Sensitivity Analysis, Estimating Derivatives and the Greeks

Estimating the sensitivity of a simulation with respect to changes in the parameter values is an important part of establishing the validity of the conclusions. If a simulation estimates an expected value at certain value of the parameters with 0.32 ± 0.05 but the derivative with respect to one parameter, say the volatility parameter σ, is 5, this indicates that a change of the volatility of only 0.02 or 2 percent would result in a change in the average of the order of 0.1. Since volatility typically changes rapidly by far more than one percent, then the apparent precision of the estimate 0.32± .005 is very misleading. Of particular importance in finance are certain derivatives of an option price or portfolio value with respect to the parameters underlying the Black Scholes model. These are called the “Greeks”, because many of them (not to mention many parameters and constants used in Statistics, Physics, Mathematics, and the rest of Science) are denoted by greek letters. Suppose V = V (S(t), t,σ, r) is the value of a portfolio or a derivative based on an asset S(t) where the volatility parameter is σ and r is the current spot interest rate. For example for a single European call option, from the Black-Scholes formula