Polynomial chaos for simulating random volatilities

  • Authors:
  • Roland Pulch;Cathrin van Emmerich

  • Affiliations:
  • Bergische Universität Wuppertal, Fachbereich Mathematik und Naturwissenschaften, Lehrstuhl für Angewandte Mathematik und Numerische Mathematik, Gauístr. 20, D-42119 Wuppertal, Germa ...;Bergische Universität Wuppertal, Fachbereich Mathematik und Naturwissenschaften, Lehrstuhl für Angewandte Mathematik und Numerische Mathematik, Gauístr. 20, D-42119 Wuppertal, Germa ...

  • Venue:
  • Mathematics and Computers in Simulation
  • Year:
  • 2009

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Abstract

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.