GARCH and volatility swaps

Abstract

This article discusses the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model. First we use a general and flexible partial differential equation (PDE) approach to determine the first two moments of the realized variance in a continuous or discrete context. Next, and also the main contribution of the paper, is a closed-form approximate solution for the so-called convexity correction, when the risk-neutral process for the instantaneous variance is a continuous time limit of a GARCH (1,1) model. Following this, we provide a numerical example using S&P 500 data.

Acknowledgments

The authors would like to thank Omar Foda (University of Melbourne), Lennart Widlund (G&W Asset Management, Sweden), Jorgen Haug (Norwegian School of Economics), Robert Trevor (Macquarie University), Jin-chuan Duan (University of Toronto), Xingmin Lu (Royal Bank of Canada), Christophe Bahadoran (Universite' Blaise-Pascal) and an anonymous referee for their very helpful comments and feedback. Robert Trevor also helped the authors with the numerical examples and the calibration process. The authors remain responsible for any errors in this paper. The discussions were mostly carried on www.wilmott.com. This is likely to be the first quantitative finance paper ever created over an internet forum.

Notes

Alireza Javaheri is a Quantitative Analyst at Citigroup Global Markets Inc. in the Multi-Asset Derivatives Research area. The opinions expressed in this article are solely those of the author and do not necessarily reflect any views by Citigroup.

The GARCH(1,1) model implies that the stock process and the volatility process contain two uncorrelated Brownian Motions. In an NGARCH process, described by (1993), we have

where c is another parameter to be estimated and creates the correlation between the two processes. However this will not affect our discussion at all and therefore we will use the GARCH(1,1) model throughout this paper. The NGARCH process is discussed for instance in (1997).

The γ in the Engle and Mezrich (1995) paper is

which is different from our definition. This is likely due to a small typo in their paper. See also Nelson (1990).

Pearson kurtosis is Fisher kurtosis plus three. The normal distribution has a Pearson kurtosis of 3 (Fisher kurtosis of 0), called mesokurtic. Distributions with Pearson kurtosis larger than 3 (Fisher higher than 0) are called leptokurtic, indicating higher peak and fatter tails than the normal distribution. Pearson kurtosis smaller than 3 (Fisher lower than 0) is termed playakurtic. Before calculating kurtosis from asset prices make sure you know if the software you are using returns Pearson or Fisher kurtosis.

For instance,

poses no problems.

See for example Haug (1997) pp. 169–70 and Wilmott (2000) pp. 299–301.

We also have

.