# Simulations

Simulations are used to mimic the behavior of real processes so as to study the effects and/or the results of an experiment under conditions that closely represent realistic situations. In the context of finance, one commonly used tool is Monte Carlo simulations. The origin of Monte Carlo^{[1]} simulations dates back to the 1940s when John Von Neumann, Stanislaw Ulam, and Nicholas Metropolis worked on the Manhattan Project at Los Alamos National Laboratory. Since its inception, Monte Carlo simulations have grown in popularity so much so that the term simulation is synonymous with the words Monte Carlo simulation. Despite its wide use in other fields, it was not until Boyle's 1977 paper that the use of Monte Carlo simulations found its application in finance.

Although not initially popular for about two decades following the publication of Boyle's article, the idea of using simulations to value derivatives began to catch on like a wild fire when the hardware (e.g., fast processing chips) started getting cheaper.

The philosophy underlying the use of simulations to solve any financial problem can be more succinctly reduced to answering the following three questions:

1. How to generate random numbers for a given distribution?

2. How to use simulations to solve the problem at hand?

3. How to reduce the number of simulated paths so as to arrive at the converged result faster (i.e. increase the speed of convergence)?

In this chapter, I will provide the answers to the above questions in the order they are posed. As an astute reader can appreciate, any variate can be uniquely transformed using probabilistic and mathematical tricks to a standard uniform variate^{[2]} and vice versa. Hence, starting with a discussion on the generation of a standard uniform number, I proceed to show how a uniform number can be transformed to a variate of any desired probability density function (pdf). I then answer the second question by showing a few examples of how simulations are used in practice. The chapter concludes with a discussion on reducing the noise associated with simulation errors – in the process answering the third question.

- [1] The term Monte Carlo was motivated by the casinos in Monte Carlo where Stanislaw Ulam's uncle used to gamble his money away. Due to the need for a code name for a confidential project that Stanislaw Ulam was part of, this moniker was created.
- [2] If
*X*has a standard uniform variate,*X*takes the form