A PDE Approach to Option Pricing Under Liquidity Risk

Abstract

We present a PDE approach for the pricing of options and other contingent claims in a liquidity risk and price impacts model. Option prices under liquidity risk are shown to be solutions of a class of semilinear degenerate parabolic partial differential equations on bounded domains. We prove the existence and uniqueness of weak solutions of this type of equations. The resulting option prices and their derivatives are H¨older continuous. We give a natural decomposition of prices into a “classical” part (without trade impact and liquidity costs) plus an error term reflecting trade impact and liquidity costs.