6.5. GARCH OPTION PRICING MODELS

While the GC model is capable of capturing implied volatility smiles and smirks at a given point in time, it assumes that volatility is constant over time and is thus inconsistent with the empirical observations we made earlier. Put differently, the GC model is able to capture the cross strike price structure but not the cross maturity structure in observed options prices. In Chapters 1 and 2 we saw that variance varies over time in a predictable fashion: High-variance days tend to be followed by high variance days and vice versa, which we modeled using GARCH and other types of models. When returns are independent, the standard deviation of returns at the -day horizon is simply times the daily volatility, whereas the GARCH model implies that the term structure of variance depends on the variance today and does not follow the simple square root rule.

We now consider option pricing allowing for the underlying asset returns to follow a GARCH process. The GARCH option pricing model assumes that the expected return on the underlying asset is equal to the risk-free rate, r, plus a premium for volatility risk, ?, as well as a normalization term. The observed daily return is then equal to the expected return plus a noise term. The noise term is conditionally normally distributed with mean zero and variance following a GARCH(1,1) process with leverage as in Chapter 2. By letting the past return feed into...