In The Last Two Decades, option pricing has witnessed an explosion of new models that each relax some of the restrictive Black-Scholes (BS) (1973) assumptions. Examples include (i) the stochastic-interest-rate option models of Merton (1973) and Amin and Jarrow (1992); (ii) the jump-diffusion/pure jump models of Bates (1991), Madan and Chang (1996), and Merton (1976); (iii)theconstant-elasticity-of-variance model of Cox and Ross (1976);(iv) the Markovian models of Rubinstein (1994) and Aït-Sahalia and Lo (1996);(v)the stochastic-volatilitymodels of Heston (1993), Hull and White (1987a), Melino and Turnbull (1990, 1995), Scott (1987), Stein and Stein (1991), and Wiggins (1987); (vi)the stochastic-volatilityandstochastic-interest-ratesmodels of Amin and Ng (1993), Bailey and Stulz (1989), Bakshi and Chen (1997a,b), and Scott (1997); and (vii) the stochastic-volatility jump-diffusion models of Bates(1996a, c), and Scott (1997). This list is by no means exhaustive, yet already overwhelming to anyone who has to choose among the alternatives. To make matters worse, the number of possible option pricing models is virtually infinite. Note that every option pricing model has to make three basic assumptions: the underlying price process (the distributional assumption), the interest rate process, and the market price of factor risks. For each of the assumptions, there are many possible choices. For instance, the underlying price can follow either a continuous-time or a discrete-time process. Among possible continuous-time processes, it can be Markov or non-Markov, a diffusion or a nondiffusion, a Poisson or a non-Poisson jump process, a mixture of jump and diffusion components with or without stochastic volatility and with or without random jumps. For the term structure of interest rates, there are similarly many choices. While the search for that perfect option pricing model can be endless, we are tempted to ask: What do we gain from each generalized feature? Is the gain, if any, from a more realistic feature worth the additional complexity or implementational costs? Can any of the relaxed assumptions help resolve known empirical biases associated with the Black-Scholes formula, such as the volatility smiles (e.g., Rubinstein (1985, 1994))? As a practical matter, that perfectly specified option pricing model is bound to be too complex for applications. Ultimately, it is a choice among misspecified models, made perhaps based on (i) “which is the least misspecified?” (ii) “which results in the lowest pricing errors?” and (iii) “which achieves the best hedging performance?” These empirical questions must be answered before the potential of recent advances in theory can be fully realized in practical applications.

The purpose of the present article is to fill in this gap and conduct a comprehensive empirical study on the relative merits of competing option pricing models.^{1} To this goal, we first develop in closed form an implementable option pricing model that admits stochastic volatility, stochastic interest rates, and random jumps, which will be abbreviated as the *SVSI-J model*. The setup is rich enough to contain almost all the known closed form option formulas as special cases, including (i) the Black-Scholes (BS) model, (ii) the stochastic-interest-rate (SI) model, (iii) the stochastic-volatility (SV) model, (iv) the stochastic-volatility and stochastic-interest-rate (SVSI) model, and (v) the stochastic-volatility random-jump (SVJ) model. The constant-volatility jump-diffusion models of Bates (1991) and Merton (1976) are special cases of the SVJ. Consequently, we concentrate our efforts on the SVSI-J and the five models just described.

Besides the obvious normative reasons, a common motivation for these new models is the abundant empirical evidence that the benchmark BS formula exhibits strong pricing biases across both moneyness and maturity (i.e., the “smile”) and that it especially underprices deep out-of-the-money puts and calls (see Bates (1996b) for an insightful review). Such evidence is clearly indicative of implicit stock return distributions that are negatively skewed with higher kurtosis than allowable in a BS log-normal distribution. Guided by this implication, the search for alternative models has mostly focused on finding the “right” distributional assumption. The SV model, for instance, offers a flexible distributional structure in which the correlation between volatility shocks and underlying stock returns serves to control the level of skewness and the volatility variation coefficient serves to control the level of kurtosis. But, since volatility in the SV is modeled as a diffusion and hence only allowed to follow a continuous sample path, its ability to internalize enough short-term kurtosis and thus to price short-term options properly is limited (unless the variation coefficient of spot volatility is unreasonably high). The jump-diffusion models, on the other hand, assert that it is the occasional, discontinuous jumps and crashes that cause the negative implicit skewness and high implicit kurtosis to exist in option prices. The fact that such jumps and crashes are allowed to be discontinuous over time makes these models more flexible than the diffusion-stochastic-volatility model, in internalizing the desired return distributions, especially at short time horizons. Therefore, the random-jump and the stochastic-volatility features can in principle improve the pricing and hedging of, respectively, short-term and relatively long-term options. The inclusion of a stochastic term structure model in an option pricing framework is, however, intended to improve the valuation and discounting of future payoffs, rather than to enhance the flexibility of permissible return distributions. Thus, while the stochastic-interest-rate feature is not expected to help resolve the cross-sectional pricing biases, it should in principle improve the pricing fit across option maturity.

We implement every model by backing out, on each day, the spot volatility and structural parameters from the observed option prices of that day. This approach is common in the existing literature (e.g., Bates (1996b)), partly out of the consideration that historical data reflect what happened in the past whereas information implicit in option prices is forward-looking. Backing out the BS model's volatility and other model's parameters daily is indeed ad hoc since volatility in the BS and the structural parameters in the other models are assumed to be constant over time. But, as this internally inconsistent treatment is how each model is to be applied, we follow this convention so as to ensure each model an equal chance.

In judging the alternative models, we employ three yardsticks. First, are the implied structural parameters consistent with those implicit in the relevant times-series data (e.g., the implied-volatility time series, and the interest-rate time series)? Much of this part of the discussion is based on Bates (1996a,c) work where he studies the relative desirability of the SV versus the SVJ models, using stock index futures and currency options. The reasoning is that if an option model is correctly specified, its structural parameters implied by option prices will necessarily be consistent with those implicit in the observed time-series data. Second, out-of-sample pricing errors give a direct measure of model misspecification. In particular, while a more complex model will generally lead to better in-sample fit, it will not necessarily perform better out of sample as any overfitting may be penalized. Third, hedging errors measure how well a model captures the dynamic properties of option and underlying security prices. In other words, in-sample and out-of-sample pricing errors reflect a model's *static* performance, while hedging errors reflect the model's *dynamic* performance. As shown later, these three yardsticks serve distinct purposes.

Based on 38,749 S&P 500 call option prices from June 1988 to May 1991, we find that the SI and the SVSI-J models do not significantly improve the performance of the BS and the SVJ models, respectively. To keep the presentation manageable, we focus on the four models of distinct interest: the BS, the SV, the SVSI, and the SVJ. Our empirical investigation leads to the following overall conclusions. First, judged on internal parameter consistency, all models are misspecified, with the SVJ the least and the BS the most misspecified. This conclusion is confirmed from several different angles. For example, according to the Rubinstein (1985) type of implied-volatility graphs, the SVJ implied volatility smiles the least across moneyness levels, followed in increasing order by the SVSI, the SV, and the BS. Second, out-of-sample pricing errors are the highest for the BS, the second highest for the SV, and the lowest for the SVJ. Overall, stochastic volatility alone achieves the first-order pricing improvement and typically reduces the BS pricing errors by 25 percent to 60 percent. However, our evidence also confirms the conjectures that (i) adding the random-jump feature improves the fit of short-term options and that (ii) including the SI feature enhances the pricing fit of long-term options. After both stochastic volatility and random jumps are modeled, the remaining pricing errors no longer exhibit clear systematic biases (e.g., across moneyness).

Two types of hedging strategy are employed to gauge the relative hedging effectiveness. First, we examine minimum-variance hedges of option contracts that rely on the underlying asset as the single hedging instrument. As argued by Ross (1995), the need for this type of hedge may arise in contexts where a perfect delta-neutral hedge may not be feasible, either because of untraded risks or because of model misspecifications and transaction costs. In the presence of more than one source of risk, single-instrument hedges can only be partial. According to results from these type of hedges, the SV outperforms all the others, while the SVJ is second. Between the other two models, the BS hedges in-the-money calls better than the SVSI, but the SVSI is better in hedging out-of-the-money calls. This hedging result is surprising as one would expect the SVSI to perform at least as well as the BS, and the SVJ to do better than the SV.

Next, we implement a conventional delta-neutral hedge, in which as many hedging instruments as there are risk sources are used to make the net position completely risk-immunized (locally). For the case of the BS, this means that only the underlying stock will be employed to hedge a call. For the SV model, however, both the price risk and volatility risk affect the value of a call, implying that an SV-based delta-neutral hedge will need a position in the underlying stock and one in a second option contract. For the SVSI, its delta-neutral hedge will involve a discount bond (to control for interest rate risk) in addition to the underlying stock and a second option contract. When such internally consistent hedges are implemented, the hedging errors for the SV, the SVSI and the SVJ are about 50 percent to 65 percent lower than those of the BS model, if each hedge is rebalanced daily. Furthermore, changing the hedge rebalancing frequency affects the BS model's hedging errors dramatically, while only affecting the other models' performance marginally. That is, after stochastic volatility is controlled for, the errors of a delta-neutral hedge seem to be relatively insensitive to revision frequency.^{2} However, like in the single-instrument hedging case, once stochastic volatility is modeled, adding the SI or the random-jump feature does not enhance hedging performance any further.

Since the delta-neutral hedge for the BS does not use a second option contract whereas it does for the other three models, this may have biased the delta-neutral hedging results against the BS model. To examine this point, we also implement the ad hoc BS delta-plus-vega neutral strategy in which the underlying stock and an option contract are used to neutralize both delta risk and vega risk (of the BS model). It turns out that in hedging out-of-the-money and at-the-money calls, this BS delta-plus-vega neutral strategy performs no worse than the other models' delta-neutral hedges. Only in hedging deep in-the-money calls do the stochastic volatility models perform better than the BS delta-plus-vega neutral strategy. This is true regardless of hedge revision frequency. Overall, hedging performance is relatively insensitive to model misspecification, since even ad hoc hedges can result in similar errors.

The rest of the article proceeds as follows. Section I develops the option pricing models. Section II provides a description of the S&P 500 option data. In Section III we present an estimation procedure, discuss the estimated parameters, and evaluate the in-sample fit of each model. Section IV assesses the extent of each model's misspecification. Sections V and VI, respectively, present the out-of-sample pricing and the hedging results. Concluding remarks are offered in Section VII. Proof of pricing equations and most formulas are provided in the Appendix.