The Remarkable Power of the Monte Carlo Method
Team Latte August 7, 2012
Ever since we published the article The Essence of Monte Carlo Methodology on this site we have been inundated with requests to elaborate further on what we explained in that article and give more examples. While we cannot go into very detailed mathematical or technical explanations of the Monte Carlo method on these pages – for that, we would refer our readers to our signature public course Certificate in Financial Engineering (CFE) – we can certainly try to elucidate a bit more on this remarkable mathematical method and why it is so powerful in solving many of the problems in physics and finance.
In short, Monte Carlo methodology is an extremely powerful method to solve problems – not just stochastic but deterministic problems as well – in physics and finance because of these two mathematical facts:
 Every definite integral can be approximated as an average;
 Every, complex and intractable, deterministic problem can be expressed as stochastic problem and solved by randomly sampling from a probability distribution.
Well, perhaps we should clarify a bit more on what exactly we mean by a "problem". A "problem" in physics or finance is a dynamical system – where the dependent and the independent variables are related via a function – which is mathematically governed by a differential equation. Be it option pricing, a mechanical pulley, the propagation of wave in a medium, a convertible bond, etc., the dynamics of all these are mathematically governed by a differential equation. And the solution to differential equations is given by integrals.
The real power of Monte Carlo method  or, the Monte Carlo integration methodology – is that it can transform a deterministic problem into a stochastic one by randomizing the independent variable, or the variables,of a given function and then allowing us to sample from a given probability distribution. This means rather than solving the integral analytically – as one would do if one is solving a differential equation analytically – we solve it by transforming the dependent variable of the function into a stochastic (random) variable and then sampling from a probability distribution.
Ordinary differential equations (ODEs), which arise when we have a single dependent variable and a single independent variable related via a function,, are, in general, easier to solve both analytically and numerically. These kinds of problems require evaluating just a single integral. However, when we have a many variable problem, where there is one dependent variable and several independent variables, related via a function of the type, , the problem involves partial differential equations (PDEs) which are, in general, more difficult to solve than ODEs. In such cases, we need to evaluate higher dimensional integrals. This is where a Monte Carlo method becomes truly powerful because solution of the PDEs involving higher dimensional integrals can become very complex.
The option pricing problem is a prime example from finance. Even though the problem is deterministic and can be solved using a differential equation we can transform the valuation problem into an expectation (under some riskneutral probability measure) and then performing the integration. Also, think of a basket option where the payoff of the option is dependent on a basket of assets, i.e. multiple variables. In such situations if we perform Monte Carlo integration we need to evaluate, say, a triple integral (if there are three assets) whereby we have three random variables to sample from some probability distribution (correlations will give rise to joint distributions). This is still far simpler than solving a partial differential equation – numerically, via lattice methods or otherwise – for the three variables with crossasset terms.
There are a large number of problems in physics and engineering, such as heat transfer, propagation of waves, etc. that require solution of partial differential equations. And Monte Carlo method can prove to be very useful and efficient numerical method in solving these problems.
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