Monte Carlo Simulation: Convergence and Variance reduction techniques for option pricing models
Monte Carlo simulation techniques are a useful tool in finance for pricing options especially when there are a large number of sources of uncertainty (in modeling terms: state variables) involved. Unlike Black Scholes formula for which closed formed solutions usually do not exist when there are three or more state variables or numerical methods like binomial option pricing models which become impractical when the sources of uncertainty increase, Monte Carlo simulation techniques can lead to convergent solutions for these derivatives.
The solution’s accuracy however is dependent on the number of trials, N, that are used. Specifically the uncertainty surrounding a possible solution is inversely related to the square root of N. This relationship of accuracy to number of trials is derived as follows:
The mean of the discounted payoff, ?, is the average of the discounted value of payoff across all trials, i.e. the estimate of the value of the instrument. The standard deviation of these discounted payoff values is denoted by ?.
The standard error of the estimate is given by ?/?N.
A 95% confidence interval around the estimate is given by:
?-1.96?/?N < estimate < ?+1.96?/?N
This shows that as the value of N increases, the range around the estimate reduces, i.e. the accuracy of the estimate or the solution from the Monte Carlo methodology increases as the number of trials increase. Specifically, in order to increase the accuracy by a factor of x, the number of trials that should be used would need to be increased by a factor of x2. For example to double the accuracy of the estimate, the number of trials that should be used to ensure this level of accuracy would need to be quadrupled.
As the number of trials increases the Monte Carlo simulation technique becomes more and more time consuming as well as computationally intensive. In order to save on computational time there are a number of variance reduction techniques that may be used. These include:
- Antithetic Variable Technique
- Control Variate Technique
- Importance Sampling
- Stratified Sampling
- Moment Matching
- Quasi Random Sequences