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Applications of Malliavin calculus to the pricing and hedging of Bermudan options

By James Newbury


The pricing of Bermudan options, which give the holder the right to buy or sell an underlying asset at a predetermined price and at a discretely spaced number of times prior to maturity, can be based on a deterministic method or on a probabilistic one. Deterministic methods such as finite differences lose their efficiency as the dimension of the problem increases, and they are therefore known to suffer from the "curse of dimensionality". Probabilistic methods enable us to overcome this problem by using Monte Carlo simulations. One particular method is the Malliavin pricing and hedging algorithm, which uses representation formulas for conditional expectation and its derivative to approximate the price and delta of a Bermudan option. This paper specifically deals with how the powerful tools of Malliavin calculus are applied in the derivation of such representation formulas, and looks at how the latter are subsequently used in the pricing and hedging algorithm.\ud \ud Key words: Bermudan option, dynamic programming principle, Malliavin\ud derivative operator, Skorohod integral, first and second variational processes,\ud representation formula, localizing function

Topics: Mathematics education
Year: 2011
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