Editing Monte Carlo methods in finance (section)

=== Sample paths for standard models ===
In finance, underlying random variables (such as an underlying stock price) are usually assumed to follow a path that is a function of a [[Brownian motion]] <sup>2</sup>. For example, in the standard [[Black–Scholes model]], the stock price evolves as

:<math> dS = \mu S \,dt + \sigma S \,dW_t. </math>

To sample a path following this distribution from time 0 to T, we chop the time interval into M units of length <math>\delta t</math>, and approximate the Brownian motion over the interval <math>dt</math> by a single normal variable of mean 0 and variance <math>\delta t</math>. This leads to a sample path of

:<math> S( k\delta t) = S(0) \exp\left( \sum_{i=1}^{k} \left[\left(\mu - \frac{\sigma^2}{2}\right)\delta t + \sigma\varepsilon_i\sqrt{\delta t}\right] \right)</math>

for each ''k'' between 1 and ''M''. Here each <math>\varepsilon_i</math> is a draw from a standard normal distribution.

Let us suppose that a derivative H pays the average value of ''S'' between 0 and ''T'' then a sample path <math>\omega</math> corresponds to a set <math>\{\varepsilon_1,\dots,\varepsilon_M\}</math> and

:<math> H(\omega) = \frac1{M} \sum_{k=1}^{M} S( k \delta t).</math>

We obtain the Monte-Carlo value of this derivative by generating ''N'' lots of ''M'' normal variables, creating ''N'' sample paths and so ''N'' values of ''H'', and then taking the average.
Commonly the derivative will depend on two or more (possibly correlated) underlyings. The method here can be extended to generate sample paths of several variables, where the normal variables building up the sample paths are appropriately correlated.

It follows from the [[central limit theorem]] that quadrupling the number of sample paths approximately halves the error in the simulated price (i.e. the error has order <math>\epsilon=\mathcal{O}\left(N^{-1/2}\right)</math> convergence in the sense of standard deviation of the solution).

In practice Monte Carlo methods are used for European-style derivatives involving at least three variables (more direct methods involving numerical integration can usually be used for those problems with only one or two underlyings. ''See'' [[Monte Carlo option model]].