6.7. SUMMARY

This chapter has surveyed some key models for pricing European options. First we introduced the famous Black-Scholes-Merton (BSM) model. The key assumption underlying the BSM model is that the underlying asset return dynamics are captured by the normal distribution with constant volatility. While the BSM model provides crucial insight into the pricing of derivative securities, the underlying assumptions are clearly violated by observed asset returns. We therefore next considered a generalization of the BSM model, which was derived from the Gram-Charlier (GC) expansion around the normal distribution. The GC distribution allows for skewness, and kurtosis and it therefore offers a more accurate description of observed returns than does the normal distribution. However, the GC model still assumes that volatility is constant over time, which we argued in earlier chapters was unrealistic. Next, we thus presented two types of GARCH option pricing models. The first type allowed for a wide range of variance specifications, but the option price had to be calculated using Monte Carlo simulation or another numerical technique as no closed-form formula existed. The second type relied on a particular GARCH specification but in return provided a closed form solution for the option price. Finally, we introduced the ad hoc implied volatility function (IVF) approach, which in essence consists of a second-order polynomial approximation to the implied volatility smile.