The most common use of Monte Carlo Simulation in finance is when one needs to calculate the expected value of a functional \(\mathbb{E} f(X)\)
Introduction
Assume \(g(\cdot)\) is the density function of random variable \(X\). Then we can express the expectation as an integral:
\[ \mathbb{E}f(X) = \int_{-\infty}^{\infty}{f(t)g(t)dt} \]
If this integral can’t be computed explicitly, then Monte Carlo simulation techniques are adopted to estimate it. The idea is to use the Law of large Numbers (LLN) to estimate the integral.
Suppose \(\{X_i\}_{i=1}^{n}\) is a sample of i.i.d. random variables with the same distribution as \(X\) and \(\mathbb{E}(X)=\mu, Var(X)=\sigma^2 < \infty\).
For the Sample Mean for random sample \(\{ f(X_1), f(X_2),...,f(X_n) \}\) defined as
\[ \overline{fX_n} = \frac{f(X_1), f(X_2),…,f(X_n)}{n} \]
we have