Polynomial chaos for simulating random volatilities Original Research Article

In financial mathematics, the fair price of options can be achieved by solutions of parabolic differential equations. The volatility usually enters the model as a constant parameter. However, since this constant has to be estimated with respect to the underlying market, it makes sense to replace the volatility by an according random variable. Consequently, a differential equation with stochastic input occurs, whose solution determines the fair price in the refined model. Corresponding expected values and variances can be computed approximately via a Monte Carlo method. Alternatively, the generalised polynomial chaos yields an efficient approach for calculating the required data. Based on a parabolic equation modelling the fair price of Asian options, the technique is developed and corresponding numerical simulations are presented.