7.1. CHAPTER OVERVIEW

In the previous chapter, we gave a brief overview of various models for pricing options. In this chapter, we turn our attention to the key task of incorporating derivative securities into the portfolio risk model, which we developed in previous chapters. Just as the nonlinear payoff function was the key feature from the perspective of option pricing in the previous chapter, it is also driving the risk management discussion in this chapter. The nonlinear payoff creates asymmetry in the portfolio return distribution, even if the return on the underlying asset follows a symmetric distribution. Getting a handle on this asymmetry is a key theme of this chapter.

The chapter is structured as follows:

1. We define the delta of an option, which provides a linear approximation to the nonlinear option price. We then present delta formulas from the various models introduced in the previous chapter.

2. We establish the delta-based approach to portfolio risk management. The idea behind this approach is to linearize the option return and thereby make it fit into the risk models discussed earlier in the book. The downside of this approach is that it ignores the key asymmetry in option payoffs.

3. We define the gamma of an option, which gives a second-order approximation of the option price as a function of the underlying asset price.

4. We use the gamma of an option to construct a quadratic model of the portfolio return distribution. We discuss two implementations of the quadratic model: one...