On the multidimensional Black–Scholes partial differential equation
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In this article, two general results are provided about the multidimensional Black–Scholes partial differential equation: its fundamental solution is derived, and it is shown how to turn it into the standard heat equation in whatever dimension. A fundamental connection is established between the multivariate normal distribution and the linear second order partial differential operator of parabolic type. These results allow to compute new closed form formulae for the valuation of multiasset options, with possible boundary crossing conditions, thus partially alleviating the « curse of dimensionality », at least in moderate dimension.
KeywordsBlack–Scholes multidimensional equation Multiasset option Dimension Parabolic Mixed derivative Multivariate normal distribution Heat equation Option on the maximum Double barrier
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