6.1. CHAPTER OVERVIEW

The previous chapters have established a framework for constructing the distribution of a portfolio of assets with simple linear payoffs for example, stocks, bonds, foreign exchange, forwards, futures, and commodities. This chapter is devoted to the pricing of options. An option derives its value from an underlying asset but its payoff is not a linear function of the underlying asset price, and so the option price is not a linear function of the underlying asset price either. This nonlinearity adds complications to pricing and risk management.

In this chapter we will do the following:

1. Provide some basic definitions.

2. Establish an option pricing formula under the simplistic assumption that daily returns on the underlying asset follow an independent normal distribution with constant variance. We will refer to this as the Black-Scholes-Merton (BSM) formula. While the BSM model provides a useful benchmark, it systematically misprices observed options. We therefore consider the following alternatives.

3. Extend the normal distribution model by allowing for skewness and kurtosis in returns. We will rely on the Gram-Charlier expansion around the normal distribution to derive an option pricing formula in this case.

4. Extend the model by allowing for time-varying variance relying on the GARCH models from Chapter 2. Two GARCH option pricing models are considered: one allows for general variance specifications but requires Monte Carlo simulation or another numerical technique; the other assumes a specific variance dynamic but provides a closed form solution for the option price.

5. Introduce the ad hoc...