# Asset Return Volatility Models

Chapter

## Abstract

Whereas the returns on financial assets can be assumed to be serially uncorrelated, although the issue of asset return predictability is still fiercely debated in the literature, see where

*e.g*. Campbell and MacKinlay (1999), it has become widely accepted that return volatility is highly autocorrelated. If the volatility process of financial asset returns were to be perfectly correlated,*i.e*. if volatility was constant over time, the variance of the underlying series could simply be calculated as$${\sigma ^2} = \frac{1}{{T - 1}}\sum\limits_{t = 1}^T {{{\left( {yt - m} \right)}^2}} ,$$

(2.1)

*y*_{ t }is the continuously compounded return on the relevant asset at time*t*,*T*is the total number of observations and the sample mean \(m = \frac{1}{T}\sum\nolimits_{t = 1}^T {yt} \) A practical question that arises is what the length of the sample should be when volatility changes over time. Conventional wisdom would suggest that the sample should consist of a large number of observations as this would improve the statistical accuracy of the volatility estimates. On the other hand, it also implies that extreme shocks to the return process that took place a relatively long time ago, and which contain little information about the current volatility level, will still have a major impact since all observations in the sample are weighted equally. An alternative could be to select a smaller sample size and use a rolling window principle, where the sample size*T*remains constant and the variance is recalculated with each new observation that becomes available. The problem of extreme values remains however as the variance estimate of equation (2.1) will drop considerably in value when observations relating to this event fall out of the sample. Exponentially weighted methods have been proposed to remedy this problem but the reported high autocorrelation of the volatility process has led to the development of more intricate time-varying volatility models which incorporate this feature.## Keywords

Stochastic Volatility Asset Return GARCH Model Stochastic Volatility Model Return Series
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## References

- 1.We refer to these models collectively as GARCH models. For surveys on the extensive GARCH literature we refer to Bollerslev, Chou and Kroner (1992), Bera and Higgins (1993), Bollerslev, Engle and Nelson (1994), Diebold and Lopez (1995) and to the comprehensive selection of influential (G)ARCH papers in Engle (1995).Google Scholar
- 2.In order to ensure that the process is covariance stationary is set to be less than one. The positivity constraints have been questioned by Nelson and Cao (1992) who give necessary and sufficient conditions to ensure a non-negative variance which are less stringent. These weaker conditions are however mainly relevant for the higher order GARCH models.Google Scholar
- 3.The development of the GARCH model was primarily motivated by the fact that in empirical applications high orders of q were needed.Google Scholar
- 4.Nelson (1991) suggested that εt follow a Generalised Error Distribution (GED) which is a family of distributions including the normal distribution for v = 2 and fatter tailed distributions for v 2.Google Scholar
- 5.SV models are reviewed in, for example, Taylor (1994), Ghysels, Harvey and Renault (1996) and Shephard (1996).Google Scholar
- 6.See Hull and White (1987), Scott (1987), Wiggins (1987) and Chesney and Scott (1989).Google Scholar
- 7.If the correlation coefficient between ε
_{t}and η*t*+1 is estimated and found to be negative, a leverage effect similar to that discussed with regard to the EGARCH model is observed. Also see the discussion and references in Harvey and Shephard (1996).Google Scholar - 9.See, for example, Jacquier, Polson and Rossi (1994), Kim, Shephard and Chib (1998), Sandmann and Koopman (1998) and Fridman and Harris (1998).Google Scholar
- 10.See e.g. French, Schwert and Stambaugh (1987) and Akgiray (1989) who examined the US stock market and Poon and Taylor (1992) for results with regard to the UK stock market.Google Scholar
- 11.For the estimation of the SV model we use the exact maximum likelihood methods as described in Appendices A and B. Computing times for estimating SV models is usually higher than for estimating GARCH models which is due to the relatively more complicated methods required for estimating SV models.Google Scholar
- 12.See for example Baillie and Bollerslev (1989). For our data set this is confirmed by the insignificant Q(12) and Q(30) statistics for the monthly squared returns in table 2. 1.Google Scholar
- 13.For recent research on SV models with fat-tailed error distributions we refer to Liesenfeld and Jung (2000) and Jacquier, Polson and Rossi (2001).Google Scholar
- 18.We note that their SV model estimate of the volatility persistence term is equal to 0.91 which explains the rapid exponential decay of the ACF (see figure 1); the high implied first-order autocorrelation coefficient for the SV model is then mainly attributable to the relatively high estimate for the variance of the log volatility process. Also see the discussion in Ghysels et al. (1996).Google Scholar
- 19.Also see the discussion in Ghysels et al. (1996) who show that ρs ”is at maximum for a normal distribution”.Google Scholar

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© Springer Science+Business Media Dordrecht 2003