Option pricing for a large trader with price impact and liquidity costs

This paper concerns the pricing of options for a large trader in a market with liquidity risk. The investor's trading action is assumed to have some lasting impact on the underlying asset, and the effect of illiquidity is modeled via trading speed (rate of change in holdings). For such a large trader, liquidity risk is quite important since the permanent price impact as well as liquidity costs incurred during trading must be taken into account. The utility maximization approach to determine option prices leads to optimal control problems. This paper shows that the value functions of these optimal control problems are the unique viscosity solutions of a fully nonlinear second-order PDE. Moreover, some illustrative examples with explicit optimal solutions and numerical results are presented.