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Financial derivatives have been growing at a phenomenal pace in recent years.Stimulated by this rapid advancement in the financial world, modern mathematical models and new mathematical methods are also emerging at an exponential rate.Among all developments, the Black-Scholes partial differential equation plays a significantly important role.While the pricing of European options are formulated as boundary value problems under the Black-Scholes framework, the valuations of American options are known as free boundary (optimal exercise boundary) problems.In this paper we consider American put options on the geometric mean of several underlying assets.Namely, we study a free boundary problem in a multi-dimensional case and analyze the property of the free boundary.