Abstracts

We present a partial differential equation (PDE) approach for the pricing of 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 PDEs on bounded domains.
We prove the existence and uniqueness of weak solutions of this type of equation. The resulting derivative prices and their gradient are smooth on certain domains. We give a natural decomposition of derivative prices into a 'classical' part (without trade impact and liquidity costs) plus an error term reflecting trade impact and liquidity costs.
We build on various liquidity models and assume that prices are affected by trades through a change in the risk-return premium. We develop the framework in probabilistic terms then define the replication problem as the solution of a backward stochastic differential equation (BSDE), to which we associate a PDE.
We show the existence and uniqueness of this PDE and prove that its solution $u$ is also the solution of the associated BSDE. We choose to work with the concept of weak solutions as opposed to viscosity solutions for the simple reason that it allows us to obtain information about the growth of the gradient $D u$ in an $L^2$-space, which would be impossible to do in the viscosity framework. Indeed, an important feature of a PDE describing a liquidity setting is that the replication strategy, given by the derivative of the option price with respect to the underlying, converges to the replication strategy in a frictionless setting when the number of options replicated is small.
The detailed analysis leads to explicit and computable bounds on the $L^2$-norms for the derivative price and its delta. Due to the analysis being highly explicit, it serves as a basis for numerical approaches to the PDE. The high regularity of the option prices suggests that numerical approaches will have a fast convergence rate.