Geometric Tools For The Valuation Of Performance-dependent Options
Free (open access)
T. Gerstner & M. Holtz
In this paper, we describe several methods for the valuation of performancedependent options. Thereby, we use a multidimensional Black–Scholes model for the temporal development of the asset prices. The martingale approach then yields the fair price as a multidimensional integral whose dimension is the number of stochastic processes in the model. The integrand is typically discontinuous, though, which makes accurate solutions difficult to achieve by numerical approaches. However, using tools from computational geometry we are able to derive a pricing formula which only involves the evaluation of smooth multivariate normal distributions. This way, performance-dependent options can efficiently be priced even for high-dimensional problems as is shown by numerical results. Keywords: option pricing, multivariate integration, hyperplane arrangements. 1 Introduction Performance-dependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. Here, we assume that the performance of an asset is determined by the relative increase of the asset price over the considered period of time. The performance of the asset is then compared to the performances of a set of benchmark assets. For each possible outcome of this comparison, a different payoff of the derivative can be realized. We use a multidimensional Black–Scholes model, see, e.g., Karatzas  for the temporal development of all asset prices required for the performance ranking. The martingale approach then yields a fair option price as a multidimensional integral whose dimension is the number of stochastic processes used in the model. In the so-called full model, the number of processes equals the number of assets.
option pricing, multivariate integration, hyperplane arrangements.