# Forecasting with Volatility Models

## Abstract

Forecasting of financial market volatility has been a major interest to both academic researchers and practitioners for many years. This level of interest is justified as the relevance of volatility forecasts is felt in many areas of financial decision making such as portfolio allocation, risk management, hedging strategies and derivatives pricing. Extensive research in this field has resulted in the proposal and development of a variety of models and procedures which are either motivated by the notion that (1) option prices contain valuable information about the future volatility of the underlying asset or that (2) historical financial market volatility calculated from fitting an appropriate volatility model can be successfully cast into the future. In Chapter 5 we further discuss the concept of implied volatility whereas in this chapter we focus on the forecasting ability of volatility models based on historical price information. The remaining sections of this chapter are then organised as follows. In the next section we show how the forecasts of the various volatility models evolve over time with emphasis on the Stochastic Volatility (SV) and Generalised Autoregressive Conditional Heteroskedasticity (GARCH) classes of volatility models. In section 4.2 we then present an empirical out-of-sample forecasting study of six international stock indices based on four different volatility models and for different forecasting horizons.

## Keywords

Random Walk Stock Market Stochastic Volatility Stock Index GARCH Model## Preview

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## References

- 1.Model specifications of the SV and GARCH models are given in Chapter 2. 2See, for example, Brailsford and Faff (1996) and Walsh and Tsou (1998).Google Scholar
- 3.We do not deduct the sample mean following Figlewski (1997), who for smaller samples finds the sample mean to be a very noisy estimate of the true mean.Google Scholar
- 5.An overview of this research is provided in, for example, Poon and Granger (2001a) who list a total 37 papers on the subject of volatility forecasting with historical price information.
^{6}Also see: McMillan, Speight and ap Gwilym (2000).Google Scholar - 8.Whereas model estimation with fat-tailed distributions often results in an improved in-sample fit, research by Claessen and Mittnik (1996) and Peters (2001) has shown that the use of non-normal distributions for out-of-sampling forecasting purposes is far less evident.Google Scholar
- 9.Other studies have considered, for example, standard deviations which mitigate the effect of extreme values; see Martens (2002) and Ebens (1999). However, variances appear to be the standard in the literature cited here; also see Poon and Granger (2001a).Google Scholar
- 10.This model is interchangeably referred to as the random walk (Heynen, 1995), historical volatility (Figlewski, 1997), historical mean (Brailsford and Faff, 1996) or historical average (Yu, 2000) model.Google Scholar
- 11.Short-term volatility is a main determinant in the forecasting formulas of daily SV and GARCH models, see section 4.1, and the same applies to daily EWMA models with near-unity estimates for φ. Ideally, parameter estimates should therefore be updated as soon as new price information becomes available.Google Scholar
- 12.Other evaluation methods include the Mean Squared Error (MSE), Mean Absolute Error (MAE) and variations thereof as well as the less popular Theil-U and the proportion of explained variability P statistics; see Poon and Granger (2001a) for a concise overview. 13Estimation results for the SV model where obtained using exact maximum likelihood methods as described in Appendices A and B. The relevant program and the data series used in this chapter can be found on the Internet at also see AppendixGoogle Scholar
- 14.We also examined the EWMA model forecasting performance for a five-year moving average with L equal to 1304 and found R2 values close to those observed for the RW model.Google Scholar
- 15.The highest average underestimation forecast error amounted to 7.72 whereas the overestimation equivalent was 2.29.Google Scholar
- 16.We also evaluated the forecasting power of the four models with the Mean Absolute Error (MAE) statistic (results available upon request) for which the SV and GARCH models ranked first and second again; the RW model which had the lowest number of overestimations also performed rather well, especially at the shorter forecasting horizons. Alternatively, asymmetric loss functions such as the LINEX loss function or the so-called mean mixed error statistics advocated by Brailsford and Faff (1996) could be used to provide further insight.Google Scholar