T. Gerstner, M. Holtz, and R. Korn.
Valuation of performance-dependent options in a Black-Scholes
In J. Appleby, D. Edelman, and J. Miller, editors, Numerical
Methods for Finance, pages 203-214. Chapman & Hall/CRC, 2007.
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In this paper, we introduce performance-dependent options as the appropriate financial instrument for a company to hedge the risk arising from the obligation to purchase shares as part of a bonus scheme for their executives. In order to determine a fair price of such options, 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. As a remedy, we derive a pricing formula which only involves the evaluation of smooth multivariate normal distributions. This way, performance-dependent options can efficiently be priced as it is shown by numerical results.