VOLATILITY: EVIDENCE FROM S&P 500 INDEX FUTURES

Transcription

1 TESTING MEAN REVERSION IN FINANCIAL MARKET VOLATILITY: EVIDENCE FROM S&P 500 INDEX FUTURES TURAN G. BALI* K. OZGUR DEMIRTAS This article presents a comprehensive study of continuous time GARCH (generalized autoregressive conditional heteroskedastic) modeling with the thintailed normal and the fat-tailed Student s-t and generalized error distributions (GED). The study measures the degree of mean reversion in financial market volatility based on the relationship between discrete-time GARCH and continuoustime diffusion models. The convergence results based on the aforementioned distribution functions are shown to have similar implications for testing mean reversion in stochastic volatility. Alternative models are compared in terms of We thank Robert Webb (the editor) and an anonymous referee for their extremely helpful comments and suggestions. We also benefited from discussions with Haim Levy, Salih Neftci, Robert Schwartz, Panayiotis Theodossiou, and Liuren Wu. An earlier version of this paper was presented at the Baruch College and the Graduate School and University Center of the City University of New York. We gratefully acknowledge the financial support from the PSC-CUNY Research Foundation of the City University of New York. *Correspondence author, Department of Economics & Finance, Baruch College, Zicklin School of Business, City University of New York, One Bernard Baruch Way, Box , New York, NY Turan_Bali@baruch.cuny.edu. Received December 2006; Accepted March 2007 Turan G. Bali is a Professor of Finance in the Department of Economics and Finance at the Zicklin School of Business at Baruch College, City University of New York in New York City, New York. K. Ozgur Demirtas is an Assistant Professor of Finance in the Department of Economics and Finance at the Zicklin School of Business at Baruch College, City University of New York in New York City, New York. The, Vol. 28, No. 1, 1 33 (2008) 2008 Wiley Periodicals, Inc. Published online in Wiley InterScience (

2 2 Bali and Demirtas their ability to capture mean-reverting behavior of futures market volatility. The empirical evidence obtained from the S&P 500 index futures indicates that the conditional variance, log-variance, and standard deviation of futures returns are pulled back to some long-run average level over time. The study also compares the performance of alternative GARCH models with normal, Student s-t, and GED density in terms of their power to predict one-day-ahead realized volatility of index futures returns and provides some implications for pricing futures options Wiley Periodicals, Inc. Jrl Fut Mark 28:1 33, 2008 INTRODUCTION This study deals with continuous-time GARCH (generalized autoregressive conditional heteroskedastic) modeling with fat-tailed distributions, and provides implications for testing mean reversion in financial market volatility based on the relationship between discrete-time GARCH and continuous-time diffusion models. Modeling and estimating the volatility of economic time series has been high on the agenda of financial economists since the early 1980s. Engle (1982) put forward the autoregressive conditional heteroskedastic (ARCH) class of models for conditional variances, which proved to be extremely useful for analyzing financial return series. Since then an extensive literature has been developed for modeling the conditional distribution of stock prices, interest rates, exchange rates, and futures prices. 1 Following the introduction of ARCH models by Engle (1982) and their generalization by Bollerslev (1986), there have been numerous refinements of this approach to estimating conditional volatility. Most of the refinements have been driven by empirical regularities in financial data. In contrast to the stochastic differential equations frequently used in the theoretical finance literature to model time-varying volatility, GARCH processes are discrete-time stochastic difference equations. The discrete-time approach has been favored by empiricists because observations are usually recorded at discrete points in time, and the likelihood functions implied by the GARCH models are usually easy to compute and maximize. By contrast, the likelihood of a nonlinear diffusion process observed at discrete intervals can be very difficult to derive, especially when there are unobservable state variables in the system. Relatively little work has been done so far on the relation between continuous time diffusion and discrete time GARCH models. Indeed, the two literatures have developed quite independently, with little attempt to reconcile the discrete and continuous time models. Nelson (1990) was the first to study 1 This vast literature on the theory and empirical evidence for ARCH modeling has been surveyed in Bollerslev, Chou, and Kroner (1992), Bollerslev, Engle, and Nelson (1994), Andersen (1994), Hentschel (1995), Pagan (1996), Duan (1997), and Bali (2000a) among others. A detailed treatment of ARCH models at a textbook level is also given by Gourieroux (1997).

3 Testing Mean Reversion 3 the continuous time its of GARCH models. Nelson and Foster (1994) derive the diffusion its of the standard GARCH, exponential GARCH, and absolute GARCH processes. Following Nelson (1990) and Nelson and Foster (1994), weak converge results have been developed by Drost and Werker (1996), Fornari and Mele (1997), Duan (1997), and Corradi (2000) for some of the GARCH processes, but clear results do not exist for others. In this article, the diffusion its of many symmetric and asymmetric GARCH models are presented, and the drift of these iting models is then used to test mean reversion in futures market volatility. The earlier research on continuous time GARCH modeling assumes that the error process is drawn from the normal density despite the mounting evidence of fat-tailed errors. One of the main contributions of this work is the development of conditions under which many GARCH models with a fat-tailed density converge weakly to Ito processes as the length of the discrete time intervals goes to zero. The former studies test for mean reversion in asset prices, but mean reversion in asset return volatility has not been tested yet. Despite a bewildering array of GARCH models, relatively little is known about how these models can be used to detect the presence of mean reversion in stochastic volatility. The present work, to our knowledge, represents the first study of testing and measuring the degree of mean reversion in volatility. The theoretical and empirical specifications of the financial return process often assume a normal distribution for modeling unexpected information shocks. However, most studies find that the normality assumption is violated, i.e., the distribution of asset returns has much thicker tails compared to the normal distribution. To take this into account, the fat-tailed standardized Student s-t list and generalized error distribution (GED) as a more flexible density function are used with degrees of freedom estimated endogenously. The convergence results based on the aforementioned distribution functions are shown to have similar implications for testing mean reversion in stochastic volatility. The geometric Brownian motion assumption utilized by standard options pricing models imply that log spot or futures price changes are identically and independently distributed (iid) normal variables, thus exhibit no moment dependencies, such as asymmetric and conditional volatility with fat-tailed distributions. However, the empirical results found from alternative specifications of the futures price process indicate the presence of significant volatility clustering and leptokurtosis in futures returns. To accommodate time-variation in volatility, recent studies derive an option pricing model with stochastic volatility. The underlying assumption in these models is that stochastic volatility follows a mean-reverting process. Without this assumption, it is extremely difficult to come up with a closed from solution for option prices. Hence, testing the presence of mean-reversion in futures market volatility has important implications

4 4 Bali and Demirtas for newly proposed option pricing models. The maximum likelihood estimates of alternative volatility/distribution specifications suggest that option pricing models with stochastic volatility may not be flexible enough to capture the extreme tails of the futures return distribution. This article is organized as follows. In the second section, continuous time its of the GARCH models with fat-tailed distributions are provided. The data and estimation methodology are then described in the next section. The empirical results are presented in the following sections, followed by a concluding section. CONTINUOUS-TIME GARCH MODELING WITH FAT-TAILED DISTRIBUTIONS In continuous time diffusion models, futures price movements are described by the following stochastic differential equation, df t mf t dt sf t dw t, (1) where F t is a futures price at time t, W t is a standard Wiener process with zero mean and variance of dt, m and s are the constant drift and diffusion parameters of the geometric Brownian motion. Applying Ito s lemma to Equation (1), the process followed by ln F t is derived: d ln F t am s2 2 bdt sdw t (2) where m s 2 /2 is the constant drift and s 2 is the constant variance of log-futures price changes. The one-factor continuous time model in Equation (2) can be extended by incorporating a stochastic volatility factor into the diffusion function: d ln F t am t s2 t 2 bdt s tdw 1,t df(s t ) f m (s t )dt f s (s t )dw 2,t (3) (4) where m t s 2 t 2 and s t are the (instantaneous) time-varying mean and volatility of futures returns, W 1,t and W 2,t are standard Brownian motion processes so that dw 1,t and dw 2,t are normally distributed with zero mean and variance of dt. In Equation (4), the stochastic volatility factor is specified with the instantaneous variance [i.e., f(s ], the log-variance [i.e., f(s t ) ln s 2 t ) s 2 t t ], or the

5 Testing Mean Reversion 5 standard deviation of futures returns [i.e., f(s t ) s t ]. f m (s t ) and f s (s t ) are the drift and diffusion functions of the volatility process, respectively. Different parameterization of f m (s t ) and f s (s t ) yields different GARCH processes in discrete time. This discussion on continuous time GARCH modeling and futures market volatility is focused on 10 parametric versions of f m (s t ) and f s (s t ). To have a more precise discussion, the discrete time GARCH models are formally defined below. ln F t ln F t am t s2 t 2 b s tz t 2 (5) AGARCH: Asymmetric GARCH model of Engle (1990) s 2 t b 0 b 1 (g s t z t ) 2 b 2 s 2 t (6) EGARCH: Exponential GARCH model of Nelson (1991) ln s 2 t b 0 b 1 [ 0z t 0 E( 0z t 0 )] b 2 ln s 2 t gz t (7) GARCH: Linear symmetric GARCH model of Bollerslev (1986) s 2 t b 0 b 1 s 2 t z 2 t b 2 s 2 t (8) GJR-GARCH: Threshold GARCH model of Glosten, Jagannathan, and Runkle (1993) s 2 t b 0 b 1 s 2 t z 2 t b 2 s 2 t gs t s 2 t z 2 t (9) St 1 if s t z t 0, and St 0 otherwise NGARCH: Nonlinear asymmetric GARCH model of Engle and Ng (1993) s 2 t b 0 b 1 s 2 t (g z t ) 2 b 2 s 2 t (10) QGARCH: Quadratic GARCH model of Sentana (1995) s 2 t b 0 b 1 s 2 t z 2 t b 2 s 2 t gs t z t (11) SQR-GARCH: Square-Root GARCH model of Heston and Nandi (1998) 2 s 2 t b 0 b 1 (gs t z t ) 2 b 2 s 2 t (12) 2 As will be shown in Appendices A C, the continuous time it of Equation (12) converges to a stochastic variance process, which is generated by the square-root diffusion popularized as a model of the short-term interest rate by Cox, Ingersoll, and Ross (1985).

6 6 Bali and Demirtas TGARCH: Threshold GARCH model of Zakoian (1994) s t b 0 b 1 s t 0z t 0 b 2 s t gs t s t z t (13) S t 1 if s t z t 0, and S t 0 otherwise TS-GARCH: The specification proposed by Taylor (1986) and Schwert (1989) s t b 0 b 1 s t 0z t 0 b 2 s t (14) VGARCH: A version proposed in Engle and Ng (1993) s 2 t b 0 b 1 (g z t ) 2 b 2 s 2 t (15) where is the length of the time interval, z t is a random variable drawn from the standardized normal distribution [z t N(0, 1)], b 0 0, 0 b 1 1, 0 b 2 1, and g 0. The diffusion parameter g allows for asymmetric volatility response to past positive and negative information shocks. Following Nelson (1990) and Nelson and Foster (1994), it can be shown that the symmetric and asymmetric GARCH models in Equations (6) (15) converge in distribution to continuous time stochastic volatility models as goes to zero. Now the properties of the stochastic difference equation system, Equations (5) (6) are considered as time is partitioned more and more finely. The parameters of the system b 0, b 1, and b 2 depend on the discrete time interval, and make both the drift term in Equation (5) and the variance of z t proportional to. Nelson s (1990) approximation scheme is applied to Equations (5) and (6): ln F k ln F (k 1) (m (k 1) s 2 (k 1) 2) s k z k s 2 k b 0 b 1 (g 1 2 s (k 1) z (k 1) ) 2 b 2 s 2 (k 1) (16) (17) z k iid where N(0, ) and (k 1) t k. We allow b 0, b 1, and b 2 to depend on because our objective is to discover which sequences {b 0, b 1, b 2 } make the 5s 2 t s 2 t 1, lnf t lnf t 1 6 process converge in distribution to the Ito process given in Equation (3) and (4) as goes to zero. We will compute the first two conditional moments, and then, after a few mild technical conditions, appeal to the theorems for weak convergence of Markov chaisn to diffusion processes by Strook and Varadhan (1979,) or by Ethier and Kurtz (1986). The first and second moments per unit of time are given by. 3 3 The continuous time it of Equations (18) and (19) yields the Ito process in Equation (3) because the drift of log-futures price changes equals m t s 2 t 2 it as goes to zero, and the instantaneous variance of futures returns equals because s 2 t 1 Var[(lnF k ln F (k 1) ) 0 (k 1) ] 1 5E[(lnF k ln F (k 1) ) 2 0 (k 1) ] (E[(lnF k ln F (k 1) ) 0 (k 1) ]) 2 6 s 2 t

7 Testing Mean Reversion 7 1 E[(ln F k ln F (k 1) ) (k 1) ] (m (k 1) s 2 (k 1) 2) 1 E[(ln F (m k 1 ) s 2 (k 1) 2) 2 s 2 k ln F (k 1) ) 2 (k 1) ] k (18) (19) 1 E[(s 2 k s 2 (k 1) ) 0 (k 1) ] 1 (b 0 b 1 g 2 ) 1 [b 1 b 2 1] (20) 1 E[(s 2 k s 2 (k 1) ) 2 0 (k 1) ] 1 (b 0 b 1 g 2 ) 2 1 [4b 2 1 g 2 2(b 0 b 1 g 2 )(b 1 b 2 1)]s 2 (k 1) 1 [2b 2 1 (b 1 b 2 1) 2 ]s 4 (k 1) (21) where (k 1) is the information set at time (k 1). To obtain the stochastic variance process in Equation (4), the following parameterization is considered for b 0, b 1, and b 2 as a function of :, (22) 1 (b 0 b 1 g 2 ) r 0, (23) 1 [b 1 b 2 1] r 1 0. (24) 1 2b 2 1 l 2 0 Equations (22) and (23) give the drift, r 0 r 1 s 2 t, of the stochastic volatility model. Equation (24) yields the standard deviation, ls t 22g 2 s 2 t, of the variance process in Equation (4) because 1 Var[(s 2 k s 2 (k 1) ) 0 (k 1) ] 1 5E[(s 2 k s 2 (k 1) ) 2 0 (k 1) ] 1 2b 2 1 s 2 (k 1) (2g 2 s 2 (k 1) ). One can easily show that Equations (22) (24) are all satisfied if b 0 r 0 g 2 l 2 B b 1 B l (E[(s 2 k s 2 (k 1) ) 0 (k 1) ]) (25) (26) b 2 1 r 1 B l (27)

8 8 Bali and Demirtas The AGARCH model of Engle (1990) in Equation (6) converges weakly to the stochastic volatility model in Equation (4) with f m (s t ) r 0 r 1 s 2 t and f s (s t ) ls t 22g 2 s 2 t. Note that there is no specific reason to present the diffusion it of AGARCH model in this section, except for illustration purposes. The diffusion its of alternative GARCH processes are presented in Appendix A. In the existing literature, diffusion its of the GARCH models are obtained under the assumption that the innovation process is normally distributed. However, Drost and Werker (1996) show that the normality assumption of an underlying continuous time GARCH(1,1) model leads to kurtosis parameters of the corresponding discrete time processes, which are greater than 3, implying heavy tails. Their results provide an explanation why fat-tailed conditional distributions are obtained, without exception, in empirical work. To capture this phenomenon, a fat-tailed innovation distribution is used in continuous time GARCH modeling. The distribution of z t t[0, v/(v 2)] is Student s-t with degrees of freedom v: f(z t ) a n 1 (28) 2 b a n 1 2 b (np) 1 2 c 1 z2 (n 1) 2 t n d where (a) is the gamma function. The t-distribution is symmetric around zero, and the first, second, and fourth moments of z t and z t are 0 xa 1 e x dx equal to: E(z t ) 0, E( 0z t 0 ) 2 a n 1 2 b a n 2 1B b n (n 1) 2 p, E(z 2 t ) n n 2, E(z4 t ) 3n 2 (n 2)(n 4) In Appendix B by applying Nelson s (1990) approximation scheme to the heavy-tailed innovation process with the statistical properties of z t and z t, the diffusion its of discrete time GARCH processes with Student s-t density are given by equations (B1) (B10). The parameter restrictions that form the relation between continuous time diffusion and discrete time GARCH models with Student s-t distribution are also given in Appendix B. Several other conditional distributions have been employed in the literature to fully capture the degree of tail fatness in speculative prices (e.g., Bali, 2003, and Bali & Weinbaum, 2007). One of these heavy-tailed distributions, the GED, has been widely used by financial economists. 4 In addition to 4 The generalized error distribution (GED) was initially introduced by Subbotin (1923), and then used by Box and Tiao (1962) to model prior densities in Bayesian estimation and by Nelson (1991) to model the distribution of stock market returns.

9 Testing Mean Reversion 9 the Student s-t distribution, the density function for the GED is utilized: f(z t ) n exp[( 1 2)0z t 0 n ] 2 [(n 1) n] (1 n) (29) [ 2( 2 v) (1 v) (3 v) ] 1 2 where. For the tail thickness parameter n 2, the GED density equals the standard normal density. For n 2 the distribution has thicker tails than the normal, whereas n 2 results in a distribution with thinner tails than normal. Applying the same approximation scheme to the fat-tailed innovation process along with the statistical properties of z t and z t for the GED density, E(z t ) 0, E( 0z t 0 ) a 2 n b, a n b a n b a 1 n b a 5 n b E(z 2 t ) 1, E(z 4 t ), a 3 2 n b The continuous time its of GARCH models are given by equations (C1) (C10) in Appendix C. The parameter restrictions that form the relation between diffusion and GARCH processes with GED density are also given in Appendix C. The continuous time stochastic volatility models presented in the Appendices A C have similar implications for testing mean reversion in financial market volatility. Table I shows the mean reversion rates based on the relation between discrete time GARCH and continuous-time diffusion models. The degree of mean reversion in stochastic volatility is measured by the negative values of r 1 for the normal distribution, k 1 for the Student s-t distribution, and u 1 for the generalized error distribution. As will be discussed, the degree (or speed) of mean reversion in futures return volatility is found to be robust across different distribution functions. 5 5 We should note that the mean reversion rates presented in here are not specific to futures return volatility. They can be used to test the presence of mean reversion in the conditional volatility of any financial return series including stock returns, interest rates, exchange rates, and commodity prices.

10 TABLE I Speed to Mean Reversion in Financial Market Volatility Models Normal distribution Student s-t distribution a GED b AGARCH r 1 1 [b 1 b 2 1] k 1 1 [ n b 1 b 2 1] u 1 1 [b 1 b 2 1] EGARCH r 1 1 (b 2 1) k 1 1 (b 2 1) u 1 1 (b 2 1) GARCH r 1 1 [b 1 b 2 1] k 1 1 [ n b 1 b 2 1] u 1 1 [b 1 b 2 1] GJR-GARCH r 1 1 [b 1 b 2 0.5g 1] k 1 1 [ n (b 1 0.5g) b 2 1] u 1 1 [b 1 b 2 0.5g 1] NGARCH r 1 1 [b 1 (1 g 2 ) b 2 1] k 1 1 [b 1 ( n g 2 ) b 2 1] u 1 1 [b 1 (1 g 2 ) b 2 1] QGARCH r 1 1 [b 1 b 2 1] k 1 1 [ n b 1 b 2 1] u 1 1 [b 1 b 2 1] SQR-GARCH r 1 1 [b 1 g 2 b 2 1] k 1 1 [b 1 g 2 b 2 1] u 1 1 [b 1 g 2 b 2 1] TGARCH r c B p b 1 b 2 1 d k 1 1 [ n b 1 b 2 1] u 1 1 [ n b 1 b 2 1] TS-GARCH r c B p b 1 b 2 1 d k 1 1 [ n b 1 b 2 1] u 1 1 [ n b 1 b 2 1] VGARCH r 1 1 (b 2 1) k 1 1 (b 2 1) u 1 1 (b 2 1) a and for the Student s-t distribution. (n 1) 2 p n n n 2 0n0 2 a n 1 2 b a n 2 1B b n b 0 (2 n) 0n for the GED. [ (3 n)] 1 2 [ (1 n)] 1 2 Note. This table displays the drift parameters of the stochastic volatility models that determine the presence of mean reversion in futures return volatility. The speed of mean reversion in volatility is measured by r 1 for the normal distribution, k 1 for the Student s-t distribution, and u 1 for the generalized error distribution (GED).

11 Testing Mean Reversion 11 DATA AND ESTIMATION The data consist of daily prices for the S&P 500 index futures. The time period of investigation extends from April 22, 1982, through December 31, 2002, giving a total of 5,232 daily observations. 6 To compute returns on the S&P 500 index futures (r t ) the formula, r t ln F t ln F t 1 is used, where F t is the value of the index futures for period t. Table II provides descriptive statistics for the daily percentage returns on S&P 500 index futures. The unconditional mean of the daily returns is about % with a standard deviation of %. The maximum and minimum values are about 17.38% and 32.73%, respectively. The skewness, excess kurtosis, first-order autocorrelation, and the Ljung-Box statistics for testing the null hypotheses of independent and identically distributed normal variates are also reported. The skewness statistic for daily returns is negative and statistically significant at the 1% level. The excess kurtosis statistic is considerably high and significant at the 1% level, implying that the distribution of futures returns has much thicker tails than the normal distribution. The fat-tail property is more dominant than skewness in the sample. The first-order autocorrelation coefficient is found to be negative and large enough to reject the null hypothesis of first-order zero correlation at the 1% level. The Ljung-Box Q 1 (12) statistic for the cumulative effect of up to 12th-order autocorrelation in the TABLE II Summary Statistics S&P 500 Index Futures April 22, 1982 December 31, 2002 # of obs. 5,232 Maximum 17.38% Minimum 32.73% Mean % SD % Skewness * Kurtosis * AC(1) * Q 1 (12) * Q 2 (12) * Note. This table shows the daily percentage returns on the S&P 500 index futures. The sample period, number of observations, maximum and minimum values, mean, standard deviation, skewness, excess kurtosis, first-order autocorrelation coefficient AC(1), and the Ljung-Box statistics for the standardized residuals Q 1 (12) and the squared standardized residuals Q 2 (12) are reported for daily returns.* Denotes the 1% level of significance. 6 At an earlier stage of the study, the daily returns on the underlying asset of index futures contracts were also used. More specifically, we used daily data on S&P 500, NASDAQ, Dow Jones Industrial Average (DJIA), and New York Stock Exchange (NYSE) indices. To save space, we do not present the empirical findings based on the S&P 500, NASDAQ, DJIA, and NYSE stock market indices. They are available upon request. We found strong mean reversion in stock return volatility and the qualitative results were robust across different data sets.

12 12 Bali and Demirtas standardized residual exceeds (1 percentile critical value from a 2 distribution with 12 df ). This provides evidence of temporal dependencies in the first moment of the distribution of returns. The Ljung-Box Q 2 (12) statistic on the squared standardized residuals provides a test of the intertemporal dependence in the variance. The Q 2 (12) statistic rejects the zero correlation null hypothesis, indicating that the distribution of the next squared return depends not only on the current return, but on several previous returns. In estimating maximum likelihood of the GARCH models, the conditional mean of futures returns is assumed to follow an autoregressive process of order one, and the error process {e t z t t } is conditionally heteroskedastic with timevarying variance, for example, given by the GARCH(1,1) process: r t v ar t 1 e t (30) s 2 t b 0 b 1 z 2 t 1s 2 t 1 b 2 s 2 t 1 (31) where z t is an independent normal N(0,1) variate, and can be viewed as an unexpected shock to the futures market. Because z t is drawn from the standard normal distribution, the density function for r t is f( ; r t ) 1 22ps 2 t0t 1 exp c 1 2 a r t m 2 t t 1 b s. s t t 1 (32) where (v, a, b 0, b 1, b 2 ) is the parameter vector, m t t 1 v a r t 1 is the conditional mean, and s 2 t0t 1 is the conditional variance of futures returns. Given the initial values of e t and s 2 t, the parameters can be estimated by maximizing the log-likelihood function over the sample period. The normal density in Equation (32) yields the following log-likelihood function: LogL normal n 2 ln(2p) n 2 ln s2 t0t a n a r 2 t v ar t 1 b. (33) s t t 1 In most empirical studies the normal density is used even though the standardized residuals obtained from ARCH-type models, which assume normality, remain leptokurtic. As shown in Table II, the excess kurtosis statistic for daily returns on S&P 500 index futures is extremely high and statistically significant, implying that the tails of the actual distribution are much thicker than the tails of the normal distribution. In light of the empirical evidence of fat-tailed errors, Bollerslev (1987) and Nelson (1991) use leptokurtic distributions such as the Student s-t distribution and the generalized error distribution, respectively. In this study, in addition to the thin-tailed normal distribution, the heavy-tailed standardized Student s-t and GED distributions, are used. t 1

13 Testing Mean Reversion 13 The conditional distribution of r t is standardized t with mean m t t 1 v ar t 1, variance, and degrees of freedom n 2, i.e., s 2 t0t 1 r t m t0t 1 e t, e t 0 t 1 f v (e t 0 t 1 ) f v (e t 0 t 1 ) a v 1 2 b a v 2 b 1 [(v 2)s 2 t0t 1] 1 2 (34) (35) where t 1 denotes the sigma-field generated by all the information up through time t 1, and f v (e t t 1 ) the conditional density function for the error term e t z t s t. It is well known that for 1/n S 0 the t-distribution approaches a normal distribution with variance s 2 t t 1, but for 1/n 0 the t-distribution has fatter tails than the corresponding normal distribution. The standardized-t density in Equation (35) gives the following log-likelihood function that can be used to estimate the model s parameters: LogL Student's-t n ln a n 1 2 b n ln a n 2 b n 2 ln[(n 2)s2 t0 1] e 2 t c 1 (v 2)s 2 t t 1 (v 1) 2 d a n 1 n 2 b a ln c 1 t 1 e 2 t d. (n 2)s 2 t t 1 (36) In addition to the heavy-tailed Student s-t distribution, the GED density is employed to model the conditional variance and the observed leptokurtosis in daily futures return data: f n (z t 0 t 1 ) nexp[( 1 2)0z t 0 n ] 2 [(n 1) n] (1 n) (37) m t0t 1 where z and (3 n) ] 1 2 t r t s t0t 1. The simplified version of the loglikelihood function for the GED density is [ 2( 2 n) (1 n) LogL GED ln(n 2) 0.5ln (3 n) 1.5ln (1 n) 0.5 a n t 1 ln s 2 t0t 1 exp[(n 2)(ln (3 n) ln2 (1 n))] a n t 1 r t m t0t s t0t 1 n (38) The log-likelihood functions implied by the conditional Student s-t and GED density yield parameter estimates, which are not excessively influenced by

14 14 Bali and Demirtas extreme observations that occur with low probability (e.g., financial market booms and crashes). In addition, the standard errors of the estimated parameters are robust, allowing for more realiable statistical inference. Another advantage is that one can formally test the empirical validity of GARCH models that assume normality. The parameters of alternative GARCH models with normal, Student s-t and GED density are estimated using the Berndt Hall Hall Hausman (BHHH, 1974) numerical algorithm and the standard errors are computed based on the outer-product of the gradients. 7 At an earlier stage of the study, for some of the models, the Broyden Fletcher Goldfarb Shanno (Broyden, 1970) maximization technique is utilized. This technique uses the inverse of the Hessian and the same BFGS method with robust errors that utilizes the sandwich estimator. 8 The qualitative results from the inverse of the Hessian and from the sandwich estimator are similar to those reported in the tables here. ESTIMATION RESULTS Table III presents the maximum likelihood estimates of the symmetric and asymmetric GARCH models for the normal, Student s-t and generalized error distributions. For all density functions, the parameters in the conditional volatility equation (b 0, b 1, b 2, g) are significant at the 1% level without any exception. A notable point in Table III is that allowing for fat-tailed disturbances consistently improves the ability of capturing the futures price dynamics. The estimation results confirm the presence of rather extreme conditionally heteroskedastic volatility effects in the futures price process (see, e.g, Bali, 1999, 2000b). For example, the symmetric GARCH parameters, b 1 and b 2, are found to be highly significant, and the sum b 1 b 2 is close to one for all density functions considered in this study. This implies the existence of substantial volatility persistence in the S&P 500 index futures. Another notable point Table III is that the asymmetry parameter, g, in the conditional variance, log-variance, and standard deviation models turns out to be highly significant. The parameter g allows for an asymmetric volatility response in the diffusion function to past positive and negative information shocks. It is estimated to be negative for all volatility models, implying that negative shocks (or unexpected decrease in futures prices) have a larger impact than positive shocks (or unexpected rise in futures prices) of the same size on conditional volatility. This is also consistent with the leverage effect that 7 The parameters and their standard errors are estimated using the econometrics software WINRATS ( and for robustness check EVIEWS ( is also used when estimating some of the GARCH specifications. 8 The interested reader may wish to consult Press, Teukolsky, Vetterling, and Flannery (1992) for Broyden (1970), Fletcher (1970), Goldfarb (1970), and Shanno (1970) (BFGS) algorithm.

15 Testing Mean Reversion 15 TABLE III Maximum Likelihood Estimates of the GARCH Models Models v a b 0 b 1 b 2 g Log-L Panel A: Normal distribution AGARCH (1.5591) ( ) (3.6488) (37.649) (205.79) ( ) EGARCH (1.8419) ( ) ( ) (24.034) (547.68) ( ) GARCH (4.9025) ( ) (24.035) (47.778) (183.85) GJRGARCH (2.3150) ( ) (28.331) (5.8949) (223.99) ( ) NGARCH (1.4731) ( ) (28.545) (24.471) (168.39) ( ) QGARCH (1.5591) ( ) (3.6488) (37.649) (205.79) ( ) SQRGARCH (1.2351) ( ) (28.531) (28.681) (139.75) ( ) TGARCH (2.4214) ( ) (80.893) (55.573) (164.88) ( ) TSGARCH (6.0469) ( ) (72.676) (67.806) (182.49) VGARCH (1.3021) ( ) (10.830) (35.196) (337.16) ( ) Models v a b 0 b 1 b 2 g n Log-L Panel B: Standardized student s-t distribution AGARCH (3.8736) ( ) (2.8904) (9.9924) (120.29) ( ) (16.313) EGARCH (4.0356) ( ) ( ) (9.3962) (430.70) ( ) (15.960) GARCH (5.5525) ( ) (5.2318) (9.5713) (130.48) (16.752) GJRGARCH (4.3930) ( ) (6.2780) (8.9876) (122.54) ( ) (16.134) NGARCH (3.7861) ( ) (6.3173) (9.3533) (84.315) ( ) (16.313) QGARCH (3.8736) ( ) (2.8904) (9.9924) (120.29) ( ) (16.313) SQRGARCH (2.2377) ( ) (3.9152) (7.2197) (58.346) ( ) (16.746) TGARCH (3.7742) ( ) (7.2743) (12.979) (144.41) ( ) (16.752) TSGARCH (5.4940) ( ) (6.0402) (11.964) (144.72) (18.910) VGARCH (2.4051) ( ) (4.4357) (7.2271) (163.96) ( ) (16.725) (Continued)

16 16 Bali and Demirtas TABLE III (continued) Models v a b 0 b 1 b 2 g n Log-L Panel C: Generalized error distribution AGARCH (4.3227) ( ) (1.0612) (12.061) (120.17) ( ) (61.752) EGARCH (4.4495) ( ) ( ) (10.178) (367.27) ( ) (65.305) GARCH (6.1334) ( ) (9.4758) (11.759) (123.44) (61.709) GJRGARCH (4.8575) ( ) (10.157) (3.3734) (125.34) ( ) (60.106) NGARCH (4.2014) ( ) (10.261) (10.339) (87.004) ( ) (61.296) QGARCH (4.3227) ( ) (1.0612) (12.061) (120.17) ( ) (61.752) SQRGARCH (2.9305) ( ) (9.7967) (7.8656) (57.093) ( ) (62.367) TGARCH (4.2296) ( ) (24.265) (15.723) (106.29) ( ) (56.038) TSGARCH (6.2885) ( ) (9.4758) (13.680) (94.450) (60.106) VGARCH (2.9582) ( ) (9.0307) (8.2229) (129.62) ( ) (64.909) Note. This table displays the maximum likelihood estimates of the GARCH models for the S&P 500 index futures. Panels A, B, and C present the estimated parameters of the thin-tailed normal, and the fat-tailed Student s-t and generalized error distributions. The parameter estimates with asymptotic t-statistics are shown in parentheses for each model. The maximized log-likelihood values are used to test the presence of asymmetry in volatility and the empirical validity of normality assumption. r t (v ar t 1 ) s t z t, e t s t z t f (s t ) U(s t 1,z t 1, 0z t 1 0 ; b 0,b 1,g) b 2 f (s t 1 ) where f (s t ) s t, s 2 t,orlns 2 t changes in futures prices are negatively related with changes in volatility. The presence of asymmetry in return volatility is formally determined on the basis of the likelihood ratio (LR) test by comparing the maximized log-likelihood values of the asymmetric and symmetric GARCH models. 9 Although not presented here, the estimated LR statistics are well above the critical value, implying that negative returns are followed by greater increases in volatility than equally large positive returns. Normally requires that the estimated degrees of freedom parameter of the Student s-t density be close to infinity, n Sq, or 1/n be equal to zero. As presented in Panel B of Table III, the estimated v is found to be in the range of 9 The likelihood ratio test (LR) statistic is calculated as LR 2[log-L* Log-L], where Log-L* is value of the log-likelihood under the null hypothesis (g 0), and Log-L is the log-likelihood under the alternative. The statistic is distributed as the chi-square, x 2, with 1 df.

17 Testing Mean Reversion to These estimates are highly significant for all GARCH processes considered here. Bollerslev (1987) notes that when testing against the null hypothesis of conditionally normal errors, i.e. 1/n 0, 1/n is on the boundary of the admissible parameter space, and the usual test statistics will likely be more concentrated towards the origin than a x 2 1 distribution. Bollerslev finds that for moderately sized samples the correct 5% critical value for the LR test statistic for null hypothesis 1/v 0 is approximately To save space, the LR statistics are not presented; they are substantially greater than the critical value for all volatility specifications, indicating that the distribution of daily index futures returns is much more leptokurtic than the corresponding normal distribution. As discussed earlier, for the tail thickness parameter n 2, the GED density equals the standard normal density. However, the estimates of v turn out to be highly significant and less than 2 for each model specification. As shown in Panel C of Table III, the degrees of freedom parameter, v, is estimated to be in the range of 1.13 to Although not presented here, comparing the maximized log-likelihood functions of the models estimated with the GED and normal distributions indicates that n is statistically different from 2, implying strong rejection of the models that utilize normally distributed innovation process. The results also suggest that the double exponential or Laplace with n 1 is a more appropriate density function. An intuitive understanding can be gained by realizing that for the normal distribution (n 2) the degree of kurtosis is equal to 3, whereas for the double exponential or Laplace distribution (n 1) the degree of kurtosis is equal to 6. TESTING MEAN REVERSION IN THE CONDITIONAL VOLATILITY OF FUTURES RETURNS In this section, the objective is to determine whether a conditional measure of the variance and standard deviation is pulled back to some long-run average level over time. To explain the concept of mean-reversion, we can think of a plot of the last period s variance, s 2, against the change in variance, s 2 t s 2 t 1 t 1. When the level of the variance is high, mean reversion tends to cause it to have a negative drift, and when it is low, mean reversion tends to cause it to have a positive drift. In other words, mean reversion in volatility implies a negative slope in the conditional mean of the volatility process because the change in variance is a function of the last period s variance as well as its lagged values. When s 2 t s 2 with the first derivative of this function being negative, df ds 2 t 1 f(s 2 t 1) t 1 0, futures return volatility follows a mean reverting process. The first-order Euler approximation of the stochastic volatility models in the Appendices A C implies that the change in variance is a function of the

18 18 Bali and Demirtas last period s variance. For example, the continuous time it of the GARCH model in Equation (A3) in Appendix A is approximated as s 2 t s 2 t (r 0 r 1 s 2 t ) gs 2 t W 2,t (39) where is the length of the time interval, W 2,t z 2,t 2 can be viewed as a normally distributed random variable with zero mean and variance of because z 2,t N(0,1), r 0 r 1 s 2 is the drift (or the conditional mean) of the volatility process, and ls 2 t t is the conditional standard deviation of the volatility process. As shown in Table I, the speed of mean reversion parameters (r 1, k 1, u 1 ) includes the GARCH parameters (b 1, b 2, g), the length of time interval, and the estimated degrees of freedom parameter (n) for the Student s-t and GED distributions. As indicated by Equation (39), mean reversion in variance (or standard deviation) of futures returns can formally be tested for the discrete time GARCH models. Futures return volatility follows a mean-reverting process if the volatility drift parameter r 1 in Equations (A1) (A10) is negative, so a test for mean reversion is a test of whether r 1 0 against the alternative that r 1 0. Rewriting the drift fuction, r, as r 1 (s 2 0 r 1 s 2 t t w) reveals that r 1 can be viewed as a measure of the speed of mean reversion in futures return volatility. The more negative r 1 is, the faster the return volatility s 2 t responds to deviations from its long-run average level, w r 0 /r 1. Table IV displays the estimated mean reversion rates based on the relationship between discrete time GARCH and continuous-time diffusion models. All of the GARCH processes with the thin-tailed normal, the fat-tailed Student s-t and GED density indicate a mean-reverting behavior of stochastic volatility. A notable point in Table IV is that the speed of mean reversion in the conditional variance, log-variance, and standard deviation is slightly bigger with the normal density compared to the estimates from the Student s-t and GED distributions. 10 For the aforementioned distribution functions, the degree of mean reversion in the conditional variance is found to be slightly greater than that in the log-variance and standard deviation. 11 Specifically, the speed of mean reversion in the variance of futures returns is estimated to be in the range of to for the normal, to for the Student s-t, and to for the GED density. The speed of mean reversion in the conditional log-variance is estimated to be for the normal, for the Student s-t and 10 Note that the AGARCH, GARCH, GJR-GARCH, NGARCH, QGARCH, SQR-GARCH, and VGARCH models are used to test mean reversion in the conditional variance of futures returns. Mean reversion in the conditional log-variance is tested based on the estimates of the EGARCH model. Mean reversion in the conditional standard deviation is tested based on the TGARCH and TS-GARCH models estimates. 11 In other words, the conditional variance of index futures responds faster than the standard deviation and log-variance to deviations from the long-run average level.

19 Testing Mean Reversion 19 TABLE IV Speed of Mean Reversion in Futures Return Volatility Based on the Relation Between GARCH and Diffusion Models Normal distribution Student s-t distribution GED Models (r 1 ) (k 1 ) (u 1 ) AGARCH EGARCH GARCH GJR-GARCH NGARCH QGARCH SQR-GARCH TGARCH TS-GARCH VGARCH Note. This table displays the speed of mean reversion in the conditional variance, log-variance, and standard deviation of daily future returns based on the relation between discrete time GARCH and continuous time diffusion models. The speed of mean reversion in futures return volatility is measured by the negative values of r 1 for the normal distribution, k 1 for the Student s-t distribution, and u 1 for the generalized error distribution (GED) for the GED density. The mean reversion rates for the standard deviation are in the range of and for the normal, to for the Student s-t and to for the GED density. Among the conditional variance models, the SQR-GARCH model of Heston and Nandi (1998) and the VGARCH model of Engle (1990) yield the highest mean reversion rates in futures return volatility. The conditional standard deviation models, the TGARCH model of Zakoian (1994) and the TS-GARCH model of Taylor (1986) and Schwert (1989), indicate similar meanreverting behavior of the volatility process. The mean reversion rates based on the parameter estimates of the TGARCH, TS-GARCH, and EGARCH models indicate that the conditional log-variance of futures returns is pulled back to a long-run average level slower than the conditional standard deviation. Although the standard errors of the parameter estimates (b 1, b 2, g, n) for each GARCH specification are presented, the standard errors of the meanreversion rates are not provided because for some GARCH specifications it is extremely difficult to find the covariance of the parameter estimates. To have some feel for the significance levels of mean-reversion rates, the standard errors for the GARCH, EGARCH, and GJR-GARCH specifications are computed. For the normal density, the standard errors are found to be , , and for the GARCH, EGARCH, and GJR-GARCH model, respectively. As shown in Table IV, the corresponding mean-reversion rates are , , and , implying statistically significance at least at the 1% level. For the Student s-t density, the standard errors are estimated to be

20 20 Bali and Demirtas , , and for the GARCH, EGARCH, and GJR-GARCH model, respectively. As reported in Table IV, the corresponding mean-reversion rates are , , and , implying statistically significance at least at the 1% level. For the GED density, the standard errors are , , and for the GARCH, EGARCH, and GJR-GARCH model, respectively. As displayed in Table IV, the corresponding mean-reversion rates are , , and , implying statistically significance at least at the 1% level. PRICING IMPLICATIONS FOR OPTIONS ON S&P 500 INDEX FUTURES Mean-reversion in the conditional volatility of futures returns implies stationarity in the stochastic volatility process. The parameter estimates also suggest strong time-variation and persistence in the conditional volatility of returns on the S&P 500 index futures. In addition, the tail-thickness parameters of the GED and Student s-t distributions indicate the presence of significant leptokurtosis in daily returns on index futures. Overall, these findings suggest that to accommodate the tail-thickness and time-variation in the futures return distribution, an option pricing model with a conditional fat-tailed density (GED or Student s-t) would be more appropriate to compute the value of call and put options. The fact that the valuation of index futures options depends on the stochastic movements of the S&P 500 index futures is not sensitive to the choice of a density function (e.g., normal, GED, Student s-t). The stochastic evolution of the futures price is largely determined by the conditional volatility process. In other words, the model that provides an accurate characterization of the future stochastic volatility of S&P 500 index futures is expected to produce more accurate estimates of index futures options. Hence, the predictive power of alternative volatility models is tested with the normal, GED, and Student s-t density for the one-day-ahead realized volatility of index futures returns. This is done by estimating the conditional variance, standard deviation, and log-variance with the normal, GED, and Student s-t distributions, and then computing the proportion of the total variation in one-day-ahead realized volatility of futures returns that can be explained by the current conditional volatilities. In recent literature developed by Andersen, Bollerslev, Diebold, and Ebens (2001), Anderson, Bollerslev, Diebold, and Labys (2003), and Barndoff- Niclson and Shephard (2001, 2002), the daily realized variance of asset returns is measured by the sum of squared intraday (e.g., 5-minute) returns. The estimate of this quadratic variation is referred to as integrated (or realized) volatility. Following the aforementioned studies, the daily integrated variance of futures

21 Testing Mean Reversion 21 returns is computed as the sum of squared 5-minute returns on the S&P 500 index futures: s 2 IV,t s 2 IV,t a D t r 2 d d 1 (40) where is the integrated variance of index futures on day t, D t is the number of 5-minute returns in day t, and r d is the futures return for 5-minute interval d. The one-day-ahead forecasting performance of the GARCH, EGARCH, and TGARCH models is measured by the following regression: s IV,t 1 b 0 b 1 s e t u t (41) where s IV, t 1 is the daily integrated volatility of futures returns on day t 1, and s e t is the estimated conditional volatility of futures returns on day t. The adjusted R 2 of this regression provides information about how well each model is able to forecast the one-day-ahead integrated volatility. Table V shows that for the normal, GED, and Student s-t distributions, the conditional log-variance model (EGARCH) performs better than the conditional variance (GARCH) and conditional standard deviation (TGARCH) models in predicting one-day-ahead realized volatility of futures returns. Also, for the EGARCH, TGARCH, and GARCH models, the conditional GED density provides a better fit than the conditional Student s-t and conditional normal density. Specifically, the adjusted R 2 values for the EGARCH model are 53.95% for GED, 48.91% for Student s-t, and 46.84% for normal distribution. The corresponding values for the asymmetric TGARCH model are 46.98%, 42.90%, and 42.07% for the GED, Student s-t, and normal distributions, respectively. The symmetric GARCH model that does not allow for asymmetric volatility response to negative and positive shocks performs the worst among all density functions considered in this study. The adjusted R 2 values for the GARCH model are 45.35% for the GED-GARCH, 40.92% for the Student s-t GARCH, and 39.27% for the normal-garch model. Table V shows that the volatility forecasts from the GED and symmetric t distributions are always superior to the forecasts from the normal distribution. Although a fairly remarkable ordering was observed in terms of adjusted R 2 values, so far the statistical significance of the differences in volatility forecasts have not been tested. Although there are some clear losers in predicting the future volatility of futures price changes (like the normal-garch model), there can be no clear winner without a more formal test. Diebold and Mariano (1995) introduce widely acceptable tests of the null hypothesis of no difference in the accuracy of two competing forecasts. One of their asymptotic test statistics is used to compare the predictive accuracy of alternative models. Specifically, the volatility forecasts of some models is compared with the normal-garch model.