An Analysis of Path-Dependent Options
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The objective of this paper is to provide an analytic theory for pricing path-dependent options of European type. General conditions for the path-dependencies are introduced, which allow a wide range of application. We present a partial differential equation describing the fair price process of a path-dependent option in a Black–Scholes world, where the classical Black–Scholes equation involves additional terms caused by the path-dependency of the option. The main result is that the problem is well posed in appropriate function spaces.
KeywordsPath-dependent options Black–Scholes Elliptic–hyperbolic partial differential equation Noncommuting sums of operators Semigroup methods
Supported by the Friedrich Naumann Foundation for Freedom and the German National Academic Foundation. We would also like to express our thanks to Frank Weber for providing us with the problem setting and the economic background in which it arises.
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