A Fully-Dynamic Closed-Form Solution for ∆-Hedging with Market Impact

We present a closed-form ∆-hedging result for a large investor whose trades generate adverse market impact. Unlike in the complete-market case, the agent no longer finds it tenable to be perfectly hedged or even within a fixed distance away from being hedged. Instead, he may find himself arbitrarily mishedged and optimally trades towards the classical Black-Scholes ∆, with trading intensity proportional to the degree of mishedge and inversely proportional to illiquidity. When combined with a recent result of Garleanu and Pedersen (2009), this suggests that ∆-hedging can be thought of as a Merton problem where the Merton-optimal portfolio is the Black-Scholes ∆-hedge. Both the discrete-time and continuous-time problems are solved. We discuss a number of applications of our result, including the equilibrium implications of our model on intraday trading patterns and stock pinning at options’ expiry. Finally, numerical simulations on TAQ data based on intraday hedging of call options suggest that this strategy is able to significantly minimize market impact cost without incuring a significant increase in mishedging error (with respect to a Black-Scholes ∆-hedging strategy).