Patrik Andersson: Efficient calculation of financial Greeks

In finance a Greek is the sensitivity of the price of a derivative, e.g. a European call option, with respect to some parameter in the model, e.g. the current price of the underlying asset. The Greeks are important both for hedging purposes and from a risk-management perspective.

In realistic models of asset prices, e.g. with stochastic volatility, there are no known formulas for these and one solution is to resort to simulation; however, calculating a Greek involves differentiating an expectation and this can be computationally difficult. A number of methods have been proposed, one of which is the so-called Malliavin calculus integration by parts.

This method allows one to write the expectation of a differentiated function as the expectation of the function itself times a stochastic weight. These stochastic weights are however not unique and so there is a freedom to choose them in an efficient way, i.e. in a way that gives a low variance.

I will make an informal introduction to Malliavin Calculus and to how it has been applied to calculate Greeks. I will then propose some ideas that may make these calculations more efficient.