Valuation of Performance‐Dependent Options

Abstract

Performance‐dependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. This paper presents a novel approach to the valuation of general performance‐dependent options. To this end, a multidimensional Black–Scholes model is used to describe the temporal development of the asset prices. The martingale approach then yields the fair price of such options as a multidimensional integral whose dimension is the number of stochastic processes used in the model. The integrand is typically discontinuous, which makes accurate solutions difficult to achieve by numerical approaches, though. Using tools from computational geometry, a pricing formula is derived which only involves the evaluation of several 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

Acknowledgement

The authors wish to thank Ralf Korn, Kaiserslautern, for the introduction to this interesting problem and for his help with the derivation of the pricing formulas.