illustrated in equations
(27) and (28). First, a continuum of instantaneous return observations must be used for the conditional
variance in equation (28) to vanish. In practice, only a discrete price record is observed, and thus an
inevitable discretization error is present. Barndorﬀ-Nielsen and Shephard [52] develop an asymptotic
theory to assess the eﬀect of this error on the RV estimate (see also Meddahi [209]). Second, market
microstructure eﬀects (e.g., price discreteness, bid-ask spread positioning due to dealer inventory control, and bid-ask bounce) contaminate the return observations, especially at the ultra-high frequency.

10
These eﬀects tend to generate spurious correlations in the return series which can be partially eliminated by ﬁltering the data prior to forming the RV estimates. However, this strategy is not a panacea
and much current work studies the optimal sampling scheme and the construction of improved realized
volatility in the presence of microstructure noise. This growing literature is surveyed by Hansen and
Lunde [165], Bandi and Russell [46], McAleer and Medeiros [205], and Andersen and Benzoni [14].
Recent notable contributions to this literature include Bandi and Russell [45], Barndorﬀ-Nielsen et
al. [49], Diebold and Strasser [121], and Zhang, Mykland, and A¨
ıt-Sahalia [262]. Related, there is the
issue of how to construct RV measures when the market is rather illiquid. One approach is to use a
lower sampling frequency and focus on longer-horizon RV measure. Alternatively the literature has
explored volatility measures that are more robust to situations in which the noise-to-signal ratio is
high, e.g., Alizadeh et al. [8], Brandt and Diebold [72], Brandt and Jones [73], Gallant et al. [151],
Garman and Klass [157], Parkinson [221], Schwert [237], and Yang and Zhang [259] consider the highlow price range measure. Dobrev [122] generalizes the range estimator to high-frequency data and
shows its link with RV measures.
Equations (27) and (28) also underscore an important diﬀerence between RV and other volatility
measures. RV is an ex-post model-free estimate of the quadratic variation process. This is in contrast
to ex-ante measures which attempt to forecast future quadratic variation using information up to
current time. The latter class includes parametric GARCH-type volatility forecasts as well as forecasts
built from stochastic volatility models through, e.g., the Kalman ﬁlter (e.g., Harvey and Shephard
[168], Harvey et al. [167]), the particle ﬁlter (e.g., Johannes and Polson [185, 186]) or the reprojection
method (e.g., Gallant and Long [152] and Gallant and Tauchen [155]).
More recently, other studies have pursued more direct time-series modeling of volatility to obtain
alternative ex-ante forecasts. For instance, Andersen et al. [21] follow an ARMA-style approach,
extended to allow for long memory features, to model the logarithmic foreign exchange rate realized
volatility. They ﬁnd the ﬁt to dominate that of traditional GARCH-type models estimated from daily
data. In a related development, Andersen, Bollerslev, and Meddahi [24, 25] exploit the general class
of Eigenfunction Stochastic Volatility (ESV) models introduced by Meddahi [208] to provide optimal
analytic forecast formulas for realized volatility as a function of past realized volatility. Other scholars
have pursued more general model speciﬁcations to improve forecasting performance. Ghysels et al.
[159] consider Mixed Data Sampling (MIDAS) regressions that use a combination of volatility measures
estimates at diﬀerent frequencies and horizons. Related, Engle and Gallo [137] exploit the information
in diﬀerent volatility measures, captured by a multivariate extension of the multiplicative error model
suggested by Engle [136], to predict multi-step volatility. Finally, Andersen et al. [20] build on the
Heterogeneous AutoRegressive (HAR) model by Barndorﬀ-Nielsen and Shephard [50] and Corsi [110]
and propose a HAR-RV component-based regression to forecast the h-steps ahead quadratic variation:
RV (t + h, h) = β 0 + β D RV (t, 1) + β W RV (t, 5) + β M RV (t, 21) + ε(t + h) .

(29)

Here the lagged volatility components RV (t, 1), RV (t, 5), and RV (t, 21) combine to provide a parsimonious approximation to the long-memory type behavior of the realized volatility series, which
has been documented in several studies (e.g., Andersen et al. [19]). Simple OLS estimation yields
consistent estimates for the coeﬃcients in the regression (29), which can be used to forecast volatility
out of sample.

11
As mentioned previously, the convergence results illustrated in equations (27) and (28) stem from
the theory of semi-martingales under conditions more general than those underlying the continuoustime diﬀusion in equation (22). For instance, these results are robust to the presence of discontinuities
in the return path as in the jump-diﬀusion SV model (15). In this case the realized volatility measure
(21) still converges to the return quadratic variation, which is now the sum of the diﬀusive integrated
volatility IV and the cumulative squared jump component:
J(s)2 .

QV (t, h) = IV (t, h) +

(30)

t−h≤s≤t

The decomposition in equation (30) motivates the quest for separate estimates of the two quadratic
variation components, IV and squared jumps. This is a fruitful exercise in forecasting applications,
since separate estimation of the two components increases predictive accuracy (e.g., Andersen et al.
[20]). Further, this decomposition is relevant for derivatives pricing, e.g., options are highly sensitive
to jumps as well as large moves in volatility (e.g., Pan [220] and Eraker [141]).
A consistent estimate of integrated volatility is the k-skip bipower variation, BV (e.g., BarndorﬀNielsen and Shephard [53]),
π
BV (t, h; k, n) =
2

n

|r t − h +
i=k+1

ih h
,
n n

||r t − h +

(i − k)h h
,
n
n

|.

(31)

Liu and Maheu [202] and Forsberg and Ghysels [147] show that realized power variation, which is
robust to the presence of jumps, can improve volatility forecasts. A well-known special case of (31)
is the ‘realized bipower variation,’ which has k = 1 and is denoted BV (t, h; n) ≡ BV (t, h; 1, n). We
can combine bipower variation with the realized volatility RV to obtain a consistent estimate of the
squared jump component, i.e.,
J(s)2 .

RV (t, h; n) − BV (t, h; n) −→ QV (t, h) − IV (t, h) =
n→∞

(32)

t−h≤s≤t

The result in equation (32) are useful to design tests for the presence of jumps in volatility, e.g.,
Andersen et al. [20], Barndorﬀ-Nielsen and Shephard [53, 54], Huang and Tauchen [172], and Mizrach
[217]. More recently, alternative approaches to test for jumps have been developed by A¨
ıt-Sahalia and
Jacod [6], Andersen et al. [23], Lee and Mykland [195], and Zhang [261].

V. Applications
The power of the continuous-time paradigm has been evident ever since the work by Merton [212] on
intertemporal portfolio choice, Black and Scholes [63] on option pricing, and Vasicek [255] on bond
valuation. However, the idea of casting these problems in a continuous-time diﬀusion context goes
back all the way to the work in 1900 by Bachelier [32].
Merton [213] develops a continuous-time general-equilibrium intertemporal asset pricing model
which is later extended by Cox et al. [112] to a production economy. Because of its ﬂexibility and
analytical tractability, the Cox et al. [112] framework has become a key tool used in several ﬁnancial

12
applications, including the valuation of options and other derivative securities, the modeling of the
term structure of risk-free interest rates, the pricing of foreign currencies and defaultable bonds.
Volatility has played a central role in these applications. For instance, an option’s payoﬀ is nonlinear in the price of the underlying asset and this feature renders the option value highly sensitive
to the volatility of underlying returns. Further, derivatives markets have grown rapidly in size and
complexity and ﬁnancial institutions have been facing the challenge to manage intricate portfolios
exposed to multiple risk sources. Risk management of these sophisticated positions hinges on volatility
modeling. More recently, the markets have responded to the increasing hedging demands of investors
by oﬀering a menu of new products including, e.g., volatility swaps and derivatives on implied volatility
indices like the VIX. These innovations have spurred an even more pressing need to accurately measure
and forecast volatility in ﬁnancial markets.
Research has responded to these market developments. We next provide a brief illustrative overview
of the recent literature dealing with option pricing and term structure modeling, with an emphasis on
the role that volatility modeling has played in these two key applications.

V.1. Options
Rubinstein [233] and Bates [55], among others, note that prior to the 1987 market crash the Black and
Scholes [63] (BS) formula priced option contracts quite accurately whereas after the crash it has been
systematically underpricing out-of-the-money equity-index put contracts. This feature is evident from
Figure 1, which is constructed from options on the S&P 500 futures. It shows the implied volatility
function for near-maturity contracts traded both before and after October 19, 1987 (‘Black Monday’).
The mild u-shaped pattern prevailing in the pre-crash implied volatilities is labeled a ‘volatility smile,’
in contrast to the asymmetric post-1987 ‘volatility smirk.’ Importantly, while the steepness and level
of the implied volatility curve ﬂuctuate day to day depending on market conditions, the curve has been
asymmetric and upward sloping ever since 1987, so the smirk remains in place to the current date,
e.g., Benzoni et al. [60]. In contrast, before the crash the implied volatility curve was invariably ﬂat
or mildly u-shaped as documented in, e.g., Bates [57]. Finally, we note that the post-1987 asymmetric
smirk for index options contrasts sharply with the pattern for individual equity options, which possess
ﬂat or mildly u-shaped implied volatility curves (e.g., Bakshi et al. [37] and Bollen and Whaley [65]).
Given the failures of the BS formula, much research has gone into relaxing the underlying assumptions. A natural starting point is to allow volatility to evolve randomly, inspiring numerous studies
that examine the option pricing implications of SV models. The list of early contributions includes
Hull and White [174], Johnson and Shanno [187], Melino and Turnbull [211], Scott [238], Stein [244],
Stein and Stein [245], and Wiggins [256]. Here we focus in particular on the Hull and White [174]
model,
dp(t) = µp dt + V (t) dWp (t)
dV (t)
= µV dt + σ V dWV (t) ,
V (t)

(33)
(34)

where Wp and WV are standard Brownian motions. In general, shocks to returns and volatility may
be (negatively) correlated, however for tractability Hull and White assume ρ = corr(dWp , dWV ) = 0.

13

14−Oct−1987

12−Oct−1988

27

27
Put IVs
Call IVs

Put IVs
Call IVs
26

25

25

24

24

23

23

22

22

21

21

20

20

19

19

18

Black−Scholes IV (percent per year)

26

18

17
−0.15

−0.1

−0.05

0
0.05
K/S−1

0.1

17
−0.15

0.15

−0.1

−0.05

0
0.05
K/S−1

0.1

0.15

Figure 1: Pre- and Post-1987 Crash Implied Volatilities. The plots depict Black-Scholes implied
volatilities computed from near-maturity options on the S&P500 futures on October 14, 1986 (the
week before the 1987 market crash) and a year later.
Under this assumption they show that, in a risk-neutral world, the premium C HW on a European call
option is the Black and Scholes price C BS evaluated at the average integrated variance V ,
V =

1
T −t

T

V (s)ds ,

(35)

t

integrated over the distribution h(V |V (t)) of V :
C HW ( p(t), V (t) ) =

C BS (V )h(V |V (t))dV .

(36)

The early eﬀorts to identify a more realistic probabilistic model for the underlying return were
slowed by the analytical and computational complexity of the option pricing problem. Unlike the BS
setting, the early SV speciﬁcations do not admit closed-form solutions. Thus, the evaluation of the
option price requires time-consuming computations through, e.g., simulation methods or numerical
solution of the pricing partial diﬀerential equation by ﬁnite diﬀerence methods. Further, the presence of
a latent factor, volatility, and the lack of closed-form expressions for the likelihood function complicate
the estimation problem.
Consequently, much eﬀort has gone into developing restrictions for the distribution of the underlying return process that allow for (semi) closed-form solutions and are consistent with the empirical
properties of the data. The ‘aﬃne’ class of continuous-time models has proven particularly useful

14
in providing a ﬂexible, yet analytically tractable, setting. Roughly speaking, the deﬁning feature of
aﬃne jump-diﬀusions is that the drift term, the conditional covariance term, and the jump intensity
are all a linear-plus-constant (aﬃne) function of the state vector. The Vasicek [255] bond valuation
model and the Cox et al. [112] intertemporal asset pricing model provide powerful examples of the
advantages of the aﬃne paradigm.
To illustrate the progress in option pricing applications built on aﬃne models, consider the return
dynamics
dp(t) = µ dt +

V (t) dWp (t) + ξ p (t) dq(t)

dV (t) = κ(V − V (t)) dt + σ V

V (t) dWV (t) + ξ V (t) dq(t) ,

(37)
(38)

where Wp and WV are standard Brownian motions with non-zero correlation ρ = corr(dWp , dWV ), q
is a Poisson process, uncorrelated with Wp and WV , with jump intensity
λ(t) = λ0 + λ1 V (t) ,

(39)

that is, Prob(dqt = 1) = λ(t) dt. The jump amplitudes variables ξ p and ξ V have distributions
ξ V (t) ; exp(ξ V )
ξ p (t) | ξ V (t) ; N (ξ p + ρξ ξ V (t), σ 2 ) .
p

(40)
(41)

Here volatility is not only stochastic but also subject to jumps which occur simultaneously with jumps
in the underlying return process. The Black and Scholes model is a special case of (37)-(41) for
constant volatility, V (t) = σ 2 , 0 ≤ t ≤ T , and no jumps, λ(t) = 0, 0 ≤ t ≤ T . The Merton [214] model
arises from (37)-(41) if volatility is constant but we allow for jumps in returns.
More recently, Heston [170] has considered a special case of (37)-(41) with stochastic volatility but
without jumps. Using transform methods he derives a European option pricing formula which may be
evaluated readily through simple numerical integration. His SV model has GARCH-type features, in
that the variance is persistent and mean reverts at a rate κ to the long-run mean V . Compared to Hull
and White’s [174] setting, Heston’s model allows for shocks to returns and volatility to be negatively
correlated, i.e., ρ < 0, which creates a leverage-type eﬀect and skews the return distribution. This
feature is consistent with the properties of equity index returns. Further, a fatter left tail in the
return distribution results in a higher cost for crash insurance and therefore makes out-of-the-money
put options more expensive. This is qualitatively consistent with the patterns in implied volatilities
observed after the 1987 market crash and discussed above.
Bates [56] has subsequently extended Heston’s approach to allow for jumps in returns and using
similar transform methods he has obtained a semi-closed form solution for the option price. The
addition of jumps provides a more realistic description of equity returns and has important option
pricing implications. With diﬀusive shocks (e.g., stochastic volatility) alone a large drop in the value of
the underlying asset over a short time span is very unlikely whereas a market crash is always possible
as long as large negative jumps can occur. This feature increases the value of a short-dated put option,
which oﬀers downside protection to a long position in the underlying asset.
Finally, Duﬃe et al. [130] have introduced a general model with jumps to volatility which embeds
the dynamics (37)-(41). In model (37)-(41), the likelihood of a jump to occur increases when volatility

15
is high (λ1 > 0) and a jump in returns is accompanied by an outburst of volatility. This is consistent
with what is typically observed during times of market stress. As in the Heston case, variance is
persistent with a mean reversion coeﬃcient κ towards its diﬀusive long-run mean V , while the total
long-run variance mean is the sum of the diﬀusive and jump components. In the special case of constant
jump intensity, i.e., λ1 = 0, the total long-run mean is V +ξ V λ0 /κ. The jump term (ξ V (t) dq(t)) fattens
the right tail of the variance distribution, which induces leptokurtosis in the return distribution. Two
eﬀects generate asymmetrically distributed returns. The ﬁrst channel is the diﬀusive leverage eﬀect,
i.e., ρ < 0, the second is the correlation between the volatility and the jump amplitude of returns
generated through the coeﬃcient ρξ . Taken together, these eﬀects increase model-implied option
prices and help produce a realistic volatility smirk.
Several empirical studies rely on models of the form (37)-(41) in option-pricing applications. For
instance, Bates [56] uses Deutsche Mark options to estimate a model with stochastic volatility and
constant-intensity jumps to returns, while Bates [57] ﬁts a jump-diﬀusion model with two SV factors
to options on S&P 500 futures. In the latter case, the two SV factors combine to help capture features
of the long-run memory in volatility while retaining the analytical tractability of the aﬃne setting
(see, e.g., Christoﬀersen et al. [101] for another model with similar features). Alternative approaches
to model long memory in continuous-time SV models rely on the fractional Brownian motion process,
e.g., Comte and Renault [108] and Comte et al. [107], while Breidt et al. [76], Harvey [166] and Deo et
al. [118] consider discrete-time SV models (see Hurvich et al. [175] for a review). Bakshi et al. [34, 37]
estimate a model similar to the one introduced by Bates [56] using S&P 500 options.
Other scholars rely on underlying asset return data alone for estimation. For instance, Andersen et
al. [15] and Chernov et al. [95] use equity-index returns to estimate jump-diﬀusion SV models within
and outside the aﬃne (37)-(41) class. Eraker et al. [142] extend this analysis and ﬁt a model that
includes constant-intensity jumps to returns and volatility.
Finally, another stream of work examines the empirical implications of SV jump-diﬀusions using a
joint sample of S&P 500 options and index returns. For example, Benzoni [59], Chernov and Ghysels
[93], and Jones [189] estimate diﬀerent ﬂavors of the SV model without jumps. Pan [220] ﬁts a model
that has jumps in returns with time-varying intensity, while Eraker [141] extends Pan’s work by adding
jumps in volatility.
Overall, this literature has established that the SV jump-diﬀusion model dramatically improves
the ﬁt of underlying index returns and options prices compared to the Black and Scholes model.
Stochastic volatility alone has a ﬁrst-order eﬀect and jumps further enhance model performance by
generating fatter tails in the return distribution and reducing the pricing error for short-dated options.
The beneﬁts of the SV setting are also signiﬁcant in hedging applications.
Another aspect related to the speciﬁcation of SV models concerns the pricing of volatility and
jump risks. Stochastic volatility and jumps are sources of uncertainty. It is an empirical issue to
determine whether investors demand to be compensated for bearing such risks and, if so, what the
magnitude of the risk premium is. To examine this issue it is useful to write model (37)-(41) in socalled risk-neutral form. It is common to assume that the volatility risk premium is proportional to
the instantaneous variance, η(t) = η V V (t). Further, the adjustment for jump risk is accomplished by
ξ
assuming that the amplitude ˜p (t) of jumps to returns has mean ˜p = ξ p + η p . These speciﬁcations
ξ
are consistent with an arbitrage-free economy. More general speciﬁcations can also be supported in a

16
general equilibrium setting, e.g., a risk adjustment may apply to the jump intensity λ(t). However, the
coeﬃcients associated to these risk adjustments are diﬃcult to estimate and to facilitate identiﬁcation
they typically are ﬁxed at zero. Incorporating such risk premia in model (37)-(41) yields the following
risk-neutral return dynamics (e.g., Pan [220] and Eraker [141]):
dp(t) = (r − µ∗ ) dt +

V (t) dWp (t) + ˜p (t) dq(t)
ξ

dV (t) = [ κ(V − V (t)) + η V V (t) ] dt + σ V

V (t) dWV (t) + ξ V (t) dq(t) ,

(42)
(43)

where r is the risk-free rate, µ∗ a jump compensator term, Wp and WV are standard Brownian motions
under this so-called Q measure, and the risk-adjusted jump amplitude variable ˜p is assumed to follow
ξ
the distribution,
˜ (t) | ξ (t) ; N (˜ + ρ ξ (t), σ 2 ) .
ξp
ξp
(44)
V
ξ V
p
Several studies estimate the risk-adjustment coeﬃcients η V and η p for diﬀerent speciﬁcations of
model (37)-(44); see, e.g., Benzoni [59], Broadie et al. [78], Chernov and Ghysels [93], Eraker [141],
Jones [189], and Pan [220]. It is found that investors demand compensation for volatility and jump
risks and these risk premia are important for the pricing of index options. This evidence is reinforced
by other studies examining the pricing of volatility risk using less structured but equally compelling
procedures. For instance, Coval and Shumway [111] ﬁnd that the returns on zero-beta index option
straddles (i.e., combinations of calls and puts that have oﬀsetting covariances with the index) are
signiﬁcantly lower than the risk-free return. This evidence suggests that in addition to market risk at
least a second factor (likely, volatility) is priced in the index option market. Similar conclusions are
obtained by Bakshi and Kapadia [36], Buraschi and Jackwerth [79], and Broadie et al. [78].

V.2. Risk-Free Bonds and their Derivatives
The market for (essentially) risk-free Treasury bonds is liquid across a wide maturity spectrum. It
turns out that no-arbitrage restrictions constrain the allowable dynamics in the cross-section of bond
yields. Much work has gone into the development of tractable dynamic term structure models capable
of capturing the salient time-series properties of interest rates while respecting such cross-sectional noarbitrage conditions. The class of so-called ‘aﬃne’ dynamic term structure models provides a ﬂexible
and arbitrage-free, yet analytically tractable, setting for capturing the dynamics of the term structure
of interest rates. Following Duﬃe and Kan [129], Dai and Singleton [114, 115], and Piazzesi [227],
the short term interest rate, y0 (t), is an aﬃne (i.e., linear-plus-constant) function of a vector of state
variables, X(t) = {xi (t), i = 1, . . . , N }:
N

y0 (t) = δ 0 +

δ i xi (t) = δ 0 + δ X X(t) ,

(45)

i=1

where the state-vector X has risk-neutral dynamics
˜ ˜
dX(t) = K(Θ − X(t))dt + Σ S(t)dW (t) .

(46)

˜
˜
In equation (46), W is an N -dimensional Brownian motion under the so-called Q-measure, K and Θ
are N × N matrices, and S(t) is a diagonal matrix with the ith diagonal element given by [S(t)]ii =

17
αi + β i X(t). Within this setting, the time-t price of a zero-coupon bond with time-to-maturity τ is
given by
P (t, τ ) = eA(τ )−B(τ ) X(t) ,
(47)
where the functions A(τ ) and B(τ ) solve a system of ordinary diﬀerential equations (ODEs); see, e.g.,
Duﬃe and Kan [129]. Semi-closed form solutions are also available for bond derivatives, e.g., bond
options as well as caps and ﬂoors (e.g., Duﬃe et al. [130]).
In empirical applications it is important to also establish the evolution of the state vector X under
the physical probability measure P, which is linked to the Q-dynamics (46) through a market price of
risk, Λ(t). Following Dai and Singleton [114] the market price of risk is often given by
Λ(t) =

S(t)λ ,

(48)

where λ is an N × 1 vector of constants. More recently, Duﬀee [127] proposed a broader ‘essentially
aﬃne’ class, which retains the tractability of standard models but, in contrast to the speciﬁcation in
equation (48), allows compensation for interest rate risk to vary independently of interest rate volatility.
This additional ﬂexibility proves useful in forecasting future yields. Subsequent generalization are in
Duarte [124] and Cheridito et al. [92].
Litterman and Scheinkman [201] demonstrate that virtually all variation in U.S. Treasury rates
is captured by three factors, interpreted as changes in ‘level,’ ‘steepness,’ and ‘curvature.’ Consistent
with this evidence, much of the term-structure literature has focused on three-factor models. One
problem with these models, however, is that the factors are latent variables void of immediate economic interpretation. As such, it is challenging to impose appropriate identifying conditions for the
model coeﬃcients and in particular to ﬁnd the ideal representation for the ‘most ﬂexible’ model, i.e.,
the model with the highest number of identiﬁable coeﬃcients. Dai and Singleton [114] conduct an extensive speciﬁcation analysis of multi-factor aﬃne term structure models. They classify these models
into subfamilies according to the number of (independent linear combination of) state variables that
determine the conditional variance matrix of the state vector. Within each subfamily, they proceed
to identify the models that lead to well-deﬁned bond prices (a condition they label ‘admissibility’)
and among the admissible speciﬁcations they identify a ‘maximal’ model that nests econometrically
all others in the subfamily. Joslin [190] builds on Dai and Singleton’s [114] work by pursuing identiﬁcation through a normalization of the drift term in the state vector dynamics (instead of the diﬀusion
term, as in Dai and Singleton [114]). Duﬃe and Kan [129] follow an alternative approach to obtain an
identiﬁable model by rotating from a set of latent state variables to a set of observable zero-coupon
yields. Collin-Dufresne et al. [104] build on the insights of both Dai and Singleton [114] and Duﬃe
and Kan [129]. They perform a rotation of the state vector into a vector that contains the ﬁrst few
components in the Taylor series expansion of the yield curve around a maturity of zero and their
quadratic variation. One advantage is that the elements of the rotated state vector have an intuitive
and unique economic interpretation (such as level, slope, and curvature of the yield curve) and therefore the model coeﬃcients in this representation are identiﬁable. Further, it is easy to construct a
model-independent proxy for the rotated state vector, which facilitates model estimation as well as
interpretation of the estimated coeﬃcients across models and sample periods.
This discussion underscores an important feature of aﬃne term structure models. The dependence
of the conditional factor variance S(t) on one or more of the elements in X introduces stochastic

18
volatility in the yields. However, when a square-root factor is present parametric restrictions (admissibility conditions) need to be imposed so that the conditional variance S(t) is positive over the range
of X. These restrictions aﬀect the correlations among the factors which, in turn, tend to worsen the
cross-sectional ﬁt of the model. Speciﬁcally, CIR models in which S(t) depends on all the elements of
X require the conditional correlation among the factors to be zero, while the admissibility conditions
imposed on the matrix K renders the unconditional correlations non-negative. These restrictions are
not supported by the data. In contrast, constant-volatility Gaussian models with no square-root factors do not restrict the signs and magnitude of the conditional and unconditional correlations among
the factors but they do, of course, not accommodate the pronounced and persistent volatility ﬂuctuations observed in bond yields. The class of models introduced by Dai and Singleton [114] falls between
these two extremes. By including both Gaussian and square-root factors they allow for time-varying
conditional volatilities of the state variables and yet they do not constrain the signs of some of their
correlations. This ﬂexibility helps to address the trade oﬀ between generating realistic correlations
among the factors while capturing the time-series properties of the yields’ volatility.
A related aspect of (unconstrained) aﬃne models concerns the dual role that square-root factors play in driving the time-series properties of yields’ volatility and the term structure of yields.
Speciﬁcally, the time-t yield yτ (t) on a zero-coupon bond with time-to-maturity τ is given by
P (t, τ ) = e−τ yτ (t) .

(49)

Thus, we have
A(τ ) B(τ )
+
X(t) .
(50)
τ
τ
It is typically assumed that the B matrix has full rank and therefore equation (50) provides a direct
link between the state-vector X(t) and the term-structure of bond yields. Further, Itˆ’s Lemma implies
o
that the yield yτ also follows a diﬀusion process:
yτ (t) = −

dyτ (t) = µyτ (X(t), t) dt +

B(τ )
Σ
τ

S(t)dW (t) .

(51)

Consequently, the (instantaneous) quadratic variation of the yield given as the squared yield volatility
coeﬃcient for yτ is
B(τ )
B(τ )
Vyτ (t) =
Σ S(t) Σ
.
(52)
τ
τ
The elements of the S(t) matrix are aﬃne in the state vector X(t), i.e., [S(t)]ii = αi +β i X(t). Further,
invoking the full rank condition on B(τ ), equation (50) implies that each state variable in the vector
X(t) is an aﬃne function of the bond yields Y (t) = {yτ j (t), j = 1, . . . , J }. Thus, for any τ there is a
set of constants aτ , j , j = 0, . . . , J, so that
J

Vyτ (t) = aτ ,0 +

aτ , j yτ j (t) .

(53)

j=1

Hence, the current quadratic yield variation for bonds at any maturity is a linear combination of the
term structure of yields. As such, the market is complete, i.e., volatility is perfectly spanned by a
portfolio of bonds.

19
Collin-Dufresne and Goldstein [103] note that this spanning condition is unnecessarily restrictive
and propose conditions which ensures that volatility no longer directly enters the main bond pricing
equation (47). This restriction, which they term ‘unspanned stochastic volatility’ (USV), eﬀectively
breaks the link between the yields’ quadratic variation and the level of the term structure by imposing
a reduced rank condition on the B(τ ) matrix. Further, since their model is a special (nested) case of
the aﬃne class it retains the analytical tractability of the aﬃne model class. Recently Joslin [190] has
derived more general conditions for aﬃne term structure models to exhibit USV. His restrictions also
produce a market incompleteness (i.e., volatility cannot be hedged using a portfolio of bonds) but do
not constrain the degree of mean reversion of the other state variables so that his speciﬁcation allows
for more ﬂexibility in capturing the persistence in interest rate series. (See also the USV conditions
in the work by Trolle and Schwartz [253]).
There is conﬂicting evidence on the volatility spanning condition in ﬁxed income markets. CollinDufresne and Goldstein (2002) ﬁnd that swap rates have limited explanatory power for returns on atthe-money ‘straddles,’ i.e., portfolios mainly exposed to volatility risk. Similar ﬁndings are in Heidari
and Wu [169], who show that the common factors in LIBOR and swap rates explain only a limited
part of the variation in the swaption implied volatilities. Moreover, Li and Zhao [197] conclude that
some of the most sophisticated multi-factor dynamic term structure models have serious diﬃculties
in hedging caps and cap straddles, even though they capture bond yields well. In contrast, Fan et al.
[143] argue that swaptions and even swaption straddles can be well hedged with LIBOR bonds alone,
supporting the notion that bond markets are complete.
More recently other studies have examined several versions of the USV restriction, again coming
to diﬀerent conclusions. A direct comparison of these results, however, is complicated by diﬀerences in
the model speciﬁcation, the estimation method, and the data and sample period used in the estimation.
Collin-Dufresne et al. [105] consider swap rates data and ﬁt the model using a Bayesian Markov Chain
Monte Carlo method. They ﬁnd that a standard three-factor model generates a time series for the
variance state variable that is essentially unrelated to GARCH estimates of the quadratic variation of
the spot rate process or to implied variances from options, while a four-factor USV model generates
both realistic volatility estimates and a good cross-sectional ﬁt. In contrast, Jacobs and Karoui [177]
consider a longer data set of U.S. Treasury yields and pursue quasi-maximum likelihood estimation.
They ﬁnd the correlation between model-implied and GARCH volatility estimates to be high. However,
when estimating the model with a shorter sample of swap rates, they ﬁnd such correlations to be small
or negative. Thompson [250] explicitly tests the Collin-Dufresne and Goldstein [103] USV restriction
and rejects it using swap rates data. Bikbov and Chernov [62], Han [164], Jarrow et al. [182], Joslin
[191], and Trolle and Schwartz [254] rely on data sets of derivatives prices and underlying interest
rates to better identify the volatility dynamics.
Andersen and Benzoni [12] directly relate model-free realized volatility measures (constructed from
high-frequency U.S. Treasury data) to the cross-section of contemporaneous bond yields. They ﬁnd
that the explanatory power of such regressions is very limited, which indicates that volatility is not
spanned by a portfolio of bonds. The evidence in Andersen and Benzoni [12] is consistent with the
USV models of Collin-Dufresne et al. [105] and Joslin [190], as well as with a model that embeds weak
dependence between the yields and volatility as in Joslin [191]. Moreover, Duarte [125] argues that the
eﬀects of mortgage-backed security hedging activity aﬀects both the interest rate volatility implied by

20
options and the actual interest rate volatility. This evidence suggests that variables that are not in the
span of the term structure of yields and forward rates contribute to explain volatility in ﬁxed income
markets. Also related, Wright and Zhou [258] ﬁnd that adding a measure of market jump volatility
risk to a regression of excess bond returns on the term structure of forward rates nearly doubles the
R2 of the regression. Taken together, these ﬁndings suggest more generally that genuine SV models
are critical for appropriately capturing the dynamic evolution of the term structure.

VI. Estimation Methods
There are a very large number of alternative approaches to estimation and inference for parametric
SV models and we abstain from a thorough review. Instead, we point to the basic challenges that
exist for diﬀerent types of speciﬁcations, how some of these were addressed in the early literature and
ﬁnally provide examples of methods that have been used extensively in recent years. Our exposition
continues to focus on applications to equity returns, interest rates, and associated derivatives.
Many of the original SV models were cast in discrete time, inspired by the popular GARCH
paradigm. In that case, the distinct challenge for SV models is the presence of a strongly persistent
latent state variable. However, more theoretically oriented models, focusing on derivatives applications, were often formulated in continuous time. Hence, it is natural that the econometrically-oriented
literature has moved in this direction in recent years as well. This development provides an added
complication as the continuous-time parameters must be estimated from discrete return data and
without direct observations on volatility. For illustration, consider a fully parametric continuous-time
SV model for the asset return r with conditional variance V and coeﬃcient vector Ψ. Most methods
to estimate Ψ rely on the conditional density f for the data generating process,
f (r(t), V (t) | I(t − 1), Ψ) = fr|V (r(t) | V (t), I(t − 1), Ψ) × fV (V (t) | I(t − 1), Ψ) ,

(54)

where I(t − 1) is the available information set at time t − 1. The main complications are readily
identiﬁed. First, analytic expressions for the discrete-time transition (conditional) density, f , or the
discrete-time moments implied by the data generating process operating in continuous time, are often
unavailable. Second, volatility is latent in SV models, so that even if a closed-form expression for f is
known, direct evaluation of the above expression is infeasible due to the absence of explicit volatility
measures. The marginal likelihood with respect to the observable return process alone is obtained
by integrating over all possible paths for the volatility process, but this integral has a dimension
corresponding to sample size, rendering the approach infeasible in general.
Similar issues are present when estimating continuous-time dynamic term structure models. Following Piazzesi [228], a change of variable gives the conditional density for a zero-coupon yield y on a
bond with time to maturity τ :
f (yτ (t) | I(t − 1), Ψ) = fX (g(yτ (t), Ψ) | I(t − 1), Ψ) × |

y

g(yτ (t), Ψ) | .

(55)

Here the latent state vector X has conditional density fX , the function g(·, Ψ) maps the observable
yield y into X, X(t) = g(yτ (t), Ψ), and y g(yτ (t), Ψ) is the Jacobian determinant of the transformation. Unfortunately, analytic expressions for the conditional density fX are known only in some special

21
cases. Further, the mapping X(t) = g(yτ (t), Ψ) holds only if the model provides an exact ﬁt to the
yields, while in practice diﬀerent sources of error (e.g., model mis-speciﬁcation, microstructure eﬀects,
measurement errors) inject a considerable degree of noise into this otherwise deterministic linkage (for
correct model speciﬁcation) between the state vector and the yields. As such, a good measure of X
might not be available to evaluate the conditional density (55).

VI.1. Estimation via Discrete-Time Model Speciﬁcation or Approximation
The ﬁrst empirical studies have estimated discrete-time SV models via a (Generalized) Method of
Moments procedure by matching a number of theoretical and sample moments, e.g., Chan et al.
[89], Ho et al. [171], Longstaﬀ and Schwartz [204], and Melino and Turnbull [211]. These models
were either explicitly cast in discrete time or were seen as approximate versions of the continuoustime process of interest. Similarly, several authors estimate diﬀusive aﬃne dynamic term structure
models by approximating the continuous-time dynamics with a discrete-time process. If the error
terms are stipulated to be normally distributed, the transition density of the discretized process is
multivariate normal and computation of unconditional moments then only requires knowledge of the
ﬁrst two moments of the state vector. This result facilitates quasi-maximum likelihood estimation. In
evaluating the likelihood function, some studies suggest using closed-form expressions for the ﬁrst two
moments of the continuous-time process instead of the moments of the discretized process (e.g., Fisher
and Gilles [145] and Duﬀee [127]), thus avoiding the associated discretization bias. This approach
typically requires some knowledge of the state of the system which may be obtained, imperfectly,
through matching the system, given the estimated parameter vector, to a set of observed zero-coupon
yields to infer the state vector X. A modern alternative is to use the so-called particle ﬁlter as an
eﬃcient ﬁltering procedure for the unobserved state variables given the estimated parameter vector.
We provide more detailed accounts of both of these procedures later in this section.
Finally, a number of authors develop a simulated maximum likelihood method that exploit the
speciﬁc structure of the discrete-time SV model. Early examples are Danielsson and Richard [117] and
Danielsson [116] who exploit the Accelerated Gaussian Importance Sampler for eﬃcient Monte Carlo
evaluation of the likelihood. Subsequent improvements were provided by Fridman and Harris [149]
and Liesenfeld and Richard [200], with the latter relying on Eﬃcient Importance Sampling (EIS).
In a second step, EIS can also be used for ﬁltering the latent volatility state vector. In general,
these inference techniques provide quite impressive eﬃciency but the methodology is not always easy
to generalize beyond the structure of the basic discrete-time SV asset return model. We discuss the
general inference problem for continuous-time SV models for which the lack of a closed-form expression
for the transition density is an additional complicating factor in a later section.

VI.2. Filtering the Latent State Variable Directly During Estimation
Some early studies focused on direct ways to extract estimates of the latent volatility state variable in
discrete-time SV asset return models. The initial approach was based on quasi-maximum likelihood
(QML) methods exploiting the Kalman ﬁlter. This method requires a transformation of the SV model
to a linear state-space form. For instance, Harvey and Shephard [168] consider a version of the Taylor’s

22
[249] discrete-time SV model,
p(t) = p(t − 1) + β +

V (t)ε(t)

log(V (t)) = α + φ log(V (t − 1)) + η(t) ,

(56)
(57)

where p is the logarithmic price, ε is a zero-mean error term with unit variance, and η is an independentlydistributed error term with zero mean and variance σ 2 .
η
Deﬁne y(t) = p(t) − p(t − 1) − β, square the observations in equation (56), and take logarithms to
obtain the measurement equation,
(t) = ω + h(t) + ξ(t) ,
(58)
where (t) ≡ log y(t)2 , h(t) ≡ log(V (t)). Further, ξ is a zero-mean disturbance term given by ξ(t) =
log(ε(t)2 ) − E[ log(ε(t)2 ) ], ω = log(σ 2 ) + E[ log(ε(t)2 ) ], and σ is a scale constant which subsumes the
eﬀect of the drift term α in equation (57). The autoregression (57) yields the transition equation,
h(t) = φh(t − 1) + η(t) ,

(59)

Taken together, equations (58) and (59) are the linear state-space transformation of the SV model (56)(57). If the joint distribution of ε and η is symmetric, i.e., f (ε, η) = f (−ε, −η), then the disturbance
terms in the state-space form are uncorrelated even if η and ε are not. A possible dependence between
ε and η allows the model to pick up some of the asymmetric behavior often observed in stock returns.
Projection of [ h(t) − Et−1 h(t) ] over [ (t) − Et−1 (t) ] yields the Kalman ﬁlter estimate of the latent
(logarithmic) variance process:
Et h(t) = Et−1 h(t) +

E{ [ h(t) − Et−1 h(t) ] × [ (t) − Et−1 (t) ] }
× [ (t) − Et−1 (t) ] ,
E{ [ (t) − Et−1 (t) ]2 }

(60)

where the conditional expectations Et−1 (t) and Et−1 h(t) are given by:
Et−1 (t) = ω + Et−1 h(t)

(61)

Et−1 h(t) = φ Et−1 h(t − 1) .

(62)

To start the recursion (60)-(62), the initial value E0 h(0) is ﬁxed at the long-run mean log(V ).
Harvey and Shephard [168] estimate the model coeﬃcients via quasi-maximum likelihood, i.e.
by treating the errors ξ and η as though they were normal and maximizing the prediction-error
decomposition form of the likelihood function obtained via the Kalman ﬁlter. Inference is valid as
long as the standard errors are appropriately adjusted. In their application they rely on daily returns
on the value-weighted U.S. market index over 1967-1987 and daily returns for 30 individual stocks over
1974-1983. Harvey et al. [167] pursue a similar approach to ﬁt a multivariate SV model to a sample
of four exchange rate series from 1981 to 1985. One major drawback of the Kalman ﬁlter approach is
that the ﬁnite sample properties can be quite poor because the error term, ξ, is highly non-Gaussian,
see, e.g., Andersen, Chung, and Sørensen [27]. The method may be extended to accommodate various
generalizations including long memory persistence in volatility as detailed in Ghysels, Harvey, and
Renault [158].
A related literature, often exploited in multivariate settings, speciﬁes latent GARCH-style dynamics for a state vector which governs the systematic evolution of a higher dimensional set of asset

23
returns. An early representative of these speciﬁcations is in Diebold and Nerlove [120], who exploit
the Kalman ﬁlter for estimation, while Fiorentina et al. [144] provide a likelihood-based estimation
procedure using MCMC techniques. We later review the MCMC approach in some detail.
The state-space form is also useful to characterize the dynamics of interest rates. Following, e.g.,
Piazzesi [227], for a discrete-time dynamic term structure model the measurement and transition
equations are
A(τ ) B(τ )
+
X(t) + ξ τ (t)
τ
τ
X(t) = µ + ΦX(t − 1) + Σ S(t)ε(t) ,

yτ (t) = −

(63)
(64)

where S(t) is a matrix whose elements are aﬃne functions of the state vector X, and A and B
solve a system of diﬀerence equations. When all the yields are observed with error (i.e., ξ τ = 0 ∀τ ,
0 ≤ τ ≤ T ), QML estimation of the system (63)-(64) via the extended Kalman ﬁlter method yields
an estimate of the coeﬃcient vector. Applications of this approach for the U.S. term structure data
include Campbell and Viceira [81], Gong and Remolona [161], and Pennacchi [225]. The extended
Kalman ﬁlter involves a linear approximation of the relation between the observed data and the state
variables, and the associated approximation error will produce biased estimates. Christoﬀersen et al.
[99] raise this concern and recommend the use of the so-called unscented Kalman ﬁlter for estimation
of systems in which the relation between data and state variables is highly non-linear, e.g., options
data.

VI.3. Methods Accommodating the Lack of a Closed-Form Transition Density
We have so far mostly discussed estimation techniques for models with either a known transition
density or one that is approximated by a discrete-time system. However, the majority of empiricallyrelevant continuous-time models do not possess explicit transition densities and alternative approaches
are necessary. This problem leads us naturally towards the large statistics and econometric literature
on estimation of diﬀusions from discretely-observed data. The vast majority of these studies assume
that all relevant variables are observed so the latent volatility or yield curve state variables, integral to
SV models, are not accounted for. Nonetheless, it may be feasible to extract the requisite estimates of
the state variable by alternate means, thus restoring the feasibility, albeit not eﬃciency, of the basic
approach. Since the literature is large and not directly geared towards genuine SV models, we focus
on methods that have seen use in applications involving latent state variables.
A popular approach is to invert the map between the state vector and a subset of the observables
assuming that the model prices speciﬁc securities exactly. In applications to equity markets this is
done, e.g., by assuming that one option contract is priced without error, which implies a speciﬁc value
(estimate) of the variance process given the model parameters Ψ. For instance, Pan [220] follows this
approach in her study of S&P 500 options and returns, which we review in more detail in Section VI.5.
In applications to ﬁxed income markets it is likewise stipulated that certain bonds are priced without
error, i.e., in equation (63) the error term ξ τ i (t) is ﬁxed at zero for a set of maturities τ 1 , . . . , τ N ,
where N matches the dimension of the state vector X. This approach yields an estimate for the latent
variables through the inverse-map X(t) = g(yτ (t), Ψ).

24
One criticism of the state vector inversion procedure is that it requires ad hoc assumptions regarding the choice of the securities that are error-free (those used to compute model-implied measures
of the state vector) vis-a-vis those observed with error (used either for estimation or to assess model
performance in an ‘out-of-sample’ cross-sectional check). In fact, the extracted state vector can be
quite sensitive to the choice of derivatives (or yields) used. Nevertheless, this approach has intuitive
appeal. Model-implied measures of the state vector, in combination with a closed-form expression
for the conditional density (55), allow for eﬃcient estimation of the coeﬃcient vector Ψ via maximum likelihood. Analytic expressions for fX in equation (55) exist in a limited number of cases. For
instance, if X is Gaussian then fX is multivariate normal, while if X follows a square-root process
then fX can be expressed in terms of the modiﬁed Bessel function (e.g., Cox et al. [113]). Diﬀerent
ﬂavors of these continuous-time models are estimated in, e.g., Chen and Scott [91], Collin-Dufresne
and Solnik [106], Duﬃe and Singleton [132], Jagannathan et al. [181], and Pearson and Sun [223]. In
more general cases, including aﬃne processes that combine Gaussian and square-root state variables,
closed-form expressions for fX are no longer available. In the rest of this section we brieﬂy review
diﬀerent methods to overcome this problem. The interested reader may consult, e.g., Piazzesi [227]
for more details.
Lo [203] warns that the common approach of estimating parameters of an Itˆ process by applyo
ing maximum likelihood to a discretization of the stochastic diﬀerential equation yields inconsistent
estimators. In contrast, he characterizes the likelihood function as a solution to a partial diﬀerential
equation. The method is very general, e.g., it applies not only to continuous-time diﬀusions but also
to jump processes. In practice, however, analytic solutions to the partial diﬀerential equations (via,
e.g., Fourier transforms) are available only for a small class of models so computationally-intensive
methods (e.g., ﬁnite diﬀerencing or simulations) are generally required to solve the problem. This is
a severe limitation in the case of multivariate systems like SV models.
For general Markov processes, where the above solution is infeasible, a variety of procedures have
been advocated in recent years. Three excellent surveys provide diﬀerent perspectives on the issue. A¨
ıtSahalia, Hansen, and Scheinkman [5] discuss operator methods and mention the potential of applying
a time deformation technique to account for genuine SV features of the process, as in Conley, Hansen,
Luttmer, and Scheinkman [109]. In addition, the A¨
ıt-Sahalia [3, 4] closed-form polynomial expansions
for discretely-sampled diﬀusions are reviewed along with the Schaumburg [235] extension to a general
class of Markov processes with L´vy-type generators. Meanwhile, Bibby, Jacobsen, and Sørensen [61]
e
survey the extensive statistics literature on estimating functions for diﬀusion-type models and Bandi
and Phillips [42] explicitly consider dealing with nonstationary processes (see also the work of Bandi
[39], Bandi and Nguyen [41], and Bandi and Phillips [43, 44]).
The characteristic function based inference technique has been particularly widely adopted due to
the natural ﬁt with the exponentially aﬃne model class which provides essentially closed-form solutions
for many pricing applications. Consequently, we dedicate a separate section to this approach.

25
VI.3.1. Characteristic Functions
Singleton [242] proposes to exploit the information contained in the conditional characteristic function
of the state vector X,
φ(iu, X(t), Ψ) = E eiu X(t+1) X(t) ,
(65)
to pursue maximum likelihood estimation of aﬃne term structure models. In equation (65) we highlight
the dependence of the characteristic function on the unknown parameter vector Ψ. When X is an
aﬃne (jump-)diﬀusion process, φ has the exponential aﬃne form,
φ(iu, X(t), Ψ) = eαt (u)+β t (u) X(t) ,

(66)

where the functions α and β solve a system of ODEs. As such, the transition density fX is known
explicitly up to an inverse-Fourier transformation of the characteristic function (65),
fX (X(t + 1) X(t); Ψ) =

1
πN

RN
+

Re e−iu X(t+1) φ(iu, X(t), Ψ) du .

(67)

Singleton shows that Gauss-Legendre quadrature with a relatively small number of quadrature points
allows to accurately evaluate the integral in equation (67) when X is univariate. As such, the method
readily delivers eﬃcient estimates of the parameter vector, Ψ, subject to an auxiliary assumption,
namely that the state vector may be extracted by assuming that a pre-speciﬁed set of security prices
is observed without error while the remainder have non-trivial error terms.
When X is multivariate the Fourier inversion in equation (67) is computationally more demanding. Thus, when estimating multi-dimensional systems Singleton suggests focusing on the conditional
density function of the individual elements of X, but conditioned on the full state vector,
fXj (Xj (t + 1)|X(t); Ψ) =

1
2π

R

e−iωIj X(t+1) φ(iω Ij , X(t), Ψ)dω ,

(68)

where the vector Ij has 1 in the jth element and zero elsewhere so that the jth element of X is
Xj (t + 1) = Ij X(t + 1). Maximization of the likelihood function obtained from fXj , for a ﬁxed j,
will often suﬃce to obtain a consistent estimate of Ψ. Exploiting more than one of the conditional
densities (68) will result in more eﬃcient Ψ estimate. For instance, the scores of multiple univariate
log-likelihood functions, stacked in a vector, yield moment conditions that allow for generalized method
of moment (GMM) estimation of the system. Alternatively, Joslin [191] proposes a change-of-measure
transformation which reduces the oscillatory behavior of the integrand in equation (67). When using
this transformation, Gauss-Hermite quadrature more readily provides a solution to the integral in (67)
even if the state vector X is multi-dimensional, thus facilitating full ML estimation of the system.
Related, several studies have pursued GMM estimation of aﬃne processes using characteristic
functions. Deﬁnition (65) yields the moment condition
E (φ(iu, X(t), Ψ) − eiu X(t+1) )z(u, X(t)) = 0 ,

(69)

where X is an N -dimensional (jump-)diﬀusion, u ∈ RN , and z is an instrument function. When X is
aﬃne, the characteristic function takes the exponential form (66). Diﬀerent choices of u and z yield a
set of moment conditions that can be used for GMM estimation and inference. Singleton [242] derives

26
the optimal instrument in terms of delivering eﬃcient estimates. Carrasco et al. [86] approximate the
optimal instrument with a set of basis functions that do not require the knowledge of the conditional
likelihood function, thus avoiding one of the assumptions invoked by Singleton. Further, they build on
Carrasco and Florens [87] to implement estimation using a continuum of moment conditions, which
yields maximum-likelihood eﬃciency. Other applications of GMM-characteristic function methods to
aﬃne (jump-) diﬀusions for equity index returns are in Chacko and Viceira [88] and Jiang and Knight
[183].
In some cases the lack of closed-form expressions for the moment condition in equation (69) can
hinder GMM estimation. In these cases the expectation in equation (69) can be evaluated by Monte
Carlo integration. This is accomplished by simulating a long sample from the discretized process for a
given value of the coeﬃcient vector Ψ. The parameter Ψ is then estimated via the simulated method
of moments (SMM) of McFadden [206] and Duﬃe and Singleton [131]. Singleton [242] proposes SMM
characteristic function estimators that exploit the special structure of aﬃne term structure models.

VI.4. Eﬃcient Estimation of General Continuous-Time Processes
A number of recent approaches oﬀer excellent ﬂexibility in terms of avoiding approximations to the
continuous-time model-implied transition density while still facilitating eﬃcient estimation of the
evolution of the latent state vector for the system.
VI.4.1. Maximum Likelihood with Characteristic Functions
Bates [58] develops a ﬁltration-based maximum likelihood estimation method for aﬃne processes. His
approach relies on Bayes’ rule to recursively update the joint characteristic function of latent variables
and data conditional on past data. He then obtains the transition density by Fourier inversion of the
updated characteristic function.
Denote with y(t) and X(t) the time-t values of the observable variable and the state vector,
respectively, and let Y (t) ≡ {y(1), . . . , y(t)} be the data observed up to time t. Consider the case in
which the characteristic function of z(t + 1) ≡ (y(t + 1), X(t + 1)) conditional on z(t) ≡ (y(t), X(t)),
is an exponential aﬃne function of X(t):
φ(is, iu, z(t), Ψ) = E eis y(t+1)+iu X(t+1) z(t)
= eα(is,iu,y(t))+β(is,iu,y(t)) X(t) ,

(70)

Next, determine the value of the characteristic function conditional on the observed data Y (t):
φ(is, iu, Y (t), Ψ) = E E eis y(t+1)+iu X(t+1) z(t)

Y (t)

= E eα(is,iu,y(t))+β(is,iu,y(t)) X(t) Y (t)
= eα(is,iu,y(t)) ψ(β(is, iu, y(t)), Y (t), Ψ) ,

(71)

where ψ(iu, Y (t), Ψ) ≡ E eiu X(t) Y (t) denotes the (marginal) characteristic function for the state
vector conditional on the observed data. Fourier inversion then yields the conditional density for the

27
observation y(t + 1) conditional on Y (t):
fy (y(t + 1) | Y (t); Ψ) =

1
2π

e−is y(t+1) φ(is, 0, Y (t), Ψ)ds .

(72)

R

The next step updates the characteristic function ψ (Bartlett [48]):
ψ(iu, Y (t + 1), Ψ) =

1
2πfy (y(t + 1) | Y (t); Ψ)

e−is y(t+1) φ(is, iu, Y (t), Ψ)ds .

(73)

R

To start the recursion, Bates initializes ψ at the unconditional characteristic function of the latent
variable X. The log-likelihood function is then given by
T

log L(Y (T ); Ψ) = log(fy (y(1); Ψ) +

log(fy (y(t) | Y (t − 1); Ψ)) .

(74)

t=2

A nice feature is that the method provides a natural solution to the ﬁltering problem. The ﬁltered
estimate of the latent state X and its variance are computed from the ﬁrst and second derivatives of
the moment generating function ψ(u, Y (t); Ψ) in equation (73), evaluated at u = 0:
E[ X(t + 1) | Y (t + 1); Ψ ] =
Var[ X(t + 1) | Y (t + 1); Ψ ] =

1
2πfy (y(t + 1) | Y (t); Ψ)
1
2πfy (y(t + 1) | Y (t); Ψ)

R

R

e−is y(t+1) φu (is, 0, Y (t); Ψ)ds

(75)

e−is y(t+1) φuu (is, 0, Y (t); Ψ)ds

− {E[ X(t + 1) | Y (t + 1) ]}2 .

(76)

A drawback is that at each step t of the iteration the method requires storage of the entire
characteristic function ψ(iu, Y (t); Ψ). To deal with this issue Bates recommends to approximate the
true ψ with the characteristic function of a variable with a two-parameter distribution. The choice
of the distribution depends on the X-dynamics while the two parameters of the distribution are
determined by the conditional mean E[ X(t + 1) | Y (t + 1); Ψ ] and variance Var[ X(t + 1) | Y (t + 1); Ψ ]
given in equations (75)-(76).
In his application Bates ﬁnds that the method is successful in estimating diﬀerent ﬂavors of the
SV jump-diﬀusion for a univariate series of daily 1953-1996 S&P 500 returns. In particular, he shows
that the method obtains estimates that are equally, if not more, eﬃcient compared to the eﬃcient
method of moments and Markov Chain Monte Carlo methods described below. Extensions of the
method to multivariate processes are theoretically possible, but they require numerical integration of
multi-dimensional functions, which is computationally demanding.
VI.4.2. Simulated Maximum Likelihood
In Section VI.2 we discussed methods for simulated ML estimation and inference in discrete-time SV
models. Pedersen [224] and Santa-Clara [234] independently develop a simulated maximum likelihood
(SML) method to estimate continuous-time diﬀusion models. They divide each interval in between
two consecutive data points Xt+1 and Xt into M sub-intervals of length ∆ = 1/M and they discretize
the X proces using the Euler scheme,
√
(77)
Xt+(i+1)∆ = Xt+i∆ + µ(Xt+i∆ )∆ + Σ(Xt+i∆ ) ∆ εt+(i+1)∆ , i = 0, . . . , M − 1 ,

28
where µ and Σ are the drift and diﬀusion terms of the X process and ε is multivariate normal with mean
zero and identity variance matrix. The transition density of the discretized process is multivariate
normal with mean µ and variance matrix ΣΣ . As ∆ goes to zero, this density converges to that of
the continuous-time process X. As such, the transition density from Xt to Xt+1 is given by
fX (Xt+1 |Xt ; Ψ) =

fX (Xt+1 |Xt+1−∆ ; Ψ)fX (Xt+1−∆ |Xt ; Ψ)dXt+1−∆ .

(78)

For suﬃciently small values of ∆ the ﬁrst term in the integrand, fX (Xt+1 |Xt+1−∆ ; Ψ), is approximated
by the transition density of the discretized process, while the second term, fX (Xt+1−∆ |Xt ; Ψ), is a
multi-step-ahead transition density that can be computed from the recursion from Xt to Xt+1−∆ .
Writing the right-hand side of equation (78) as a conditional expectation yields
fX (Xt+1 |Xt ; Ψ) = EXt+1−∆ |Xt fX (Xt+1 |Xt+1−∆ ; Ψ) .

(79)

The expectation in equation (79) can be computed by Monte Carlo integration over a large number
of paths for the process X, simulated via the Euler scheme (77). As ∆ vanishes, the Euler scheme is
consistent. Thus, when the size of the simulated sample increases the sample average of the function
fX , evaluated at the random draws of Xt+1−∆ , converges to the true transition density. Application
of the principles in Bladt and Sørensen [64] may well be useful in enhancing the eﬃciency of the
simulation scheme and hence the actual eﬃciency of the inference procedure in practice.
Brandt and Santa-Clara [75] apply the SML method to estimate a continuous-time model of the
joint dynamics of interest rates in two countries and the exchange rate between the two currencies.
Piazzesi [228] extends the SML approach for jump-diﬀusion processes with time-varying jump intensity.
She considers a high-frequency policy rule based on yield curve information and an arbitrage-free bond
market and estimates the model using 1994-1998 data on the Federal Reserve target rate, the six-month
LIBOR rate, and swap yields.
An important issue is how to initialize any unobserved component of the state vector, X(t), such as
the volatility state at each observation to provide a starting point for next Monte Carlo integration step.
This may be remedied through application of the particle ﬁlter, as mentioned earlier and discussed
below in connection with MCMC estimation. Another possibility is, as also indicated previously, to
extract the state variable through inversion from derivatives prices or yields assumed observed without
pricing errors.
VI.4.3. Indirect inference
There are also other method-of-moments strategies to estimate ﬁnitely-sampled continuous-time processes of a general type. One prominent approach approximates the unknown transition density for
the continuous-time process with the density of a semi-nonparametric (SNP) auxiliary model. Then
one can use the score function of the auxiliary model to form moment conditions for the parameter
vector Ψ of the continuous-time model. This approach yields the eﬃcient method of moments estimator (EMM) of Gallant and Tauchen [154], Gallant et al. [150], and Gallant and Long [152], and the
indirect inference estimator of Gouri´roux et al. [162] and Smith [243].
e
To ﬁx ideas, suppose that the conditional density for a continuous-time return process r (the
‘structural’ model) is unknown. We intend to approximate the unknown density with a discretetime model (the ‘auxiliary’ model) that is tractable and yet suﬃciently ﬂexible to accommodate the

29
systematic features of the actual data sample well. A parsimonious auxiliary density for r embeds
ARMA and EGARCH leading terms to capture the conditional mean and variance dynamics. There
may be residual excess skewness and kurtosis that elude the ARMA and EGARCH forms. As such,
the auxiliary density is rescaled using a nonparametric polynomial expansion of order K, which yields
gK (r(t)|x(t); ξ) =

[PK (z(t), x(t))]2
2
R [PK (z(t), x(t))] φ(u)du

ν + (1 − ν ) ×

φ(z(t))
√
,
h(t)

(80)

where ν is a small constant, φ(.) is the standard normal density, x(t) contains lagged return observations, and
z(t) =

r(t) − µ(t)
h(t)

,

(81)
s

u

µ(t) = φ0 + c h(t) +

φi r(t − 1) +
i=1

δ i ε(t − 1) ,

(82)

i=1

p

log h(t) = ω +

β i log h(t − 1) +
i=1

(1 + α1 L + ... + αq Lq ) [ θ1 z(t − 1) + θ2 (b(z(t − 1)) −


Kz

Kz

i

PK (z, x) =

ai (x)z =
i=0



i=0

Kx

aij xj  z i ,

a00 = 1 ,

2/π) ] ,

(83)
(84)

|j|=0

Here j is a multi-index vector, xj ≡ (xj1 , . . . , xjM ), and |j | ≡ M jm . The term b(z) is a smooth
1
m=1
M
(twice-diﬀerentiable) function that closely approximates the absolute value operator in the EGARCH
variance equation.
In practice, the representation of PK is given by Hermite orthogonal polynomials. When the order
K of the expansion increases, the auxiliary density will approximate the data arbitrarily well. If the
structural model is indeed the true data generating process, then the auxiliary density will converge to
that of the structural model. For a given K, the QML estimator ˆ for the auxiliary model coeﬃcient
ξ
satisﬁes the score condition
T
1
∂ log gK (r(t) | x(t); ˆ
ξ)
= 0.
(85)
T
∂ξ
t=1

Suppose now that the structural model is correct and Ψ0 is the true value of its coeﬃcient vector.
Consider a series {r(t; Ψ), x(t; Ψ)}, t = 1, . . . , T (T ), simulated from the structural model. Then we
expect that the score condition (85) holds when evaluated by averaging over the simulated returns
rather than over the actual data:
mT (T ) (Ψ0 , ˆ =
ξ)

1
T (T )

T (T )
t=1

∂ log gK (r(t, Ψ0 ) | x(t, Ψ0 ); ˆ
ξ)
≈ 0.
∂ξ

(86)

When T and T (T ) tend to inﬁnity, condition (86) holds exactly.
ˆ
Gallant and Tauchen [154] propose the EMM estimator Ψ deﬁned via
ˆ
Ψ = arg min mT (T ) (Ψ, ˆ WT mT (T ) (Ψ, ˆ ,
ξ) ˆ
ξ)
Ψ

(87)

30
ˆ
where the weighting matrix WT is a consistent estimate of the inverse asymptotic covariance matrix
of the auxiliary score function, e.g., the inverse outer product of the SNP gradient:
1
ˆ −1
WT =
T

T
t=1

∂ log gK (r(t) | x(t); ˆ
ξ)
∂ξ

∂ log gK (r(t) | x(t); ˆ
ξ)
∂ξ

.

(88)

An important advantages of the technique is that EMM estimates achieve the same degree of
eﬃciency as the ML procedure, when the score of the auxiliary model asymptotically spans the score
of the true model. It also delivers powerful speciﬁcation diagnostics that provide guidance in the model
selection. Gallant and Tauchen [154] show that the EMM estimator is asymptotically normal. Further,
under the assumption that the structural model is correctly speciﬁed, they derive a χ2 statistic for
ˆ ξ)
the test of over-identifying restrictions. Gallant et al. [150] normalize the vector mT (T ) (Ψ, ˆ by its
standard error to obtain a vector of score t-ratios. The signiﬁcance of the individual score elements is
often informative of the source of model mis-speciﬁcation, with the usual caveat that failure to capture
one characteristic of the data may result in the signiﬁcance of a moment condition that pertains to
a coeﬃcient not directly related to that characteristic (due to correlation in the moment conditions).
Finally, EMM provides a straightforward solution to the problem of ﬁltering and forecasting the latent
return variance process V , i.e., determining the conditional densities f (V (t) | x(t), Ψ) and f (V (t +
j) | x(t), Ψ), j ≥ 0. This is accomplished through the reprojection method discussed in, e.g., Gallant
and Long [152] and Gallant and Tauchen [155]. In applications to dynamic term structure models,
the same method yields ﬁltered and forecasted values for the latent state variables.
The reprojection method assumes that the coeﬃcient vector Ψ is known. In practice, Ψ is ﬁxed at
ˆ
the EMM estimate Ψ. Then one simulates a sample of returns and latent variables from the structural
model and ﬁts the auxiliary model on the simulated data. This is equivalent to the ﬁrst step of
the EMM procedure except that, in the reprojection step, we ﬁt the auxiliary model assuming the
structural model is correct, rather than using actual data. The conditional density of the auxiliary
model, estimated under the null, approximates the unknown density of the structural model:
ˆ
gK (r(t + j)|x(t); ˜ ≈ f (r(t + j)|x(t); Ψ), j ≥ 0 ,
ξ)

(89)

where ˜ is the QML estimate of the auxiliary model coeﬃcients obtained by ﬁtting the model on
ξ
simulated data. This approach yields ﬁltered estimates and forecasts for the conditional mean and
variance of the return via
ˆ
E r(t + j) | x(t); Ψ

=

ˆ
V ar r(t + j) | x(t); Ψ

=

y gK (y|x(t); ˜ dy ,
ξ)
ˆ
y − E r(t + j) | x(t); Ψ

(90)
2

gK (y|x(t); ˜ dy .
ξ)

(91)

An alternative approach consists in ﬁtting an auxiliary model for the latent variable (e.g., the return
conditional variance) as a function of current and lagged returns. It is straightforward to estimate
such model using data on the latent variable and the associated returns simulated from the structural
ˆ
model with the EMM coeﬃcient Ψ. Also in this case the auxiliary model density approximates the
true one, i.e.,
V
ˆ
gK (V (t + j)|x(t); ˜ ≈ f V (V (t + j)|x(t); ψ), j ≥ 0 .
ξ)
(92)

31
This approach yields a forecast for the conditional variance process,
ˆ
E V (t + j) | x(t); Ψ

=

V
v gK (v|x(t); ˜ dv .
ξ)

(93)

In sum, reprojection is a simulation approach to implement a non-linear Kalman-ﬁlter-type technique,
which yields eﬀective forecasts for the unobservable state vector.
The indirect inference estimator by Gouri´roux et al. [162] and Smith [243] is closely related to the
e
EMM estimator. Indirect inference exploits that the following two quantities should be close when
the structural model is correct and the data are simulated at the true parameter Ψ0 : (i) the QML
estimator ˆ for the auxiliary model computed from actual data; (ii) the QML estimator ˆ
ξ
ξ(Ψ) for the
auxiliary model ﬁtted on simulations from the structural model. Minimizing the distance between ˆ
ξ
ˆ
and ξ(Ψ) in an appropriate metric yields the indirect inference estimator for Ψ. Similar to EMM,
asymptotic normality holds and a χ2 test for over-identifying restrictions is available. However, the
indirect inference approach is computationally more demanding, because ﬁnding the value of Ψ that
minimizes the distance function requires re-estimating the auxiliary model on a diﬀerent simulated
sample for each iteration of the optimization routine. EMM does not have this drawback, since the
EMM objective function is evaluated at the same ﬁtted score at each iteration. Nonetheless, there
may well be circumstances where particular auxiliary models are of primary economic interest and
estimation based on the corresponding moment conditions may serve as a useful diagnostic tool for
model performance in such directions.
Several studies have used EMM to ﬁt continuous-time SV jump-diﬀusion models for equity index
returns, e.g., Andersen et al. [15], Benzoni [59], Chernov and Ghysels [93], and Chernov et al. [94, 95].
Andersen and Lund [28] and Andersen et al. [16] use EMM to estimate SV jump-diﬀusion models for
the short-term interest rate. Ahn et al. [1, 2], Brandt and Chapman [71], and Dai and Singleton [114]
ﬁt diﬀerent ﬂavors of multi-factor dynamic term structure models. Andersen et al. [27] document the
small-sample properties of the eﬃcient method of moments estimator for stationary processes, while
Duﬀee and Stanton [128] study its properties for near unit-root processes.
A. Ronald Gallant and George E. Tauchen at Duke University have prepared well-documented
general-purpose EMM and SNP packages, available for download at the web address ftp.econ.duke.edu
in the directories pub/get/emm and pub/get/snp. In applications it is often useful to customize the
SNP density to allow for a more parsimonious ﬁt of the data under investigation. For instance,
Andersen et al. [15, 16], Andersen and Lund [28], and Benzoni [59] rely on the SNP density (80)-(84).
VI.4.4. Markov Chain Monte Carlo
The MCMC method provides a Bayesian solution to the inference problem for a dynamic asset pricing
model. The approach treats the model coeﬃcient Ψ as well as the vector of latent state variables
X as random variables and computes the posterior distribution f (Ψ, X|Y ), conditional on certain
observable variables Y , predicted by the model. The setting is suﬃciently general to deal with a
wide range of situations. For instance, X and Y can be the (latent) volatility and (observable) return
processes as is the case of an SV model for asset returns. Or X and Y can be the latent state vector
and observable yields in a dynamic term structure model.

32
The posterior distribution f (Ψ, X|Y ) is the main tool to draw inference not only on the coeﬃcient
Ψ but also on the latent vector X. Since f (Ψ, X|Y ) is unknown in closed-form in relevant applications,
MCMC relies on a simulation (a Markov Chain) from the conditional density f (Ψ, X|Y ) to compute
mode, mean, and standard deviations for the model coeﬃcients and state variables via the Monte
Carlo method.
The posterior f (Ψ, X|Y ) is analytically untractable and extremely high-dimensional, so that simulation directly from f (Ψ, X|Y ) is typically infeasible. The MCMC approach hinges on the CliﬀordHammersley theorem, which determines conditions under which the posterior f (Ψ, X|Y ) is uniquely
determined by the marginal posterior distributions f (Ψ|X, Y ) and f (X|Ψ, Y ). In turn, the posteriors
f (Ψ|X, Y ) and f (X|Ψ, Y ) are determined by a set or univariate posterior distributions. Speciﬁcally,
denote with Ψ(i) the ith element of the coeﬃcient Ψ, i = 1, . . . , K, and with Ψ(−i) the vector consisting of all elements in Ψ except for the ith one. Similarly denote with X(t) the tth row of the state
vector, t = 1, . . . , T , and with X(−t) the rest of the vector. Then the Cliﬀord-Hammersley theorem
allows to characterize the posterior f (Ψ, X|Y ) via K + T univariate posteriors,
f (Ψ(i)|Ψ(−i), X, Y ),

i = 1, . . . , K

(94)

f (X(t)|X(−t), Ψ, Y ),

t = 1, . . . , T .

(95)

The construction of the Markov Chain relies on the so-called Gibbs sampler. The ﬁrst step of
the algorithm consists in choosing initial values for the coeﬃcient and the state, Ψ0 and X0 . When
(one of or both) the multi-dimensional posteriors are tractable, the Gibbs sampler generates values
Ψ1 and X1 directly from f (Ψ|X, Y ) and f (X|Ψ, Y ). Alternatively, each element of Ψ1 and X1 is
drawn from the univariate posteriors (94)-(95). Some of these posteriors may also be analytically
intractable or eﬃcient algorithms to draw from these posteriors may not exist. In such cases the
Metropolis-Hastings algorithm ensures that the simulated sample is consistent with the posterior
target distribution. Metropolis-Hastings sampling consists of an accept-reject procedure of the draws
from a ‘proposal’ or ‘candidate’ tractable density, which is used to approximate the unknown posterior
(see, e.g., Johannes and Polson [186]).
Subsequent iterations of Gibbs sampling, possibly in combination with the Metropolis-Hastings
sampling, yield a series of ‘sweeps’ {Ψs , Xs }, s = 1, . . . , S, with limiting distribution f (Ψ, X|Y ).
A long number of sweeps may be necessary to ‘span’ the whole posterior distribution and obtain
convergence due to the serial dependence of subsequent draws of coeﬃcients and state variables.
When the algorithm has converged, additional simulations provide a sample from the joint posterior
distribution.
The MCMC approach has several advantages. First, the inference automatically accounts for
parameter uncertainty. Further, the Markov Chain provides a direct and elegant solution to the
smoothing problem, i.e., the problem of determining the posterior distribution for the state vector X
conditional on the entire data sample, f (X(t) | Y (1), . . . , Y (T ), Ψ), t = 1, . . . , T . The limitation on the
approach is largely that eﬃcient sampling schemes for the posterior distribution must be constructed
for each speciﬁc problem at hand which by nature is case speciﬁc and potentially cumbersome or
ineﬃcient. Nonetheless, following the development of more general simulation algorithms, the method
has proven ﬂexible for eﬃcient estimation of a broad class of important models.

33
One drawback is that MCMC does not deliver an immediate solution to the ﬁltering problem,
i.e., determining f (X(t) | Y (1), . . . , Y (t), Ψ), and the forecasting problem, i.e., determining f (X(t +
j) | Y (1), . . . , Y (t), Ψ), j > 0. However, recent research is overcoming this limitation through the use
of the ‘particle ﬁlter.’ Bayes rule implies
f (X(t + 1) | Y (1), . . . , Y (t + 1), Ψ) ∝ f (Y (t + 1) | X(t + 1), Ψ) f (X(t + 1) | Y (1), . . . , Y (t), Ψ) , (96)
where the symbol ∝ denotes ‘proportional to.’ The ﬁrst density on the right-hand side of equation
(96) is determined by the SV model and it is often known in closed form. In contrast, the second
density at the far-right end of the equation is given by an integral that involves the unknown ﬁltering
density at the prior period, f (X(t) | Y (1), . . . , Y (t), Ψ):
f (X(t + 1) | Y (1), . . . , Y (t), Ψ) =

f (X(t + 1) | X(t), Ψ) f (X(t) | Y (1), . . . , Y (t), Ψ) dX(t) .

(97)

The particle method relies on simulations to construct a ﬁnite set of weights wi (t) and particles X i (t),
i = 1, . . . , N , that approximate the unknown density with a ﬁnite sum,
N

wi (t)δ X i (t) .

f (X(t) | Y (1), . . . , Y (t), Ψ) ≈

(98)

i=1

where the Dirac function δ X i (t) assigns mass one to the particle X i (t). Once the set of weights and
particles are determined, it is possible to re-sample from the discretized distribution. This step yields
a simulated sample {X s (t)}S which can be used to evaluate the density in equation (97) via Monte
s=1
Carlo integration:
f (X(t + 1) | Y (1), . . . , Y (t), Ψ) ≈

1
S

S

f (X(t + 1) | X s (t), Ψ) .

(99)

s=1

Equation (99) solves the forecasting problem while combining formulas (96) and (99) solves the ﬁltering
problem. The challenge in practical application of the particle ﬁlter is to identify an accurate and
eﬃcient algorithm to construct the set of particles and weights. We point the interested reader to
Kim et al. [193], Pitt and Shephard [226], and Johannes and Polson [186] for a discussion on how to
approach this problem.
The usefulness of the MCMC method to solve the inference problem for SV models has been evident
since the early work by Jacquier et al. [179], who develop an MCMC algorithm for the logarithmic SV
model. Jacquier et al. [180] provide extensions to correlated and non-normal error distributions. Kim
et al. [193] and Chib et al. [96] develop simulation-based methods to solve the ﬁltering problem, while
Chib et al. [97] use the MCMC approach to estimate a multivariate SV model. Elerian et al. [135] and
Eraker [140] discuss how to extend the MCMC inference method to a continuous-time setting. Eraker
[140] uses the MCMC approach to estimate an SV diﬀusion process for interest rates, while Jones [188]
estimates a continuous-time model for the spot rate with non-linear drift function. Eraker et al. [142]
estimate an SV jump-diﬀusion process using data on S&P 500 return while Eraker [141] estimates a
similar model using joint data on options and underlying S&P 500 returns. Li et al. [196] allow for
L´vy-type jumps in their model. Collin-Dufresne et al. [104] use the MCMC approach to estimate
e
multi-factor aﬃne dynamic term structure model using swap rates data. Johannes and Polson [185]
give a comprehensive survey of the still ongoing research on the use of the MCMC approach in the
general nonlinear jump-diﬀusion SV setting.

34

VI.5. Estimation from Option Data
Options’ payoﬀs are non-linear functions of the underlying security price. This feature renders options
highly sensitive to jumps in the underlying price and to return volatility, which makes option data
particularly useful to identify return dynamics. As such, several studies have taken advantage of
the information contained in option prices, possibly in combination with underlying return data, to
estimate SV models with or without discontinuities in returns and volatility.
Applications to derivatives data typically require a model for the pricing errors. A common
approach is to posit that the market price of an option, O∗ , normalized by the underlying observed
security price S ∗ , is the sum of the normalized model-implied option price, O/S ∗ , and a disturbance
term ε (e.g., Renault [230]):
O(S ∗ , V, K, τ , Ψ)
O∗
=
+ ε,
(100)
S∗
S∗
where V is the latent volatility state, K is the option strike price, τ is time to maturity, and Ψ is
the vector with the model coeﬃcients. A pricing error ε could arise for several reasons, including
measurement error (e.g., price discreteness), asynchroneity between the derivatives and underlying
price observations, microstructure eﬀects, and perhaps most importantly speciﬁcation error. The
structure imposed on ε depends on the choice of a speciﬁc ‘loss function’ used for estimation (e.g.,
Christoﬀersen and Jacobs [98]). Several studies have estimated the coeﬃcient vector Ψ by minimizing
the sum of the squared option pricing errors normalized by the underlying price S ∗ , as in equation
(100). Others have focused on either squared dollar pricing errors, or squared errors normalized by
the options market price (instead of S ∗ ). The latter approach has the advantage that a $1 error on
an expensive in-the-money option carries less weight than the same error on a cheaper out-of-themoney contract. The drawback is that giving a lot of weight to the pricing errors on short-maturity
deep-out-of-the-money options could bias the estimation results. Finally, the common practice of
expressing option prices in terms of their Black-Scholes implied volatilities has inspired other scholars
to minimize the deviations between Black-Scholes implied volatilities inferred from model and market
prices (e.g., Mizrach [216]). An alternative course is to form a moment-based loss function and follow a
GMM- or SMM-type approach to estimate Ψ. To this end moment conditions stem from distributional
assumptions on the pricing error ε (e.g., E[ε] = 0) or from the scores of a reduced-form model that
approximates the data.
In estimating the model, some researchers have opted to use a panel of options consisting of
contracts with multiple strikes and maturities across dates in the sample period. This choice brings
a wealth of information on the cross-sectional and term-structure properties of the implied volatility
smirk into the analysis. Others rely on only one option price observation per time period, which shifts
the focus to the time-series dimension of the data. Some studies re-estimate the model on a daily
basis rather than seeking a single point estimate for the coeﬃcient Ψ across the entire sample period.
This ad hoc approach produces smaller in-sample pricing errors, which can be useful to practitioners,
but at the cost of concealing speciﬁcation ﬂaws by over-ﬁtting the model, which tends to hurt outof-sample performance. The diﬀerent approaches are in part dictated by the intended use of the
estimated system as practitioners often are concerned with market making and short-term hedging
while academics tend to value stable relations that may form the basis for consistent modeling of the
dominant features of the system over time.

35
Early contributions focus on loss functions based on the sum of squared option pricing errors and
rely entirely on option data for estimation. This approach typically yields an estimate of the model
coeﬃcient Ψ that embeds an adjustment for risk, i.e., return and volatility dynamics are identiﬁed
under the risk-neutral rather than the physical probability measure. For instance, Bates [56] considers
an SV jump-diﬀusion model for Deutsche Mark foreign currency options and estimates its coeﬃcient
vector Ψ via nonlinear generalized least squares of the normalized pricing errors with daily option data
from January 1984 to June 1991. A similar approach is followed by Bates [57] who ﬁts an SV model
with two latent volatility factors and jumps using daily data on options on the S&P 500 futures from
January 1988 to December 1993. Bakshi et al. [34] focus on the pricing and hedging of daily S&P
500 index options from June 1988 to May 1991. In their application they re-calibrate the model on
a daily basis by minimizing the sum of the squared dollar pricing errors across options with diﬀerent
maturities and strikes. Huang and Wu [173] explore the pricing implications of the time-changed
L´vy process by Carr and Wu [84] for daily S&P 500 index options from April 1999 to May 2000.
e
Their L´vy return process allows for discontinuities that exhibit higher jump frequencies compared
e
to the ﬁnite-intensity Poisson jump processes in equations (37)-(41). Further, their model allows for
a random time change, i.e., a monotonic transformation of the time variable which generates SV in
the diﬀusion and jump components of returns. In contrast, Bakshi et al. [35] ﬁt an SV jump-diﬀusion
model by SMM using daily data on long-maturity S&P 500 options (LEAPS).
More recent studies have relied on joint data on S&P 500 option prices and underlying index
returns, spanning diﬀerent periods, to estimate the model. This approach forces the same model to
price securities in two diﬀerent markets and relies on information from the derivatives and underlying
securities to better pin down model coeﬃcients and risk premia. For instance, Eraker [141] and Jones
[189] ﬁt diﬀerent ﬂavors of the SV model (with and without jumps, respectively) by MCMC. Pan
[220] follows a GMM approach to estimate an SV jump-diﬀusion model using weekly data. She relies
on a single at-the-money option price observation each week, which identiﬁes the level of the latent
volatility state variable (i.e., at each date she ﬁxes the error term ε at zero and solves equation (100) for
V ). A¨
ıt-Sahalia and Kimmel [7] apply A¨
ıt-Sahalia’s [4] method to approximate the likelihood function
for a joint sample of options and underlying prices. Chernov and Ghysels [93] and Benzoni [59] obtain
moment conditions from the scores of a SNP auxiliary model. Similarly, other recent studies have
found it useful to use joint derivatives and interest rate data to ﬁt dynamic term structure models,
e.g., Almeida et al. [9], and Bikbov and Chernov [62].
Finally, a diﬀerent literature has studied the option pricing implications of a model in which asset
return volatility is a deterministic function of the asset price and time, e.g., Derman and Kani [119],
Dupire [134], Rubinstein [233], and Jackwerth and Rubinstein [176]. Since volatility is not stochastic
in this setting, we do not review these models here and point the interested reader to, e.g., Dumas et
al. [133] for an empirical analysis of their performance.

VII. Future Directions
In spite of much progress in our understanding of volatility new challenges lie ahead. In recent years
a wide array of volatility-sensitive products has been introduced. The market for these derivatives
has rapidly grown in size and complexity. Research faces the challenge to price and hedge these new

36
products. Moreover, the recent developments in model-free volatility modeling have eﬀectively given
empirical content to the latent volatility variable, which opens the way for a new class of estimation
methods and speciﬁcation tests for SV systems. Related, improved volatility measures enable us to
shed new light on the properties and implications of the volatility risk premium. Finally, more work
is needed to better understand the linkage between ﬂuctuations in economic fundamentals and lowand high-frequency volatility movements. We conclude this chapter by brieﬂy reviewing some open
issues in these four areas of research.

VII.1. Volatility and Financial Markets Innovation
Volatility is a fundamental input to any ﬁnancial and real investment decision. Markets have responded
to investors’ needs by oﬀering an array of volatility-linked instruments. In 1993 the Chicago Board
Option Exchange (CBOE) has introduced the VIX index, which measures the market expectations of
near-term volatility conveyed by equity-index options. The index was originally computed using the
Black-Scholes implied volatilities of eight diﬀerent S&P 100 option (OEX) series so that, at any given
time, it represented the implied volatility of a hypothetical at-the-money OEX option with exactly 30
days to expiration (see Whaley [257]). On September 22, 2003, the CBOE began disseminating price
level information using a revised ‘model-free’ method for the VIX index. The new VIX is given by the
price of a portfolio of S&P 500 index options and incorporates information from the volatility smirk
by using a wider range of strike prices rather than just at-the-money series (see Britten-Jones and
Neuberger [77]). On March 26, 2004, trading in futures on the VIX Index started on the CBOE Futures
Exchange (CFE) while on February 24, 2006, options on the VIX began trading on the Chicago Board
Options Exchange. These developments have opened the way for investors to trade on option-implied
measures of market volatility. The popularity of the VIX prompted the CBOE to introduce similar
indices for other markets, e.g., the VXN NASDAQ 100 Volatility Index.
Along the way, a new over-the-counter market for volatility derivatives has also rapidly grown in
size and liquidity. Volatility derivatives are contracts whose payments are expressed as functions of
realized variance. Popular examples are variance swaps, which at maturity pay the diﬀerence between
realized variance and a ﬁxed strike price. According to estimates by BNP Paribas reported by the
Risk [192] magazine, the daily trading volume for variance swaps on indices reached $4-5 million in
vega notional (measured in dollars per volatility point) in 2006, which corresponds to payments in
excess of $1 billion per percentage point of volatility on an annual basis (Carr and Lee [82]). Using
variance swaps hedge fund managers and proprietary traders can easily place huge bets on market
volatility.
Finally, in recent years credit derivatives markets have evolved in complexity and grown in size.
Among the most popular credit derivatives are the credit default swaps (CDS), which provide insurance
against the risk of default by a particular company. The buyer of a single-name CDS acquires the
right to sell bonds issued by the company at face value when a credit event occurs. Multiple-name
contracts can be purchased simultaneously through credit indices. For instance, the CDX indices
track the credit spreads for diﬀerent portfolios of North American companies while the iTraxx Europe
indices track the spreads for portfolios of European companies. At the end of 2006 the notional amount
of outstanding over-the-counter single- and multi-name CDS contracts stood at $19 and $10 trillion,

37
respectively, according to the September 2007 Bank for International Settlements Quarterly Review.
These market developments have raised new interesting issues for research to tackle. The VIX
computations based on the new model-free deﬁnition of implied volatility used by the CBOE requires
the use of options with strike prices that cover the entire support of the return distribution. In
practice, liquid options satisfying this requirement often do not exist and the CBOE implementation
introduces random noise and systematic error into the index (Jiang and Tian [184]). Related, the VIX
implementation entails a truncation, i.e., the CBOE discards illiquid option prices with strikes lying
in the tails of the return distribution. As such, the notion of the VIX is more directly linked to that
of corridor volatility (Andersen and Bondarenko [26]). In sum, robust implementation of a model free
measure of implied volatility is still an open area of research. Future developments in this direction
will also have important repercussions on the hedging practices for implied-volatility derivatives.
Pricing and hedging of variance derivatives is another active area of research. Variance swaps
admit a simple replication strategy via static positions in call and put options on the underlying asset,
similar to model-free implied volatility measures (e.g., Britten-Jones and Neuberger [77] and Carr and
Madan [83]). In contrast, it is still an open area of research to determine the replication strategy for
derivatives whose payoﬀs are non-linear function of realized variance, e.g., volatility swaps, which pay
the square-root of realized variance, or call and put options on realized variance. Carr and Lee [82] is
an interesting paper in this direction.
Limited liability gives shareholders the option to default on the ﬁrm’s debt obligation. As such,
a debt claim has features similar to a short position in a put option. The pricing of corporate debt
is therefore sensitive to the volatility of the ﬁrms’ assets: higher volatility increases the probability
of default and therefore reduces the price of debt and increases credit spreads. The insights and
techniques developed in the SV literature could prove useful in credit risk modeling and applications
(e.g., Jacobs and Li [178], Tauchen and Zhou [248] and Zhang et al. [260]).

VII.2. The Use of Realized Volatility for Estimation of SV Models
Another promising line of research aims at extracting the information in RV measures for the estimation of dynamic asset pricing models. Early work along these lines includes Barndorﬀ-Nielsen and
Shephard [51], who decompose RV into actual volatility and realized volatility error. They consider
a state-space representation for this decomposition and apply the Kalman ﬁlter to estimate diﬀerent
ﬂavors of the SV model. Moreover, Bollerslev and Zhou [68] and Garcia et al. [156] build on the
insights of Meddahi [210] to estimate SV diﬀusion models using conditional moments of integrated
volatility. More recently, Todorov [252] generalizes the analysis for the presence of jumps.
Related, recent studies have started to use RV measures to test the implications of models previously estimated with lower-frequency data. Since RV gives empirical content to the latent quadratic
variation process, this approach allows for a direct test of the model-implied restrictions on the latent volatility factor. Recent work along these lines includes Andersen and Benzoni [12], who use
model-free RV measures to show that the volatility spanning condition embedded in some aﬃne term
structure models is violated in the U.S. Treasury market. Christoﬀersen et al. [100] note that the
Heston square-root SV model implies that the dynamics for the standard deviation process are conditionally Gaussian. They reject this condition by examining the distribution of the changes in the

38
square-root RV measure for S&P 500 returns.

VII.3. Volatility Risk Premium
More work is needed to better understand the link between asset return volatility and model risk
premia. Also in this case, RV measures are a fruitful source of information to shed new light on the
issue. Among the recent studies that pursue this venue is Bollerslev et al. [66], who exploit the moments
of RV and option-implied volatility to gauge a measure of the volatility risk premium. Todorov [251]
explores the variance risk premium dynamics using high-frequency S&P 500 index futures data and
data on the VIX index. He ﬁnds the variance risk premium to vary signiﬁcantly over time and to
increase during periods of high volatility and immediately after big jumps in underlying returns. Carr
and Wu [85] provide a broader analysis of the variance risk premium for ﬁve equity indices and 35
individual stocks. They ﬁnd the premium to be large and negative for the indices while it is much
smaller for the individual stocks. Further, they also ﬁnd the premium to increase (in absolute value)
with the level of volatility. Additional work on the volatility risk premium embedded in individual
stock options is in Bakshi and Kapadia [36], Driessen et al. [123], and Duarte and Jones [126]. Other
studies have examined the linkage between volatility risk premia and equity returns (e.g., Bollerslev
and Zhou [69]) and hedge-fund performance (e.g., Bondarenko [70]). New research is also examining
the pricing of aggregate volatility risk in the cross-section of stock returns. For instance, Ang et al. [30]
ﬁnd that average returns are lower on stocks that have high sensitivities to innovations in aggregate
volatility and high idiosyncratic volatility (see also the related work by Ang et al. [31], Bandi et al.
[?], Chen [90], and Guo et al. [163]). This evidence is consistent with the ﬁndings of the empirical
option pricing literature, which suggests that there is a negative risk premium for volatility risk.
Intuitively, periods of high market volatility are associated to worsened investment opportunities and
tend to coincide with negative stock market returns (the so-callled leverage eﬀect). As such, investors
are willing to pay higher prices (i.e., accept lower expected returns) to hold stocks that do well in
high-volatility conditions.

VII.4. Determinants of Volatility
Finally, an important area of future research concerns the linkage between asset return volatility and
economic uncertainty. Recent studies have proposed general equilibrium models that produce lowfrequency ﬂuctuations in conditional volatility, e.g., Campbell and Cochrane [80], Bansal and Yaron
[47], McQueen and Vorkink [207], and Tauchen [246]. Related, Engle and Rangel [139] and Engle et
al. [138] link macroeconomic variables and long-run volatility movements. It is still an open issue,
however, to determine the process through which news about economic fundamentals are embedded
into prices to generate high-frequency volatility ﬂuctuations. Early research by Schwert [236] and
Shiller [241] has concluded that the amplitude of the ﬂuctuations in aggregate stock volatility is diﬃcult
to explain using simple models of stock valuation. Further, Schwert [236] notes that while aggregate
leverage is signiﬁcantly correlated with volatility, it explains a relatively small part of the movements
in stock volatility. Moreover, he ﬁnds little evidence that macroeconomic volatility (measured by
inﬂation and industrial production volatility) helps predict future asset return volatility. Model-free
realized volatility measures are a useful tool to further investigate this issue. Recent work in this

39
direction includes Andersen et al. [22] and Andersen and Bollerslev [17], who explore the linkage
between news arrivals and exchange rates volatility, and Andersen and Benzoni [13], who investigate
the determinants of bond yields volatility in the U.S. Treasury market. Related, Balduzzi et al.
[38] and Fleming and Remolona [146] study the reaction of trading volume, bid-ask spread, and price
volatility to macroeconomic surprises in the U.S. Treasury market, while Brandt and Kavajecz [74] and
Pasquariello and Vega [222] focus instead on the price discovery process and explore the implications
of order ﬂow imbalances (excess buying or selling pressure) on day-to-day variation in yields.

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Meredith A. Crowley

WP-07-21

Health Capital and the Prenatal Environment:
The Effect of Maternal Fasting During Pregnancy
Douglas Almond and Bhashkar Mazumder

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
Maude Toussaint-Comeau

WP-07-24

A Leverage-based Model of Speculative Bubbles
Gadi Barlevy

WP-08-01

Displacement, Asymmetric Information and Heterogeneous Human Capital
Luojia Hu and Christopher Taber

WP-08-02

BankCaR (Bank Capital-at-Risk): A credit risk model for US commercial bank charge-offs
Jon Frye and Eduard Pelz

WP-08-03

Bank Lending, Financing Constraints and SME Investment
Santiago Carbó-Valverde, Francisco Rodríguez-Fernández, and Gregory F. Udell

WP-08-04

Global Inflation
Matteo Ciccarelli and Benoît Mojon

WP-08-05

Scale and the Origins of Structural Change
Francisco J. Buera and Joseph P. Kaboski

WP-08-06

Inventories, Lumpy Trade, and Large Devaluations
George Alessandria, Joseph P. Kaboski, and Virgiliu Midrigan

WP-08-07

School Vouchers and Student Achievement: Recent Evidence, Remaining Questions
Cecilia Elena Rouse and Lisa Barrow

WP-08-08

4

Working Paper Series (continued)
Does It Pay to Read Your Junk Mail? Evidence of the Effect of Advertising on
Home Equity Credit Choices
Sumit Agarwal and Brent W. Ambrose

WP-08-09

The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
Sumit Agarwal and Robert Hauswald

WP-08-10

Consumer Choice and Merchant Acceptance of Payment Media
Wilko Bolt and Sujit Chakravorti

WP-08-11

Investment Shocks and Business Cycles
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

WP-08-12

New Vehicle Characteristics and the Cost of the
Corporate Average Fuel Economy Standard
Thomas Klier and Joshua Linn

WP-08-13

Realized Volatility
Torben G. Andersen and Luca Benzoni

WP-08-14

Revenue Bubbles and Structural Deficits: What’s a state to do?
Richard Mattoon and Leslie McGranahan

WP-08-15

The role of lenders in the home price boom
Richard J. Rosen

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
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder

WP-08-20

Public Investment and Budget Rules for State vs. Local Governments
Marco Bassetto

WP-08-21

Why Has Home Ownership Fallen Among the Young?
Jonas D.M. Fisher and Martin Gervais

WP-09-01

Why do the Elderly Save? The Role of Medical Expenses
Mariacristina De Nardi, Eric French, and John Bailey Jones

WP-09-02

Using Stock Returns to Identify Government Spending Shocks
Jonas D.M. Fisher and Ryan Peters

WP-09-03

5

Working Paper Series (continued)
Stochastic Volatility
Torben G. Andersen and Luca Benzoni

WP-09-04

6
Full text of Working Papers (Federal Reserve Bank of Chicago) : Stochastic Volatility, Working Paper 2009-04

FRASER
