Option prices under liquidity risk as weak solutions of semilinear diffusion equations

Matthias Albrecht Fahrenwaldt; Alexandre F. Roch

doi:10.1007/s00030-017-0435-0

Option prices under liquidity risk as weak solutions of semilinear diffusion equations

Matthias Albrecht Fahrenwaldt, Alexandre F. Roch

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Abstract

Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.

Original language	English
Article number	12
Number of pages	32
Journal	Nonlinear Differential Equations and Applications
Volume	24
Issue number	2
Early online date	18 Feb 2017
DOIs	https://doi.org/10.1007/s00030-017-0435-0
Publication status	Published - Apr 2017

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abstract = "Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.",

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AB - Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.

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Option prices under liquidity risk as weak solutions of semilinear diffusion equations

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