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Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes JING-ZHI HUANG Penn State University and New York University LIUREN WU Fordham University This version: March 30, 2003

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Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes JING-ZHI HUANG Penn State University and New York University LIUREN WU Fordham University This version: March 30, 2003 We are grateful to Rick Green (the editor), an anonymous referee, Menachem Brenner, Peter Carr, Robert Engle, Steve Figlewski, Martin Gruber, Jean Helwege, and Rangarajan Sundaram for helpful comments and discussions. We also thank seminar participants at Baruch College, the University of Notre Dame, and Washington University in St. Louis for helpful comments. Smeal College of Business, Penn State University, Univ Park, PA 16802; tel: (814) ; fax: (814) ; Stern School of Business, New York University, New York, NY 10012; (212) ; jhuang0. Graduate School of Business, Fordham University, 113 West 60th Street, New York, NY 10023; tel: (212) ; fax: (212) ; ABSTRACT This article analyzes the specifications of option pricing models based on time-changed Lévy processes. We classify option pricing models based on (i) the structure of the jump component in the underlying return process, (ii) the source of stochastic volatility, and (iii) the specification of the volatility process itself. Estimation of a variety of model specifications indicates that, to capture the behavior of the S&P 500 index options, one needs to incorporate a jump component with infinite activity and generate stochastic volatilities from two separate sources: the jump component and the diffusion component. Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes The seminal work of Black and Scholes (1973) has spawned an enormous literature on option pricing and also played a key role in the tremendous growth of the derivatives industry. However, the model has been known to systematically misprice equity index options. While various extensions of the Black and Scholes model have been proposed and tested, researchers are still facing the challenge of finding a model that can capture both the time series and cross-sectional across both the option strike and maturity behavior of index options. Obviously such a model would be very valuable to participants of option markets. Perhaps equally important, this line of research would also help us understand the dynamics of the underlying return process given the cross-sectional feature of option price data. In this article we synthesize the ongoing efforts in searching for the true underlying return process by performing a specification analysis of option pricing models within a new general framework. We then apply this analysis to S&P 500 index options and empirically investigate some open issues regarding the specification of the index return process. The empirical option pricing literature has documented three anomalies or inconsistencies with the Black and Scholes (1973) model in the data. First, the model assumes that the underlying asset return is normally distributed. However, the cross-sectional behavior of the equity index options along the strike price dimension indicates that the conditional index return distribution under the risk-neutral measure is not normally distributed. In particular, the risk-neutral distribution for the index return inferred from the options data is highly skewed to the left; see, for example, Aït-Sahalia and Lo (1998), Jackwerth and Rubinstein (1996), and Rubinstein (1994) for empirical evidence from the S&P 500 index options. To generate return non-normality and hence to reduce the mispricing of the Black-Scholes model along the strike dimension, one response of the literature is to incorporate a jump component into the underlying asset return process (e.g. Merton (1976)). Second, the assumption of a constant return volatility made in the Black and Scholes model has also been shown to be violated in practice. For instance, empirical studies have documented so called volatility clustering and the leverage effect. The former refers to the observation that while stock re- turns are approximately uncorrelated, the return volatility exhibits strong serial dependence (e.g., Ding, Engle, and Granger (1993) and Ding and Granger (1996)). The latter stylized fact refers to observed negative correlation between stock returns and return volatilities (Black (1976)). To accommodate these stylzed facts, one direction taken in the literature is to allow return volatility to be stochastic (e.g., Heston (1993), Hull and White (1987)). The third stylized empirical fact that cannot be explained by the Black-Scholes model is the maturity pattern of the model pricing bias along the strike dimension mentioned earlier. It has been recognized that this bias across strike (so called volatility smile/smirk) is most significant at short maturities and then flattens out as option maturity increases (e.g. Bates (1996)). More recently, Carr and Wu (2002a) document that volatility smirk in the S&P 500 index options persists even as option maturity increases up to the observable horizon of two years. This evidence implies that the conditional nonnormality of the index returns does not die away with increasing horizon, in contrast to the implication of the classic central limit theorem. The literature tries to accommodate this maturity pattern by introducing jumps into the (underlying asset) return process as well as allowing for stochastic volatility with mean reversion. The rational behind this is while jumps can generate non-normal returns at very short horizons, a persistent stochastic volatility process can slow down the convergence of the return distribution to normality as maturity increases. On balance, the consensus from the empirical option pricing literature is that in order to capture the behavior of equity index options as well as the index returns, we need stochastic volatility jumpdiffusion models models that include both stochastic return volatility and jumps in the return process. Existing stochastic volatility jump-diffusion models of option pricing are often specified within the jump-diffusion affine framework of Duffie, Pan, and Singleton (2000). Recent examples include Bakshi, Cao, and Chen (1997), Bates (1996, 2000), Das and Sundaram (1999), Pan (2002), and Scott (1997). In these models, the underlying asset return innovation is generated by a jump-diffusion process. The diffusion component captures small and frequent market moves. The jump component, which is assumed to follow a compound Poisson process as in Merton (1976), captures the rare and large events. This is because the number of jumps within any finite time interval is assumed to be finite in the compound Poisson model. The empirical estimates for the Poisson arrival rate are usually small, averaging about one jump for every one or two years in equity indices (e.g. Andersen, Benzoni, 2 and Lund (2002)). This is not surprising since these models implicitly assume that the market movements can be characterized either as small diffusive moves or as rare large events. In practice, however, one often observes much more frequent discontinuous movements of different sizes in equity indices. These high frequency jumps are difficult to capture using a compound Poisson model. Another notable feature of the existing option pricing models is that the stochastic volatility is often assumed to come solely from the diffusion component of the underlying return process. Even in models that incorporate jumps, the arrival rate of the jump events is assumed to be either a constant or a linear function of the diffusion variance. However, such specifications are mainly driven by analytical tractability. In practice, the variation in return volatility can be driven by stochastic diffusion variance as well as by variation in the arrival rates of jumps. How these two components of stochastic volatility vary over time and relatively to each other is purely an empirical issue. In this paper, we examine a sample of S&P 500 index options to determine what type of jump structure best captures the index movement. We also investigate whether arrival rates of jump events depend on the diffusion variance or depend on different factors. The specification analysis and empirical study in this paper are based on Carr and Wu (2002b), who propose a theoretical framework of option pricing with time-changed Lévy processes. A Lévy process is a continuous time stochastic process with independent stationary increments, analogous to iid innovations in a discrete setting. In general, a Lévy process can be decomposed into a diffusion component and a jump component. In addition to the Brownian motion and the compound Poisson jump process used widely in the traditional option pricing literature, the class of Lévy processes also includes other jump processes that exhibit higher jump frequencies and hence may better capture the dynamics of equity indices than the compound Poisson process. Heuristically, a time change is a monotonic transformation of the time variable. Stochastic volatility can be generated by applying random or locally deterministic time changes to (the original time variable of) individual components of a Lévy process. In particular, stochastic volatility can be generated by applying different time changes to the diffusion and the jump components of a Lévy process. As a consequence, time-changed Lévy processes include a rich class of jump-diffusion stochastic volatility models. Furthermore, option pricing models in this new framework can have the same analytical tractability as those in the affine framework of Duffie, Pan, and Singleton (2000). 3 Within the class of time-changed Lévy processes, we classify model specifications into three separate but interrelated dimensions: (i) the choice of a jump component, (ii) the identification of the sources for stochastic volatility, and (iii) the specification of the volatility process itself. Such a classification scheme encompasses almost all existing option pricing models in the literature and provides a framework for future modeling efforts. Based on this framework, we design and estimate a series of models using S&P 500 index options data and test the relative goodness-of-fit of each specification. The specification analysis focuses on addressing two important questions on model design. (Q1) What type of jump structure best describes the underlying price movement and the return innovation distribution? (Q2) Where does stochastic volatility come from? To our knowledge, this paper represents the first extensive empirical study in option pricing based on the framework of time-changed Lévy processes. The empirical analysis in this paper focuses on the performance of twelve option pricing models generated by a combination of three jump processes and four stochastic volatility specifications. The three jump processes include the standard compound Poisson jump process used in Merton (1976), the variance-gamma jump model (VG) of Madan, Carr, and Chang (1998), and the log stable model (LS) of Carr and Wu (2002a). Unlike the compound Poisson jump model (which generates a finite number of jumps within any finite time interval), both VG and LS allow an infinite number of jumps within any finite interval and hence are better suited to capture highly frequent discontinuous movements. These three different jump structures are used to answer question Q1 posed above. The four stochastic volatility specifications considered in our empirical analysis include traditional ones such as those used in Bates (1996) and Bakshi, Cao, and Chen (1997), where the diffusion component of the total return variance is stochastic but the jump component is constant. However, we also introduce new specifications that allow stochastic volatility to be generated separately from the jump component and the diffusion component. This is motivated by question Q2 posed earlier. Our estimation results show that, in capturing the behavior of the S&P 500 index options, models based on VG and LS outperform those based on compound Poisson processes. This performance ranking is robust to variations in the stochastic volatility specification and holds for both in-sample and out-of-sample tests. These results suggest that the market prices index options as if there are many (actually infinite) discontinuous price movements (jumps) of different magnitudes in the S&P 4 500 index. This implication is in favor of incorporating high frequency jumps such as VG and LS in the underlying asset return process. The LS model is especially useful in capturing the maturity pattern of the volatility smirk in equity index options. 1 The estimation results also indicate that variations in the index return volatility come from two separate sources: the instantaneous variance of the diffusion component and the arrival rate of the jump component. One implication of this finding is that the intensities of both small and large index movements vary over time and they vary separately. Furthermore, the model parameter estimates indicate that the diffusion volatility and the jump volatility behave differently (in the risk-neutral measure). In particular, while the former is more volatile, the latter exhibits much more persistence. As a result, the behavior of short term options is influenced more by the randomness from the diffusive movements, whereas the behavior of long term options is mostly influenced by the randomness in the arrival rate of jumps. The above specification of stochastic volatility is also consistent with empirical evidence from time series that return volatilities are driven by multiple factors (e.g., Cont and da Fonseca (2002)). Using multi volatility factors obviously increases the flexibility of a model in capturing the time series behavior of the index option prices. Another implication of the above results is that a model specification with stochastic volatility driven by both diffusion and jumps can also improve the model performance cross-sectionally. The reason is as follows. Under such a specification, diffusion volatility and jump volatility two components of the total return variance are driven by independent random sources so their relative weight in the return variance varies along the option maturity dimension. In fact as mentioned above, our empirical evidence shows that the jump component dominates the behavior of long term options. This implies that non-normality of the (risk-neutral) return distribution will not simply reduce as the option maturity increases since jumps are the main source of non-normality. Namely, there will be a persistent volatility smirk across the option maturity. And this is consistent with the maturity pattern of volatility smirk documented in equity index options. To summarize, empirical results from our specification analysis of option pricing models based on Lévy processes provide further evidence for stochastic volatility jump-diffusion models. However, 1 Notice that the central limit theorem does not apply in this model since the return variance is infinite in the log stable model. As a result, the return distribution remains to be non-normal even as the time horizon increases. Namely, the model allows a persistent deviation from normality and therefore can capture the maturity pattern of the volatility smirk. 5 one can improve the model by including high frequency jumps in the underlying return process and allowing the stochastic return volatility to be driven independently by diffusion and jumps. Time change is a standard technique for generating new processes in the theory of stochastic processes. There is a growing literature on applying the technique to finance problems, which perhaps goes back to Clark (1973). He suggests that a random time change be interpreted as a cumulative measure of business activity. Ané and Geman (2000) provide empirical evidence of this interpretation. Examples of other applications include Barndorff-Nielsen and Shephard (2001), Carr, Geman, Madan, and Yor (2001), and Geman, Madan, and Yor (2001). The remainder of this paper is organized as follows. The first section constructs option pricing models through time changing Lévy processes. Section II addresses the data and estimation issues. Section III compares the empirical performance of different model specifications. Section IV analyzes the remaining structures in the pricing errors for different models. Section V concludes with suggestions for future research. I. Model Specifications In this section, we generate candidate option pricing models by modeling the underlying asset return process as time-changed Lévy processes. Under our classification scheme, each model specification requires the specification of the following aspects: (i) the jump component in the return process; (ii) the source for stochastic volatility; and (iii) the dynamics of the volatility process itself. We consider 12 model specifications, under which the characteristic function of log returns has a closed-form solution. We then price options via an efficient fast Fourier transform (FFT) algorithm. 6 A. Dynamics of the Underlying Price Process Formally, let (Ω,F,(F t ) t 0,Q) be a complete stochastic basis and Q be the risk-neutral probability measure. Suppose that the logarithm of the underlying stock price (index level) process, (S t ;t 0), follows a time-changed Lévy process under Q as the following: ( lns t = lns 0 + (r q)t + σw T d t 1 ) ( 2 σ2 Tt d + J j T t ) ξtt j, (1) where r denotes the instantaneous interest rate and q the dividend yield, 2 σ is a positive constant, W is a standard Brownian motion, and J denotes a compensated pure Lévy jump martingale process, which [ ] we will elaborate later. The vector T t Tt d,tt j denotes potential stochastic time changes applied to the two Lévy components W t and J t. By definition, the time change T t is an increasing, right-continuous process with left limits satisfying the usual regularity conditions. 3 While stochastic time change has much wider applications, our focus here is its role in generating stochastic volatilities. For this purpose, we further restrict T t to be continuous and differentiable with respect to t. In particular, let [ v(t) v d (t),v (t)] j = Tt / t. (2) Then, v d (t) is proportional to the instantaneous variance of the diffusion component, while v j (t) is proportional to the arrival rate of the jump component. Following Carr and Wu (2002b), we label v(t) as the instantaneous activity rate. Intuitively speaking, one can regard t as the calendar time and T t as the business time at calendar time t. A more active business day, captured by a higher activity rate, generates higher volatility for asset returns. The randomness in business activity generates randomness in volatility. Note that in equation (1), we apply stochastic time changes only to the diffusion and jump martingale components, but not to the instantaneous drift. The reason is that the equilibrium interest rate and dividend yield are defined on the calendar time, not on business event time. Furthermore, we apply 2 Bakshi, Cao, and Chen (1997) also consider the role played by stochastic interest rates but find that the impact on option pricing is minimal. Here we treat both r and q as deterministic. 3 T t is finite Q-a.s. for all t 0 and that T t as t. 7 separate time changes on the diffusion martingale component and on the jump martingale component, allowing potentially different time-variation in the intensities (activity rates) of small and large events. Also note that in this article, volatility is used as a generic term capturing the financial activities of an asset. It is not used as a statistical term for standard deviation. Just like in Heston (1993) and in many other papers, we model the stochastic volatility from the diffusion component by specifying a stochastic process for v d (t), which is proportional to the instantaneous variance of the diffusion component. In ad

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