230 The Basics of financial economeTrics
Application of GARCH Models to Option Pricing
An option is a derivative instrument that grants the buyer of the option the
right to buy (call) or sell (put) the underlying at the strike price at or before
the option expiration date. Options are called European options if they can
be exercised only at the expiration date, American options if they can be
exercised at any time. The most well-known model for pricing European
options is the Black-Scholes model.^10 In this model, the return distribution
is assumed to be a normal distribution and the price of a European call and
put option is given by a closed form that depends on only one unknown
parameter: the variance of the return. The assumption of the Black-Scholes
model is that the volatility is constant.
Instead of assuming a constant volatility (as in the case of the Black-
Scholes model) what has been proposed in the literature is the use of either a
stochastic volatility model or a GARCH model. Stochastic volatility models
were first proposed by Hull and White.^11 The approach using a GARCH
model has been proposed in a number of studies.^12
As we have seen, standard GARCH models assume residuals are nor-
mally distributed. When fitting GARCH models to a return series, it is
often found that the residuals tend to be heavy tailed. One reason is that
the normal distribution is insufficient to describe the residual of return
distributions. In general, the skewness and leptokurtosis observed for
financial data cannot be captured by a GARCH model assuming residu-
als are normally distributed. To allow for particularly heavy-tailed con-
ditional (and unconditional) return distributions, GARCH processes
with non normal distribution have been considered.^13 Although asset
(^10) Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabili-
ties,” Journal of Political Economy 81 (1973): 637–659.
(^11) John Hull and Alan White, “The Pricing of Options on Assets with Stochastic
Volatilities,” Journal of Finance 42 (1987): 281–300.
(^12) See, for example, Jaesun Noh, Robert F. Engle, and Alex Kane, “Forecasting Volatility
and Option Prices of the S&P 500 Index,” Journal of Derivatives 2, no. 1 (1994): 17–30;
Jan Kallsen and Murad S. Taqqu, “Option Pricing in ARCH-Type Models,” Mathemat-
ical Finance 8 (1998): 13–26; Christian M. Hafner and Helmut Herwartz, “Option
Pricing under Linear Autoregressive Dynamics Heteroscedasticity and Conditional
Leptokurtosis,” Journal of Empirical Finance 8 (2001): 1–34; Christian M. Hafner and
Arle Preminger, “Deciding between GARCH and Stochastic Volatility Using Strong
Decision Rules,” Journal of Statistical Planning and Inference 140 (2010): 791–805; and
Jeroen Rombouts and Lars Stentoft, “Multivariate Option Pricing with Time Varying
Volatility and Correlations,” Journal of Banking and Finance 35 (2011): 2267–2281.
(^13) Stefan Mittnik, Marc S. Paolella, and Svetlozar T. Rachev, “Unconditional and
Conditional Distributional Models for the Nikkei Index,” Asia-Pacific Financial
Markets 5, no. 2 (1998): 99–128.