Chapter 4

Monte Carlo Methods*

The two preceding chapters dealt with probability and statistics. The former involves the generation of random variables from known distributions. The second deals with estimation of distribution parameters from actual data. With estimated distributions in hand, we can proceed to the next step, which is the simulation of random variables for the purpose of risk management.

Such simulations, called Monte Carlo (MC) simulations, are central to financial engineering and risk management. They allow financial engineers to price complex financial instruments. They allow risk managers to build the distribution of portfolios that are too complex to model analytically.

Simulation methods are quite flexible and are becoming easier to implement with technological advances in computing. Their drawbacks should not be underestimated, however. For all their elegance, simulation results depend heavily on the model's assumptions: the shape of the distribution, the parameters, and the pricing functions. Risk managers need to be keenly aware of the effect that errors in these assumptions can have on the results.

This chapter shows how Monte Carlo methods can be used for risk management. Section 4.1 introduces a simple case with just one source of risk. Section 4.2 shows how to apply these methods to construct value at risk (VAR) measures, as well as to price derivatives. Multiple sources of risk are then considered in Section 4.3.

4.1 SIMULATIONS WITH ONE RANDOM VARIABLE ...