In this tutorial we will investigate ways we can reduce the variance of results from a Monte Carlo simulation method when valuing financial derivatives.
Monte Carlo simulations is a way of solving probabilistic problems by numerically simulating many possible scenarios so that you may calculate statistical properties of the outcomes, such as expectations, variances of probabilities of certain outcomes. In the case of Financial Derivatives, this gives us a handy tool for which to price complex derivatives for which and analytical formulae is not possible.
Unfortunately, although a great method for approximating option values with complex payoffs or high dimensionality, in order to get an acceptably accurate estimate we must perform a large number of simulations M. Instead we can lean on Variance Reduction methods which work on the same principles as that of hedging an option position. i.e. the variability of a hedged option portfolio will have a smaller variance that that of it's unhedged counterpart.
The mathematic notation and examples are from Les Clewlow and Chris Strickland's book Implementing Derivatives Models.
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