Mean-correction and Higher Order Moments for a Stochastic Volatility Model with Correlated Errors

In an efficient stock market, the log-returns and their time-dependent variances are often jointly modelled by stochastic volatility models (SVMs). Many SVMs assume that errors in log-return and latent volatility process are uncorrelated, which is unrealistic. It turns out that if a non-zero correlation is included in the SVM (e.g., Shephard (2005)), then the expected log-return at time t conditional on the past returns is non-zero, which is not a desirable feature of an efficient stock market. In this paper, we propose a mean-correction for such an SVM for discrete-time returns with non-zero correlation. We also find closed form analytical expressions for higher moments of log-return and its lead-lag correlations with the volatility process. We compare the performance of the proposed and classical SVMs on S&P 500 index returns obtained from NYSE.


Introduction
Over the last few decades different aspects of stock price movements in discrete time have been the focus of numerous research avenues. Suppose P t denotes the price of a stock at time t, then the continuously compounded return or log-return (here onwards referred to as return) of the stock is defined as r t = log(P t /P t−1 ). A stock market is said to be efficient if the price of a stock contains every available information about it. In such a market the risk involved in investing on a stock is measured by the standard deviation of r t , often termed as the volatility of the stock in finance literature. It has been noted that volatility varies over time (Engle 1982). Stochastic Volatility Models (SVMs) is a popular class of models for describing the time-varying volatility of stock returns (Shephard 2005).
Although there are a plethora of SVMs for describing the stock returns, one of the simplest yet most popular discrete-time SVM is given by Taylor (1982), where the return process r t is a non-linear product of two independent stochastic processes, viz. an i.i.d. error process ǫ t , and a latent volatility process h t , which is further modelled as an AR(1). That is, where α = E(h t ) is the long-range volatility, φ is the stationarity parameter, σ measures the variability of the volatility process h t , and ǫ t and η t are uncorrelated i.i.d. N(0, 1) errors.
Hereafter this model will be referred as SV M 0 .
As in (1), many of the new generation SVMs which are being used in the finance literature assume that ǫ t and η t are independent N(0, 1) errors. In reality, however, ǫ t and η t are often correlated (Harvey & Siddique 1999). Though discrete-time SVMs with non-zero corr(ǫ t , η t ) have been developed earlier and are being used, they assume that h t+1 (instead of h t as in (1)) depends on η t via AR(1) (see e.g. Meyer & Yu (2000) and Berg, Meyer & Yu (2004)). In this paper, we focus on the SVM presented in (1) with correlated errors (denoted as SV M ρ ).
It turns out that introducing a non-zero correlation between η t and ǫ t in (1) has an adverse effect on the admissibility of the SVM from an efficient market's viewpoint. In particular, the conditional expectation of r t given the past data, E[r t | F t−1 ], is not zero, where F t−1 is the space (σ-field) generated with r 1 , ..., r t−1 . This zero conditional expectation of the return is a necessary requirement for an efficient market hypothesis (EMH) (see Yu (2005) for a review).
In this paper, we propose a mean-correction for SV M ρ -model (1) with correlated errors, such that E[r t | F t−1 ] becomes zero and the corrected SVM would satisfy EMH. The proposed mean-corrected model is denoted by SV M ρµ . Further, Black (1976) mentioned that, usually, the amount of increment in volatility due to price fall is larger than the magnitude of reduction in the volatility due to price increase. In turn, this indicates the volatility of positive returns, var(r t |r t > 0), is less than the volatility of the negative returns, var(r t |r t < 0) resulting in skewness in return distribution. Moreover, the kurtosis quantifies the proportion of extreme values, that occur during crashes, explained by the model. We find the closed form expressions for the higher-order moments and the lead-lag correlation of the underlying return process. These descriptive statistics indicate the influence of past/future volatility on today's return.
The remainder of the article is organized as follows. Section 2 presents the main results: SV M ρµ -the mean-corrected SVM with non-zero correlation that satisfies EMH, and the closed form analytical expressions for the higher order moments and lead-lag correlation for the proposed model. For the returns of S&P 500 NSYE, Section 3 presents a comparison between the standard zero correlation model (1) and the ones with non-zero correlation.
Finally Section 4 outlines the concluding remarks and a few possible future directions.

Main Results
For this section, we assume that the error terms ǫ t and η t in (1) have not only a constant correlation ρ and i.i.d. N(0, 1) marginals, but they also follow a bivariate normal distribution.
The proposed mean-corrected model (SV M ρµ ) contains an additional term µ, i.e., Theorem 1 establishes the value of µ for which the proposed mean-corrected model (2) gives zero conditional expectation E[r t | F t−1 ] and hence satisfy EMH. Later in this section, we derive closed form expressions for the higher-order moments, i.e., variance, skewness, and kurtosis of r t , and lead-lag correlations between r t and h t±k .
Proof The conditional expected return E[r t | F t−1 ] = 0 gives Since (ǫ t , η t ) follows a standard bivariate normal with correlation ρ, the condition distribution of ǫ t | η t is given by N (ρη t , 1 − ρ 2 ). This conditional normal distribution and the moment generating function (mgf) of a normal distribution simplifies the third term in (4) as and the second term to Hence the final expression for µ follows from (4)-(6). ✷ Yu (2005) tried to compute E[r t | F t−1 ], but the final expression appears to be incorrect.
Note that the proposed mean-correction (in Theorem 2.1) makes the model (2)

Higher-order moments
For additional key features on the distribution of returns, we estimate higher order moments, in particular, variance, skewness and kurtosis conditional on F t−1 .
Theorem 2 For SV M ρµ in (2), if Theorem 2.1 holds, then the variance of returns conditional on F t−1 is given by Proof Following the definition of variance, (4)- (6)).
The final result follows by substituting the value of µ from Theorem 2.1. ✷ The expressions of the conditional mean and variance are the most crucial components in finding the skewness and kurtosis statistics. For SV M ρµ in (2), under the same conditions as in Theorem 2.2, the skewness conditional on F t−1 is measured by The proof of (8) starts with µ 3 = E[r 3 t | F t−1 ], and proceeds in the exact same manner as in Theorems 2.1 and 2.2. Similarly the closed form expression of kurtosis can also be found as µ 4 /(V ar(r t | F t−1 )) 2 , where As expected all four descriptive statistics found here depends heavily on corr(ǫ t , η t ) = ρ.
On a closer inspection of these statistics, we see that ρ = 0 (i.e., the classical SVM by (Taylor 1982)) gives µ = 0, µ 3 = 0, Note that the simplified expressions found here are consistent with the ones reported by Ghysels, Harvey & Renault (1996), and hence the proposed model SV M ρµ (2) is a general-ization of the classical model SV M 0 in (1). Next we investigate the conditional (on F t−1 ) dependence between the current returns and past, current and future volatility.

Lead-lag correlations
In this section, we wish to estimate three quantities: (1) dependence between the current returns and current volatility, corr(r t , h t |F t−1 ), (2) the potential influence of current returns on future volatility, corr(r t , h t+k |F t−1 ), and (c) the influence of past volatility on current returns, corr(r t , h t−k |F t−1 ). Though empirical estimation of such quantities is not uncommon, e.g., in Bollerslev, Litvinova & Tauchen (2006), our aim is to find closed analytical expression for these descriptive measures under SV M ρµ specification.
Since var(r t |F t−1 ) is given by (7) and var(h t |F t−1 ) = σ 2 /(1 − φ 2 ), we only need to find the expressions for the conditional covariances. First, we recall that under the proposed model, the conditional means are E(r t |F t−1 ) = 0 and E(h t |F t−1 ) = α. Now, if we assume which further implies that cov(r t , h t+k ) = φ k σ rh for k ≥ 1. By applying the key mathematical techniques (i.e., properties of expectation, normal mgf and the expansion of h t = α + σ ∞ j=1 η t−j φ j ) used in proving results of Section 2.1, one can easily show that Clearly, both the lead (cov(r t , h t+k )) and lag (cov(r t , h t−k )) covariances are smaller than the contemporaneous covariance cov(r t , h t−k |F t−1 ). The contemporaneous correlation can be interpreted as feedback effect of volatility change on future returns, whereas the impact of return change on future volatility is termed as leverage effect. Bekaert & Wu (2000) found that volatility feedback effect is stronger than leverage effect. The closed form expressions we have derived above provide a theoretical proof of the mentioned findings under SV M ρµ specification. Moreover, Bollerslev et al. (2006) have empirically observed that the lagcorrelation with lag h is smaller than lead correlation with lead h which we have established theoretically. Further note that all these covariances and hence correlations vanish if ρ = corr(ǫ t , η t ) = 0. Next, we compare the goodness of fit of the three stochastic volatility models, SV M 0 (classical -with zero correlation), SV M ρ (with correlation ρ) and SV M ρµ (mean-corrected with correlation ρ), for a real data on returns.
3 Example: S&P 500 NYSE In this paper, we compare the performance of the three models ( , For SV M ρ , the mean and variance of the conditional distribution of r t changes to and the conditional distribution of h t remains the same. Similarly, the implementation of the mean-corrected model SV M ρµ is characterized by updating the mean and variance of the conditional distribution of r t to The parameters of interest are (α, φ, ρ, σ) = Θ (say). We use the same prior (including the hyperparameters) for α, φ and σ as in Meyer & Yu (2000), and a non-informative Unif (−1, 1) prior for the correlation parameter. The posterior of Θ and H = {h t , h t−1 , ...} given the data {r t , r t−1 , ...} is obtained via JAGS. We set the total length of chains to be 180,000, out of which 30,000 was the burn-in, and from the remaining 150,000 posterior realizations (with the thinning of every 50 th realization) were used (i.e., 3000 realizations in total) to obtain the plug-in estimates of the parameters. The thinning process facilitates a safeguard against the chain dependency in the sampling process. Figure Table 1). Table 1 shows that the posterior estimates of the parameters in SV M 0 , SV M ρ and SV M ρµ are similar. Further, the near-unity estimate of φ indicates presence of strong volatility clustering. The estimate of the correlation parameter ρ is small yet positive, which is  (1987) and Campbell & Hentschel (1992). This may be taken as an indication of no significant effect of current return on future volatility. We now compare the three models using the descriptive measures (mean, variance, skewness and kurtosis), three lead-lag correlations, mean deviance over the posterior distribution, and the mean square prediction error (MSPE): T t=1r 2 t /T . The deviance function, suggested by (Dempster 1974), is D(Θ) = −2 log f (r | Θ, H) + 2 log g(r), where f (r | Θ, H) is the likelihood for a given realization of Θ and H, and g(r) is the normalizing constant. Table 2 presents the plug-in values of these "goodness of fit" measures for the three models.
Since ρ ≈ 0.1 (very small), the estimated mean is also small µ = −4.05 · 10 −6 . Thus MSPE values indicate that the basic SVM provides better prediction among the three models.
Though the numerical results presented through the S& P 500 NYSE example do not provide sufficient evidence for SV M ρµ giving additional information than SV M 0 , it certainly is the generalization of SV M 0 and an example with large ρ = corr(ǫ t , η t ) might have given more convincing evidence.

Concluding Remarks
In this paper, we have proposed a mean-correction for the SVM with correlation between ǫ t and η t . This mean-correction step enables the conditional expected return to be zero, which is a necessary condition for a good SVM (i.e., a model that adhere to the EMH). We have also found the closed form analytical expression for the higher moments of returns and lead-lag correlation between the return and volatility.
From S&P500 example, we see that most of the empirical observations on statistical properties of returns are reflected through all the three models. However, SV M ρµ gives a slightly better fit to the data (in terms of average deviance) compared to the classical model SV M 0 as well as SV M ρ . A close look at this research endeavour generates several interesting and challenging research problems.
First, the estimated error correlation ρ turns out to be positive despite the fact that return and its volatility move in opposite directions (Nelson 1991 Second, the observed kurtosis from the data is not completely explained by the model based estimates of kurtosis. Indeed, the significant difference between empirical kurtosis and the model based estimates again suggests non-normality of the return error distribution. The problem can be tackled in two ways-(1) introducing jumps in returns or (2)  Although continuous time stochastic volatility has been studied extensively in the literature, the comparatively new discrete-time SVM brings out new interesting features such as leverage effect and feedback eefect which occurs due to lagged reaction between return and its volatility. In this paper we have established that the empirically observed pattern of leverage effect and lagged correlations (Bollerslev et al. 2006) are explained by SV M ρµ . In particular, we have shown that the correlation between current return and future volatility is maximum in magnitude at lead 0 (or contemporaneously) and the future leverage effects disappear exponentially with the lead time. Indeed, strong volatility clustering effect in-dicates more persistent leverage effect. It may also be noted that the existing practice of assuming h t+1 = α + φ(h t − α) + ση t (instead of h t ) and corr(ǫ t , η t ) = ρ for a correct SVM specification would not support the empirical observation on contemporaneous correlation.
Mean correction to the contemporaneously correlated SVM has a very important application. Zero conditional expected returns is also referred to as the martingale difference property, which is a necessary and sufficient condition for no arbitrage -which further leads to the existence of option pricing kernel (Back (1991), Shephard & Andersen (2009)). That is, we believe that the proposed mean-correction strategy can also be used in option pricing.