Contingent Claims in an Illiquid Market

Consistent with trading in an illiquid market, we assume that a large trader drives up the stock price as she buys and pushes it down as she sells. The effect of this price impact on the replication of a European contingent claim for the large trader is considered and a generalized nonlinear Black-Scholes pricing partial differential equation for computing this cost is obtained. The pricing PDE indicates that one of the main effects of the price impact is the resulting endogenous stochastic volatility for the stock return. The existence and uniqueness of a classical solution to such an equation under certain conditions is established. This implies that the large trader can still perfectly replicate the contingent claim (but with a higher cost). The replicating strategy involves an initial discrete trade followed by continuous trading. It turns out that unlike in the presence of transaction costs, super-replication in the presence of price impact incurs larger costs than replication. Compared to the case without price impact, the large trader generally buys more the stock and borrows more (shorts and lends more) to replicate an out-of-the-money call (put), but buys less the stock and borrows less (shorts and lends less) to replicate an in-the-money call (put).