The authors identify the optimal trading strategy to implement when hedging a short call position by trading the underlying stock through a limit order book (LOB) under various degrees of liquidity. The optimal strategy considers liquidating the terminal portfolio, hedging risk, minimizing incurred costs, and exploiting price friction, but its effectiveness is largely dependent on the availability of liquidity as characterized by the LOB depth and resilience.

The optimal hedging strategy depends largely on the resilience speed of the limit order book, which is measured by the time it takes new orders to replenish the LOB and resolve the spread between the best ask (bid) and the fundamental price. In the extreme case of infinite resilience, trading has no impact on price. As resilience approaches infinity, the optimal strategy can deviate from the Black–Scholes hedging model in the presence of a shallow LOB that requires patient trading to minimize illiquidity or feedback effects.

Zero resilience speed means the price impact from trading is permanent; thus, there is an incentive to manipulate the stock price by effecting lasting price changes. A price impact that does not dissipate allows for liquidity costs to be offset by lasting gains in the underlying equity. A limit order book with resilience, indicating only temporary price impacts, has liquidity costs that cannot be offset by the price manipulation strategy.

The volatility of the underlying asset and gamma of the derivative should be considered in relation to the depth of the LOB because greater volatility requires more shares to hedge the position.

The detailed work of this research is useful to such institutional traders as hedge funds and high-frequency or automated algorithm-based traders. More inexperienced traders may also benefit from paying attention to the trade-off between the liquidity risk of trading and the market risk of hedging in an illiquid market.

The authors develop a model to simulate hedging a short call position by trading the underlying stock through an electronic limit order book. Geometric Brownian motion and 100 trades are used to simulate the stock activity under multiple scenarios of various liquidity levels to determine the optimal hedging strategy for different market conditions.

The model considers the size and time decay of the price impact caused by subsequent trades executed through the LOB. A continuous but simplified uniform price distribution of shares throughout the LOB supports the stated assumptions that the size impact of trades is linear and the time decay of the impact is exponential. The authors also assume a risk-free rate of zero.

Liquidity is determined by the depth of the LOB, which is measured by the order size needed to cause a marginal price change, and the resilience of the LOB to replenish itself with new orders.

The optimal trading strategy is identified as the strategy that maximizes terminal wealth of the portfolio based on changes in stock price, changes in the derivative value, and the price impact of trading. The costs of liquidating the portfolio at the terminal date are assumed to be negligible and thus ignored.

The risk tolerance characteristics unique to each trader will influence which hedging strategy is optimal for that trader. When traders are averse to market exposure from being mishedged, they should hedge more aggressively; otherwise, managing the liquidity costs is more of a priority.

In addition, the model recognizes a terminal value while ignoring the costs of liquidating the portfolio. Although modeling several forms of transaction costs and fees can create unnecessary noise, liquidation costs may be relevant when low LOB resilience suggests a price manipulation strategy. If entering a stock position through a limit order book has a manipulation effect on price, exiting the position may negate the benefit by causing a price change of similar magnitude.