Closed-Form Solutions for Option Hedging with Market Impact

Abstract

We present a highly-intuitive closed-form delta-hedging result for a large investor whose trades generate adverse market impact. Unlike the complete-market or proportional-transaction-cost cases, the agent no longer finds it tenable to be perfectly hedged or even within a fixed distance of being hedged. Instead, he may find himself arbitrarily mishedged and optimally trades towards the classical Black-Scholes delta, with trading intensity proportional to the degree of mishedge and inversely proportional to illiquidity. When viewed in light of a recent result of Garleanu and Pedersen (2009), this suggests that delta-hedging can be thought of as a Merton problem where the Merton-optimal portfolio is the Black-Scholes delta hedge. Both the discrete-time and continuous-time problems are solved. We discuss a number of applications of our result, including the 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 reduce market impact cost with respect to Black-Scholes delta hedging.