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

## Abstract

Forecasts of financial market volatility play a crucial role in financial decision making and the need for accurate forecasts is apparent in a number of areas. All investors face the decision whether or not to hedge the risks associated with their investments. Portfolio diversification strategies can be applied to reduce the total risk of a portfolio but exposures can be even further reduced by means of hedging strategies. Investors will then base their hedging decisions on their risk perception over the remaining investment horizon; the more volatile the market the more inclined investors will be to hedge their exposures. Instruments that are commonly used for this purpose are financial derivatives such as options and futures. The pricing of these sometimes very intricate instruments largely depends on the risk associated with the underlying asset and volatility is therefore also a key input parameter is many derivatives pricing models. The issue of accurate volatility forecasts is therefore firmly positioned at the centre of financial decision making. Unfortunately, it is notoriously difficult to predict volatility accurately 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^{1}.

## Keywords

Stochastic Volatility Implied Volatility Stock Index Forecast Horizon Stochastic Volatility Model## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A comprehensive survey of the findings in the volatility forecasting performance literature is given in Poon and Granger (2001a, 2001b).Google Scholar
- 2.See e.g. Akgiray (1989), Dimson and Marsh (1990) and Walsh and Tsou (1998) for an overview.Google Scholar
- 3.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).Google Scholar
- 4.SV models are reviewed in, for example, Taylor (1994), Ghysels, Harvey and Renault (1996) and Shephard (1996).Google Scholar
- 5.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 vari¬ance equation has been undertaken in, for example, our study of the Stochastic Volatility in Mean model in Chapter 3.Google Scholar
- 8.Hull and White (1987) found that when volatility was stochastic biases in the Black-Scholes model were smallest for near-the-money and close to maturity options. Also see Feinstein (1995).Google Scholar
- 9.See: Fleming, Ostdiek and Whaley (1995) and Blair et al. (2001). Also, it was pointed out to us that ignoring the wildcard option implicit in the OEX options, see Fleming and Whaley (1994) and Whaley (1993), and the fact that index option prices may include a volatility risk premium, see Bakshi and Kapadia (2001), could further contribute to an upwardly biased implied volatility measure.Google Scholar
- 10.The construction of the VIX index is described in more detail by Whaley (1993) and by Fleming, Ostdiek and Whaley (1995), who regard it as a useful proxy for expected stock market volatility.Google Scholar
- 11.Note that the graph of the squared return series is truncated at a value of 100 which only affects the observation of 19 October 1987 that has a value of 561.214.Google Scholar
- 14.See Campbell, Lo and MacKinlay (1997, Chapter 3).Google Scholar
- 15.For the VIX data set the equivalent volatility is even higher at 26.0%.Google Scholar
- 16.On the basis of the same criteria Oomen (2001) selected a sampling frequency for the FTSE 100 stock index of 25 minutes. We also considered a sample that excluded the overnight returns which had little effect on our results.Google Scholar
- 18.For all forecasting horizons volatility forecasts of the SVX+ and SIV model are near per¬fectly correlated with sample correlation coefficients of 0.99.Google Scholar
- 19.The standard SV model forecasts and the forecasts obtained with the SVX+ and SIV model have sample correlation coefficients of 0.74 and 0.76, respectively.Google Scholar
- 20.Had we used the VIX index directly the realised volatility process would have been overestimated on average, see footnote 15. Also see Fleming (1998) who reports that the Standard & Poor’s 100 implied volatilities calculated separately from call and put options contain relevant information with regard to future volatility but are nevertheless upwardly biased.Google Scholar
- 21.By comparison, the average annualised volatility of the 4 January 1988 to 3 January 1996 sample amounts to 13.2% and that of the evaluation sample to 19.9% when based on intraday volatility.Google Scholar
- 22.It might be of interest for future research to investigate alternative methods for combining SV model volatility forecasts and option implied volatility. For example, in the GARCH literature Donaldson and Kamstra (2001) develop a model with trading volume as regime-switching variable. They find that implied volatility dominates during high volume periods and that GARCH is more important when trading volume is low.Google Scholar