Crash Modelling, Value at Risk and Optimal Hedging

Philip Hua and Paul Wilmott

Abstract

In this paper we present a new model for pricing and hedging a portfolio of derivatives that takes into account the effect of an extreme movement in the underlying. We make no assumptions about the timing of this 'crash' or the probability distribution of its size, except that we put an upper bound on the latter. The pricing and hedging follow from the assumption that the worst scenario actually happens i.e. the size and time of the crash are such as to give the option its worst value. The optimal static hedge follows from the desire to make the best of this worst value. There are many applications for this crash modelling, we shall focus on using the model to evaluate the Value at Risk for a portfolio of options.

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