Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics

Chapter Twelve

Sensitivity Estimation

The aim of most Monte Carlo simulations is to estimate the expected value of a function of several random variables. On the one hand, this function depends on the sample path, which we may associate with a random event ω; informally, we may think of ω as a sequence of (pseudo)random numbers. On the other hand, the function depends on some relevant parameters as well. If, for the sake of simplicity, we focus on a single parameter α, we may denote the function implemented by the simulation program as f(α, ω). When taking the expected value, we obtain another function

(12.1)

depending on α alone. Monte Carlo is a way to find an estimate (α). In many practical settings, we are also interested in the sensitivity of g(·) with respect to its argument α. Formally, we would like to find

(12.2)

The standard example in financial engineering is the computation of the option greeks, which measure the sensitivity of option prices with respect to parameters like the current price of the underlying asset, volatility, etc. More generally, we might need the sensitivity with respect to parameters that are the decision variables of a stochastic optimization problem. ...