T. Gerstner and M. Holtz.
Geometric tools for the valuation of performance-dependent options.
In M. Costantino and C. Brebbia, editors, Computational Finance
and its Application II, pages 161-170, London, 2006. WIT Press.
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In this paper, we describe several methods for the valuation of performance-dependent 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 it is shown by numerical results.