Unformatted text preview: TI 2000104/4 Tinbergen Institute Discussion Paper Forecasting the Variability of Stock F orecasting Index Returns with Stochastic Volatility Models and Implied Volatility Eugenie Hol Siem Jan Koopman Tinbergen Institute
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Eugenie Hol and Siem Jan Koopman Department of Accounting and Finance, University of Birmingham Department of Econometrics, Free University Amsterdam November 13, 2000 In this paper we compare the predictive abilility of Stochastic Volatility (SV) models to that of volatility forecasts implied by option prices. We develop an SV model with implied volatility as an exogeneous variable in the variance equation which facilitates the use of statistical tests for nested models we refer to this model as the SVX model. The SVX model is then extended to a volatility model with persistence adjustment term and this we call the SVX+ model. This class of SV models can be estimated by quasi maximum likelihood methods but the main emphasis will be on methods for exact maximum likelihood using Monte Carlo importance sampling methods. The performance of the models is evaluated, both within sample and outofsample, for daily returns on the Standard & Poor's 100 index. Similar studies have been undertaken with GARCH models where ndings were initially mixed but recent research has indicated that implied volatility provides superior forecasts. We nd that implied volatility outperforms historical returns insample but that the latter contains incremental information in the form of stochastic shocks incorporated in the SVX models. The outofsample volatility forecasts are evaluated against daily squared returns and intradaily squared returns for forecasting horizons ranging from 1 to 10 days. For the daily squared returns we obtain mixed results, but when we use intradaily squared returns as a measure of realised volatility we nd that the SVX+ model produces the most accurate outofsample volatility forecasts and that the model that only utilises implied volatility performes the worst as its volatility forecasts are upwardly biased. Abstract KEYWORDS: Forecasting, Implied Volatility, Monte Carlo likelihood method, Stochastic volatility, Stock indices. 1 Introduction
Forecasts of nancial market volatility play a crucial role in nancial decision making and the need for accurate forecasts is apparent in a number of areas, such as option pricing, hedging strategies, portfolio allocation and ValueatRisk calculations. Unfortunately, it is notoriously di cult to accurately predict volatility and the problem is exacerbated by the fact that realised volatility has to be approximated as it is inherently unobservable. Due to its critical role the topic of volatility forecasting has however received much attention and the resulting literature is considerable. One of the main sources of volatility forecasts are historical parameteric volatility models such as Generalised Autoregressive Conditional Heteroscedasticity (GARCH) and Stochastic Volatility (SV) models. The parameters in these models are estimated with historical data and subsequently used
Corresponding author: Siem Jan Koopman, Department of Econometrics, Free University, De Boelelaan 1105, NL1081 HV Amsterdam. Email [email protected] 1 to construct outofsample volatility forecasts. The high degree of intertemporal volatility persistence observed by these models suggests that the variability of stock index returns is highly predictable and that past observations contain valuable information for the prediction of future volatility. Studies comparing the forecasting abilities of the various volatility models have been undertaken for a number of stock indices and the general consensus appears to be that those models that attribute more weight to recent observations outperform others1 . Little e ort has however been made to compare exante volatility forecasts produced by GARCH models with those of SV models2. An alternative information source for volatility prediction is found in implied volatility which is calculated from option prices in combination with a certain option pricing model. Early empirical studies by Latane and Rendelman (1976), Chiras and Manaster (1978) and Beckers (1981) have indicated that implied volatility, when compared with historical standard deviations, can be regarded as a good predictor of future volatility. Implied volatility is also often referred to as the market's volatility forecast and is forward looking, as opposed to historical based methods which are by de nition backward looking. Provided that the option market is e cient and that the option pricing model has been correctly speci ed, the information content of implied volatility should therefore subsume that of all other variables in the information set. The question whether the most accurate volatility forecasts are produced by implied volatility, rather than by the historically based volatility models, was rst addressed by Day and Lewis (1992) who developed a GARCH model with embedded implied volatility. Contrary to theory, their results indicated that GARCH models provided better volatility forecasts than implied volatility but that the latter might contain additional information as the best forecasts were obtained using both sources of information. Canina and Figlewski (1993) even found "little or no correlation at all between implied volatility and subsequent realized volatility" and favoured a simple historical volatility measure. Findings in the early nineties were therefore mixed and the assumed comprehensive information content of implied volatility appeared questionable as Lamoureux and Lastrapes (1993) were also unable to reject the hypothesis that predictions based on GARCH models contained incremental information about future volatility. Recent studies by Christensen and Prabhala (1998), Fleming (1998) and Blair, Poon and Taylor (2000) are however much more supportive as all present evidence that the most accurate volatility forecasts for returns on the Standard & Poor's 100 stock index are based on implied volatility. Moreover, their research strongly suggests that historical data contains little or no incremental information about future volatility3 . Thusfar the issue of comparitive forecasting ability has however not been studied in the context of SV models. In recent years this class of volatility model has received considerable attention in the literature and it can now be regarded as a competitive alternative to GARCH models eventhough its empirical application has been limited. In this paper we examine the predictive ability of the SV model and compare its volatility forecasts with those of implied volatility. For this purpose we introduce an SV model which incorporates implied volatility as an exogeneous variable in the variance equation. This model, which we will refer to as the Stochastic Volatility with eXogeneous variables (SVX) model, allows us to perform statistical tests for nested models. We evaluate the predictive performance for daily returns on the Standard & Poor's 100 index and as a measure of implied volatility we use the VIX index of the Chicago Board Options Exchange (CBOE). In addition, we compare the exante forecasting ability of the di erent methods over a ve year evaluation period for forecasting horizons ranging from 1 to 10 trading days. As measures of realised volatility we consider both daily squared returns and intradaily squared returns. The SV class of models considered in this paper are estimated using exact maximum likelihood
produced more accurate volatility forecasts than GARCH models. 3 It has been suggested, most notably by Blair et.al (2000), that the earlier ndings are due to measurement errors in the calculated implied volatility measure.
1 See e.g. Akgiray (1989), Dimson and Marsh (1990) and Walsh and Tsou (1998) for an overview. 2 An exception is Heynen (1995) who examined a variety of international stock indices and found that SV models 2 methods which are based on Monte Carlo simulation techniques such as importance sampling and antithetics. More accurate estimates of the likelihood function are obtained when the number of simulations is increased. Therefore the estimates can be as accurate as desired at the cost of computer time. The remainder of this paper is structured as follows. In the next section we discuss the various model speci cations while in section 3 we present the relevant estimation methods. The data and insample estimation results are presented in section 4. In section 5 we give details of our forecasting methodology and the outofsample forecasting results are presented in section 6. In the nal section we conclude and provide a summary. 2 Model Speci cations
Generalised Autoregressive Conditional Heteroscedasticity (GARCH) models have thusfar been the most frequently applied class of timevarying volatility model. Since its introduction by Engle (1982) and subsequent generalisation by Bollerslev (1986) this model has been extended in numerous ways which usually involved alternative formulations for the volatility process4 . Although the Stochastic Volatility (SV) model has been recognised as a viable alternative to the GARCH model, the latter is still the standard in empirical applications5. This is mainly due to the problems which arise as a consequence of the intractability of the likelihood function of the SV model which prohibits its direct evaluation. However, in recent years considerable progress has been made in this area which does not only encourage further empirical research but also enables the development of various extensions of the SV model. One of the possible extensions involves the inclusion of exogenous variables in the variance equation which we will discuss in this paper the resulting model we refer to as the SVX model. Volatility models are usually de ned by their rst two moments, the mean and the variance equation. The general notation for the mean equation of timevarying volatility models is given by yt = t + t "t "t NID(0 1) t = 1 ::: T (1) where yt denotes the return series of interest and t its conditional mean6 . The disturbance term "t is assumed to be identically and independently distributed with zero mean and unit variance. In addition, the assumption of normality is added. A common notation for the variance equation of the SV class of volatility models is given by
2 t = 2 exp(ht ) (2) and it is therefore de ned as the product of a positive scaling factor 2 and the exponential of the stochastic process ht . For the standard SV model this process is speci ed as ht = ht;1 + t t NID(0 1) (3) 4 For surveys on GARCH models we refer to Bollerslev, Chou and Kroner (1992), Bera and Higgins (1993), Bollerslev, Engle and Nelson (1994) and Diebold and Lopez (1995). 5 SV models are reviewed in, for example, Taylor (1994), Ghysels, Harvey and Renault (1996) and Shephard (1996). 6 For SV models the conditional mean is usually assumed to be equal to zero or is modelled prior to estimation of the volatility process. Simultaneous estimation of the mean and variance equation has been undertaken in, for example, Koopman and Hol Uspensky (2000). where the degree of volatility persistence is measured by the parameter which is restricted to a positive value smaller than one in order to ensure the stationarity of the volatility process, so 0 < < 1. Further, it is assumed that the disturbance term t is mutually uncorrelated with the error term "t 3 in the mean equation (1), both contemporaneously and at all lags. The SV model with embedded implied volatility is labelled the SVX model and we could specify its stochastic process as ht = ht;1 + xt + t (4) t NID(0 1) where xt denotes the contemporaneous implied volatility measure in logarithmic squared form, so xt = ln 2 t. The value for in the SVX model is restricted to be less than one in absolute values, IV i.e. ;1 < < 1. The problem with this speci cation is that it includes an entire lag structure of the implied volatility measure which becomes apparent when we rewrite the volatility process in logarithmic terms as ln 2 = ln 2 + ht t = ln 2 + ht;1 + xt + t = (1 ; ) ln 2 + ln 2;1 + xt + t t and if we repeatedly substitute for the lagged volatility process we observe that ln 2 = ln t
2 + xt + t; X
1 In comparison, the equivalent notation for the SV model, with ht as de ned in equation (3), can be written as t;1 Xi 2 2 ln t = ln + t;i : Inclusion of these multiple lagged implied volatility measures lead to a downwardly biased value for when iPpositive. For a negative parameter the estimate for will be asymptotically upwardly s biased, as 1 i < 0 for ;1 < < 0. Obviously, the size of this bias depends on the estimated value i=1 for the persistence parameter . If is close to zero and insigni cant, i.e. if all volatility information is impounded in the implied volatility measure, will only be marginally biased. For the GARCH class of models the issue of a multiple lagged implied volatility structure was pointed out by Amin and Ng (1997), who suggested a persistence adjustment term. A similar structure can be implemented for the SVX model by de ning ht as follows ht = ht;1 + (1 ; L)xt + t (5) t NID(0 1) which by recursive substitution of the logarithmic variance equation leads to ln 2 = (1 ; ) ln 2 + ln 2;1 + (1 ; L)xt + t t t = ln
2 i=1 i xt;i + t; X
1 i i=0 t;i : i=0 + xt + t; X
1 i and therefore omits the implied volatility lag structure. By de ning ht as in equation (5) we therefore obtain an alternative SVX model, which we will refer to hereafter as the SVX model with persistence adjustment, or the SVX+ model. Finally, we also consider a deterministic volatility model that only utilises implied volatility as a source of volatility information. This model we obtain by imposing the restrictions = 0 and 2 = 0 on equation (4) or (5), and therefore ht = xt (6) with ln 2 = ln 2 + xt : t This last of our four models we term the VX model as the volatility process does not have a separate error term and is solely determined by exogeneous variables. 4 i=0 t;i 3 Model Estimation
In this section we show how the parameters of the SVX class of models can be estimated by simulated maximum likelihood using importance sampling. Further, we show how to compute the conditional mean and variance of the volatility process ht . First we show that a quasimaximum likelihood method can also be used. Consider model (1), (2) and (3) which we can transform by taking logs of squared yt 's, that is yt = ln yt2 , to obtain the model yt = + ht + ut = ln 2 ut ln 2 1 with ht given by (3) for t = 1 : : : T . We take t as zero, implying that the yt process remains unmodelled. The resulting model is within the class of linear state space models for an introduction to state space models we refer to Harvey (1993, Chapter 4). Note that the disturbance term of the model for yt is nonGaussian. Nevertheless, the Kalman lter and the associated smoothing algorithm produce the minimum mean square linear estimator for ht see Harvey (1993, section 4.3). By assuming that the disturbance ut is normally distributed with mean and variance set equal to the mean and variance of a ln 2 variable, we obtain socalled quasi maximum likelihood estimates of the 1 unknown parameters , and when maximising the Gaussian likelihood function with respect to these parameters. This estimation procedure is proposed by Harvey, Ruiz and Shephard (1994) and is implemented in the computer package STAMP of Koopman et.al (2000). The inclusion of explanatory variables in the logvolatility process does not complicate matters further. The logsquared transformation does not a ect the logvolatility processes of the SVX models as de ned in equations (4) and (5). Therefore the Kalman lter can still be applied to the resulting linear model. However the regression coe cient need to be estimated additionally. In the following we will develop exact maximum likelihood methods for the estimation of the parameters of SV and SVX models. Quasimaximum likelihood methods will not be used in the empirical studies of sections 4 and 6 since we prefer to use exact likelihood methods. Let y = (y1 : : : yT )0 and = ( 1 : : : T )0 where observation yt is modelled as (1) and its logvolatility is given by 2 t = + ht t = exp( t ) with signal ht modelled as (3), (4) or (5), for t = 1 : : : T . Note that 2 = exp( ). Further we shall collect the parameters which are not included in the state vector below in the parameter vector . The SVX model (1), (2) and (4) in state space form is given by 3.1 Quasimaximum likelihood of SV and SVX models 3.2 Exact maximum likelihood of SVX models using importance sampling p(yj )= T Y t=1 N(0 2 ) t with t = 0, 2 = exp( t ) = exp( + ht ) = 2 exp(ht ). The state vector collects the components t of the logvolatility and is given by t = ( ht )0 . The socalled transition equation for the state vector is given by 2 301 10 0 6 0 1 0 7 t+B 0 C t 5 @0 A t+1 = 4 0 xt 5 where disturbances t are distributed as NID(0 1). The initial state vector 1 is given by 0 1 80 1 2 39 >0 > <B C 6 0 0 B C N @0A 40 7= 0 @A: 5> >0 2 2 0 0 =(1 ; ) h1 for some arbitrary large value for . Finally, the parameter vector is given by 0 =B @ 1 C: A This completes the model speci cation in state space form. It follows that p(yj )= T Y This representation of SV and SVX models can be regarded as a nonlinear state space model. The aim now is to estimate the parameter vector by exact maximum likelihood. This requires the evaluation of the loglikelihood function. A convenient expression for the loglikelihood is developed below. In section 3.2.2 we provide some computational details required for estimation. The state vector t elements, which include the regression coe cients and and the stochastic logvolatility process ht , are estimated using signal extraction methods which are brie y discussed in section 3.2.3. t=1 N(0 exp t ) t = (1 0 1) t : 3.2.1 Loglikelihood evaluation The construction of the exact likelihood for the SV model using the Monte Carlo likelihood approach of Shephard and Pitt (1997) and Durbin and Koopman (1997) can be modi ed for the SVX model. The nonlinear relation between the logvolatility ht and the observation equation is not altered in the SVX case only the speci cation for ht is di erent. Similar considerations are discussed by Chib, Nardari and Shephard (1998) in a Bayesian context. The same modi cation can be used for the SVX+ model since we merely replace the explanatory variable xt by (1 ; L)xt . The loglikelihood function for the SVX model can be computed via the Monte Carlo technique of importance sampling. The likelihood function can be expressed as L( ) = p(yj ) = p(y j )d = p(yj Z Z )p( j )d : (7) An e cient way of evaluating the likelihood is by using importance sampling see Ripley (1987, Chapter 5). We require a simulation device to sample from an importance density p( jy ) which we ~ prefer to be as close as possible to the true densitity p( jy ). An obvious choice for the importance density is the conditional Gaussian density since in this case it is relatively straightforward to sample from p( jy ) = g( jy ). An approximating Gaussian model for the SVX model is developed in ~ the appendix. The simulation smoother of de Jong and Shephard...
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