Quantitative Finance

8Approximating Stochastic Processes: Monte Carlo Simulations and Generating and Generating Random Variables

8.1 Introduction

This chapter is divided into two parts. The first part discusses the Monte Carlo simulation methods for pricing options, and the second part deals with methods used for simulating and generating random variables. The Monte Carlo simulation method is another technique that can also be used to price options. It has been extensively studied by many authors which includes [32, 38, 81, 101, 140] and several others.

It is very important to know how to simulate a random experiment to find out what expectations one may have about the results of the phenomenon. We shall see later that the central limit theorem (CLT) and the Monte Carlo method allow us to draw conclusions about the expectations even if the distributions involved are very complex. We assume that given a Uniform (0,1) random number generator, any software can produce a uniform random variable. The typical name for such a uniform random number is RAND. For more current references and a very efficient way to generate exponential and normal random variables without going to uniform, we refer the reader to [174].

We begin the first part of this chapter with a discussion of the Plain vanilla Monte Carlo method.

8.2 Plain Vanilla Monte Carlo Method

Pricing options using the Monte Carlo simulation method is tractable since it departs from the regular geometric Brownian motion assumption for the stock ...