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

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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 languageEnglish
Article number12
Number of pages32
JournalNonlinear Differential Equations and Applications
Volume24
Issue number2
Early online date18 Feb 2017
DOIs
StatePublished - Apr 2017

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Liquidity
Weak solution
Partial differential equations
Asymptotic representation
Semilinear equations
Hedging
Parabolic partial differential equations
Existence and uniqueness of solutions
Semilinear
Diffusion equation
Replication
Non-negative
Converge
Zero
market

Cite this

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title = "Option prices under liquidity risk as weak solutions of semilinear diffusion equations",
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.",
author = "Fahrenwaldt, {Matthias Albrecht} and Roch, {Alexandre F.}",
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