Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility

Abstract

In this paper we compare the predictive abilility o f S t o c hastic Volatility (S V) 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 c a l l t h e S V X + 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 out-of-sample, 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 in-sample but that the latter contains incremental information in the form of stochastic shocks incorporated in the SVX models. The out-of-sample 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 out-of-sample volatility forecasts and that the model that only utilises implied volatility performes the worst as its volatility forecasts are upwardly biased.

The effect of inflation uncertainty on inflation: Stochastic volatility in mean model within a dynamic framework

“…A choice for the importance density is conditional upon density function since in the case of Gaussian it is relatively straightforward to sample from p y = θ; ψ ð Þ= g y= θ; ψ ð Þ using simulation smoothers such as the ones developed by De Jong and Shephard (1995) and Durbin and Koopman (2002). Hol and Koopman (2000) and Asaf (2006) provide a guideline for the construction of an importance model and the likelihood function for the SV model using this model. One may also visit Koopman and Uspensky (2002) for details.…”

The inflation and inflation uncertainty relationship for Turkey: a dynamic framework

“…The likelihood function of the approximating Gaussian model can be calculated via the Kalman Filter. Guidelines for the construction of an importance model and the likelihood function for the SV model are given by Hol and Koopman (2000), Durbin and Koopman (2002) and Asaf (2006). 10…”

“…GARCH models also include past conditional variances, σ 2 t−i , along with past squared innovations, ε 2 t−i . Similarly, EGARCH models include the logarithm of past conditional variances 2 Since including explanatory variables at time t could give a biased estimator for ln σ 2 t , we do not allow the variance equation to include any variable at time t. Moreover, as the term ln σ * 2 can be regarded as the constant term in the logarithm of the volatility equation (h t ), the logarithm of the volatility process does not include an additional intercept term (Hol and Koopman 2000). 3 A full discussion of the SVMs and their estimation procedures are provided in the appendix.…”

Assessing the performance of a prediction error criterion model selection algorithm in the context of ARCH models

“…Pagan and Schwert (1990) use statistical criteria to compare the in-sample and out-of-sample performance of parametric and nonparametric ARCH models. Besides, Heynen and Kat (1994) investigate the predictive performance of ARCH and stochastic volatility models and Hol and Koopman (2000) compare the predictive ability of stochastic volatility and implied volatility models. Andersen et al (1999) applied heteroscedasticityadjusted statistics to examine the forecasting performance of intraday returns.…”