What is Monte Carlo Simulation?
Monte Carlo (MC) Simulation is a mathematical tool that allows users to estimate the solution of complex analytical problems. It is widely used in finance to price complex, and sometimes not-so-complex derivatives. The name comes from the gaming tables of the Monte Carlo Casino, as the technique relies on the generation of random numbers.
“We can replace a large part of mathematical knowledge with a Monte Carlo simulator”
Nassim Taleb, Fooled By Randomness
To get an idea of how the technique works, consider a six sided dice. If you roll the dice, you will get a number in the range 1 to 6. The average of these six numbers is 3.5. This is merely the sum of the six numbers divided by 6 [(1+2+3+4+5+6)/6 = 21/6 = 3.5]. This is in fact the best expectation of the outcome of a roll, before you roll the dice, assuming the dice is not rigged. This can be thought of as the “fair price” of the dice. However how can you tell if the dice IS rigged? The easiest way is to roll the dice many times (10,000 times seems to be the number of choice in finance) and average the answer. If the average is not very close to 3.5, then it is a reasonable assumption that the dice are dodgy, and the price is not “fair”. This is what is done in MC simulation. An instrument is priced many times (10,000…) and the average of the prices is the “price” of the instrument. It should be stressed that when using MC simulation to generate risk numbers, a much greater number of simulations must be taken. This is to ensure that the estimated risk values are not confused with prices estimation errors that are a hallmark of MC simulation prices.