T. Gerstner and M. Holtz.
Valuation of performance-dependent options.
Applied Mathematical Finance, 15(1):1-20, 2008.
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Performance-dependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. In this paper, we present a novel approach for the valuation of general performance-dependent options. To this end, we use a multidimensional Black-Scholes model 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, we are able to derive a pricing formula 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 it is shown by numerical results.