Nonlinear (Option) Risk Models*
The previous chapter focused on linear risk models (e.g., hedging using contracts such as forwards and futures whose values are linearly related to the underlying risk factors). Because linear combinations of normal random variables are also normally distributed, linear hedging maintains normal distributions, which considerably simplifies the risk analysis.
Nonlinear risk models, however, are much more complex. In particular, option values can have sharply asymmetrical distributions. Even so, it is essential for risk managers to develop an ability to evaluate this type of risk, because options are so widespread in financial markets. Since options can be replicated by dynamic trading, this also provides insights into the risks of active trading strategies.
In a previous chapter, we have seen that market losses can be ascribed to the combination of two factors: exposure and adverse movements in the risk factor. Thus a large loss could occur because of the risk factor, which is bad luck. Too often, however, losses occur because the exposure profile is similar to a short option position. This is less forgivable, because exposure is under the control of the portfolio manager.
The challenge is to develop measures that provide an intuitive understanding of the exposure profile. Section 14.1 introduces option pricing and the Taylor approximation. It starts from the Black-Scholes formula that was presented in Chapter 8. Partial derivatives, also ...