Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {T.~Gerstner and M.~Holtz and R.~Korn},
  title = {Valuation of Performance-Dependent Options in a
		  {B}lack-{S}choles Framework},
  booktitle = {Numerical Methods for Finance},
  pages = {203--214},
  publisher = {Chapman \& Hall/CRC},
  editor = {J. Appleby and D. Edelman and J. Miller},
  year = {2007},
  annote = {proceedings,ALM},
  pdf = { 1},
  abstract = {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.}