The Essence of Monte Carlo Methodology
June 20, 2012
In finance, just as in physics and engineering, we use Monte Carlo methods quite extensively in solving problems. For valuation of financial derivatives Monte Carlo simulation methodology has become the dominant numerical technique within the banks and financial institutions. One of the important reasons that banks have been able to come up with ultra-complex and sophisticated financial products to sell to the investors over the last decade and a half is because the quants – the financial engineers – working within these banks have found it rather easy to value and price these products using Monte Carlo simulation techniques.
So what is this Monte Carlo method?
Monte Carlo method is a mathematical technique that allows us to solve complex (as in, "complicated") and difficult problems that cannot be solved analytically. An analytical solution means that you can solve an equation and find the value of the dependent variable in terms of one or more independent variables. It is a deterministic process of finding the solution to an algebraic, differential or an integral equation. However, many problems – such as those in statistical physics or option pricing in finance – cannot be solved analytically. They are too complicated. In that case we employ the Monte Carlo method.
What do we mean by "cannot be solved analytically"?
Say, we have a function we want to evaluate the integral with an upper bound of 5 and a lower bound of 0. That is, we want to evaluate the integral:
Remember, when you find the value of an integral, you are essentially finding the area under the curve traced by the function. In the above case, the integral gives the area under the curve in a plane given by in the domain [0, 5]. Another way of looking at an integral is to think of it as an average value. Finding an integral is like averaging over the function in that domain.
This is easy. Simply use the rule:
After this, simply substitute the upper and lower bounds and evaluate the integral.
The above is a deterministic process and using a rule, an algorithm or a formula we are able to solve the integral – that is, find the area under the curve - easily without much effort.
However, what if we did not know the above rule of integration? Or worse, the function was enormously complex and there was no rule or formula like the above to help us do the integration. In short, the function is “analytically intractable”. In that case, we would have to use a Monte Carlo method to solve it.
So what is this Monte Carlo method?
Consider a function that is analytically intractable and we want to evaluate the integral
Since it is analytically – deterministically – intractable, we need to convert the above deterministic problem to a stochastic (random) one.
Draw a random variable in the interval [a, b], i.e. , where is the lower bound of the integral and is the upper bound of the integral. The curve starts from (on the axis) and ends at (again, on the axis). In the previous example, above, was 0 and was 5.
The random variable, is drawn from a known probability distribution (such as a Gaussian distribution) which has a known density function . Then, if we consider the following estimator, , we can write the “expected” value of the integral as
Now, the problem has become stochastic (random) but tractable because we can sample from a known probability distribution and estimate the value, even though that would not be an exact value. The estimator helps in preserving the sampling probability distribution. Here, without knowing anything about the analytic behaviour of the function we are able to evaluate the integral, and find the area under the curve. This is the essence of Monte Carlo methodology
Reference: For more on the above refer to the excellent financial mathematics book, The Mathematics of Derivatives Securities with applications in MATLAB, Mario Cerrato, John Wiley & Sons (2012)
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