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Towards Exact Solutions Of The American Style Options

Fredrick Michael

posted on 09 December 2016

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Exact Solutions of the American style call or put option on the same footing as the European (Vanilla) call or put option have not been derived. That is a function that describes the call option say as depending on the underlying & portfolio hedge & portfolio time dependent riskless asset increase in value & yet inclusive of the uncertainty due to early exercise. In this letter we present several approaches towards deriving such a function that are conceptually inspired by and are based on current methods of discrete probabilistic decision trees and their diffusion approximation, & from methods of applying boundary conditions that are here additionally random in time. We derive under a certain set of limiting assumptions i.e. the random in time boundary conditions & their statistics (utilizing for illustration the simplest) and furthermore for flux allowing VN boundaries & flux absorbing Dirichlet boundaries a closed form solution.

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Note: Shortly after upload, noticed typos. Replaced manuscript with edited version. I look forward to feedback & comments or suggestions.