In this tutorial we will investigate ways we can reduce the variance of results from a Monte Carlo simulation method when valuing financial derivatives by using Delta-based control variates.
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, to get an acceptably accurate estimate we must perform many 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.
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00:00 Intro
01:55 Control Variate & Hedging
06:20 What are the benefits?
09:30 Gamma-based control variates
14:42 Slow python implementation
20:54 Fast python implementation
27:08 Visualisation of Convergence
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