Hedging Through a Limit Order Book with Varying Liquidity

Abstract

Market impact is one of the most important aspects of real-world trading. It is typically a major component of transaction costs, and trading strategies are carefully optimized to minimize it. But most of the research has focused on one-off trades, not the repeated series of transactions required in delta-hedging an options position. In this article, the authors explore optimal hedging in an illiquid environment for a large trader whose trades absorb a nonmarginal fraction of the standing orders in the limit order book. The trader needs to take account of both the immediate price impact of each trade and the speed at which the limit order book is restored afterward—the market’s resilience. These two aspects of market dynamics together can produce a permanent effect of a big trade and also a temporary one that dissipates over time. Depending on the parameters of the liquidity process, the optimal strategy can be the canonical Black–Scholes delta hedge (in the limiting case of infinite liquidity) or one that is distinctly less volatile (when market impact is large, but there is some resilience in prices) or even one that entails price manipulation (when there is no resilience and the price impact is permanent).