The Econometrics of Option Pricing
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The Econometrics of Option Pricing¤
Ren?e Garcia
Universit?e de Montr?eal, CIRANO, CIREQ
Eric Ghysels
University of North Carolina and CIRANO
?E
ric Renault
Universit?e de Montr?eal, CIRANO, CIREQ
First draft: November 2001
This version: August 1, 2003
Keywords: Stock PriceDynamics, Multivariate Jump-Di?usionModels, Latent variables, Stochastic
Volatility, Objective and Risk Neutral Distributions, Nonparametric Option Pricing, Discretetime
Option Pricing Models, Risk Neutral Valuation, Preference-free Option Pricing.
JEL Classiˉcation: C1,C5,G1
¤Address for correspondence: D?epartement de Sciences ?E conomiques, Universit?e de Montr?eal, C.P.
6128, Succ. Centre-Ville, Montr?eal, Qu?ebec, H3C 3J7, Canada. The ˉrst and the last authors gratefully
acknowledge ˉnancial support from the Fonds de la Formation de Chercheurs et l'Aide μa la Recherche du
Qu?ebec (FCAR), the Social Sciences and Humanities Research Council of Canada (SSHRC), the Network
of Centres of Excellence MITACS and the Institut de Finance Math?ematique de Montr?eal (IFM2). The
second author thanks CIRANO for ˉnancial support.
1 Introduction and overview
The growth of the option pricing literature parallels the spectacular developments of derivative
securities and the rapid expansion of markets for derivatives in the last three decades.
Writing a survey of option pricing models appears therefore like a formidable task. To
delimit our focus we will put emphasis on the more recent contributions since there are
already a number of surveys that cover the earlier literature. For example, Bates (1996b)
wrote an excellent review, discussing many issues involved in testing option pricing models.
Ghysels, Harvey and Renault (1996) and Shephard (1996) provide a detailed analysis of
stochastic volatility modelling, while Renault (1997) explores the econometric modelling
of option pricing errors. More recently, Sundaresan (2000) surveys the performance of
continuous-time methods for option valuation. The material we cover obviously has many
seminal contributions that pre-date the most recent work. Needless to say that due credit
will be given to the seminal contributions related to the general topic of estimating and
testing option pricing models. A last introductory word of caution: our survey deals almost
exclusively with studies that have considered modelling the return process of a stock index
and determining the price of European options written on this index.
One of the main advances that marked the econometrics of option pricing in the last
ˉve years has been the use of price data on both the underlying asset and options to jointly
estimate the parameters of the process for the underlying and the risk premia associated
with the various sources of risk. Even if important progress has been made regarding
econometric procedures, the lesson that can be drawn from the numerous investigations,
both parametric and nonparametric, in continuous time or in discrete time, is that the
empirical performance still leaves much room for improvement. The empirical option pricing
literature has revealed a considerable divergence between the risk-neutral distributions
estimated from option prices after the 1987 crash and conditional distributions estimated
from time series of returns on the underlying index. Three facts clearly stand out. First,
the implied volatility extracted from at-the-money options di?ers substantially from the
realized volatility over the lifetime of the option. Second, risk neutral distributions feature
substantial negative skewness which is revealed by the asymmetric implied volatility curves
when plotted against moneyness. Third, the shape of these volatility curves changes over
time and maturities, in other words the skewness and the convexity are time-varying and
maturity-dependent. Our survey will therefore explore possible explanations for the divergence
between the objective and the risk neutral distributions. Modelling of the dynamics
of the underlying asset price is an important part of the puzzle, while another essential
element is the existence of time-varying risk premia. The last issue stresses the potentially
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explicit role to be played by preferences in the pricing of options, a departure from the
central tenet of the preference-free paradigm.
The main approach to modelling stock returns at the time prior surveys were written,
was a continuous time stochastic volatility (henceforth SV) di?usion process possibly
augmented with an independent jump process in returns. Heston (1993) proposed a SV
di?usion model for which one could derive analytically an option pricing formula. Soon
thereafter, see e.g. Du±e and Kan (1996), it was realized that Heston's model belonged to
a rich class of a±ne jump di?usion processes for which one could obtain similar results.
Du±e, Pan and Singleton (2000) discuss equity and ˉxed income derivatives pricing for
the general class of a±ne jump di?usions. The evidence regarding the empirical ˉt of the
a±ne class of processes is mixed, see e.g. Dai and Singleton (2000), Chernov, Gallant,
Ghysels and Tauchen (2003), Ghysels and Ng (1998) for further discussion. There is a
consensus that single volatility factor models, a±ne (like Heston (1993)) or non-a±ne (like
Hull and White (1987) or Wiggins (1987)), do not ˉt the data (see Andersen, Benzoni and
Lund (2002), Benzoni (1998), Chernov, Gallant, Ghysels and Tauchen (2003), Pan (2002),
among others). How to expand single volatility factor di?usions to mimic the data generating
process remains unsettled. Several authors augmented a±ne SV di?usions with jumps,
see Andersen, Benzoni and Lund (2001), Bates (1996a), Chernov, Gallant, Ghysels and
Tauchen (2003), Eraker, Johannes and Polson (2001), Pan (2002), among others. Bakshi,
Cao and Chen (1997), Bates (2000) Chernov, Gallant, Ghysels and Tauchen (2003) and
Pan (2002) show, however, that SV models with jumps in returns are not able to capture all
the empirical features of observed option prices and returns. Bates (2000) and Pan (2002)
argue that the speciˉcation of the volatility process should include jumps, possibly correlated
with the jumps in returns. Chernov, Gallant, Ghysels and Tauchen (2003) maintain
that a two-factor non-a±ne logarithmic SV di?usion model without jumps yields a superior
empirical ˉt compared to a±ne one-factor or two factor SV processes, or SV di?usions
with jumps. Alternative models were also proposed in recent years: they include volatility
models of the Ornstein-Uhlenbeck type but with L?evy innovations (Barndor?-Nielsen and
Shephard, 2001) and stochastic volatility models with long memory in volatility (Breidt,
Crato and de Lima (1998)) and Comte and Renault (1998)).
The statistical ˉt of the underlying process and the econometric complexities associated
with it should not be the only concern, however. An important issue for option pricing is
whether or not the models deliver closed-form solutions. We will therefore discuss if and
when there exists a trade-o? between obtaining a good empirical ˉt or a closed-form option
pricing formula.The dynamics of the underlying fundamental asset cannot be related to
option prices without additional assumptions or information. One possibility is to assume
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that the risks associated with stochastic volatility or jumps are idiosyncratic and not priced
by the market. There is a long tradition of this, but most recent empirical work clearly
indicates there are prices for volatility and jump risk (see e.g. Andersen, Benzoni and Lund
(2002), Chernov and Ghysels (2000), Pan (2002), among others). One can simply set values
for these premia and use the objective parameters to derive implications for option prices as
in Andersen, Benzoni and Lund (2001). A more informative exercise is to use option prices
to calibrate the parameters under the risk neutral process given some version of a nonlinear
least-squares procedure as in Bakshi, Cao and Chen (1997) and Bates (2000). An even more
ambitious program is to use both the time series data on stock returns and the panel data
on option prices to characterize the dynamics of returns with stochastic volatility and with
or without jumps as in Chernov and Ghysels (2000), Pan (2002), Poteshman (2000) and
Garcia, Lewis and Renault (2001).
Whether one estimates the objective probability distribution, the risk neutral or both,
there are many challenges in estimating the parameters of di?usions. The presence of latent
volatility factors make maximum likelihood estimation computationally infeasible. This is
the area where probably the most progress has been made in the last few years. Several
methods have been designed for the estimation of continuous time dynamic state-variable
models with the pricing of options as a major application. Simulation-based methods have
been most successful in terms of empirical implementations. That includes the indirect
inference and e±cient methods of moments of Gouri?eroux, Monfort and Renault (1993)
and Gallant and Tauchen (1996) respectively, and several procedures discussed by Johannes
and Polson (2002) as well as A?3t-Sahalia, Hansen and Scheinkeman (2002) in thisHandbook.
Another approach is to use implied state methods. While Pastorello, Patilea and Renault
(2003) base an indirect inference approach on Black-Scholes implied volatilities, Pan (2002)
uses the Fourier transform to derive a set of moment conditions pertaining to implied states.
Renault and Touzi (1996), Patilea and Renault (1997) and Renault (1997) propose iterative
and recursive procedures which extend the EM (expectation-maximization) methodology
to maximum likelihood contexts where it usually does not apply. Pastorello, Patilea and
Renault (2003) propose a general methodology of iterative and recursive estimation in
structural non-adaptive models which nests all the previous implied state approaches.
Nonparametric methods have also been used extensively. Several studies aimed at
recovering the risk-neutral probabilities or state-price densities implicit in option or stock
prices. For instance, Rubinstein (1996) proposed an implied binomial tree methodology to
recover risk-neutral probabilities which are consistent with a cross-section of option prices.
A?3t-Sahalia and Lo (1998) use a kernel estimator of the volatility function in a Black-
Scholes type model. Stutzer (1996) uses an approach called canonical valuation which
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uses past return data and possibly but not necessarily option price data to estimate the
payo? distribution at expiration. Another approach consists in estimating directly the
option pricing function with nonparametric methods. Hutchinson, Lo and Poggio (1994),
Broadie, Detemple, Ghysels and Torrμes (2000a,b), and Garcia and Gen?cay (2000) follow
this route. An important issue with the model-free nonparametric approaches is that the
recovered risk-neutral probabilities are not always positive and one may consider adding
constraints on the pricing function or the state-price densities. For example, A?3t-Sahalia
and Duarte (2003) impose monotonicity and convexity restrictions using a nonparametric
method based on locally polynomial estimators.
Bates (2000), among others, shows that risk-neutral distributions recovered from option
prices before and after the crash of 1987 are fundamentally di?erent whereas the objective
distributions do not show such structural changes. Before the crash, both the risk neutral
and the actual distributions look roughly lognormal. After the crash, the risk-neutral
distribution is left skewed and leptokurtic. A possible explanation for the di?erence is
a large change in the risk aversion of the average investor. Since risk aversion can be
recovered empirically from the risk neutral and the actual distributions, A?3t-Sahalia and
Lo (2000), Jackwerth (2000) and Rosenberg and Engle (2002) estimate preferences for
the representative investor using simultaneously S&P500 returns and options prices for
contracts on the index. Preferences are recovered based on distance criteria between the
model risk neutral distribution and the risk neutral distribution implied by option price
data.
Another approach of recovering preferences is to set up a representative agent model and
estimate the preference parameters from the ˉrst-order conditions using a GMM approach.
While this has been extensively done with stock and Treasury bill return data (see Hansen
and Singleton (1982), Epstein and Zin (1991) among others), it is only recently that Garcia,
Luger and Renault (2003) estimated preference parameters in a recursive utility framework
using option prices. In this survey we will discuss under which statistical framework option
pricing formulas are preference-free and risk-neutral valuation relationships (Brennan, 1979)
hold in a general stochastic discount factor framework (Hansen and Richard (1987)). When
these statistical restrictions do not hold, it will be shown that preferences play a role. Bates
(2001) argues that the overall industrial organization of the stock index option markets is
not compatible with the idealized construct of a representative agent. He therefore proposes
an equilibrium analysis with investor heterogeneity.
Apart from statistical model ˉtting, there are a host of other issues pertaining to the
implementation of models in practice. A recent survey by Bates (2003) provides an overview
of the issues involved in empirical option pricing, especially the questions surrounding data
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selection, estimation or calibration of the model and presentation of results.
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