Abstracts

Short-selling is the selling of a financial security which the investor does not own. The trading is an important tool for portfolio management to hedge the downside risk of the security price. In this paper, we formulate a short-selling strategy and seek an optimal timing of short covering as an optimal stopping problem. The aim is to study how the optimal trading strategy of the short-seller is influenced by various features of the stock borrowing market, including the random time recall from broker-dealers, the loan fee payment and the short interest rebate. We characterize the optimal timing of short covering depending on the conditions that lead to different cost and benefit of keeping the position. We find that the short-seller should stop earlier in the presence of broker's recall and should stop immediately when the recall risk is high during an up market. Moreover, the value function can become negative due to a forced termination at the recall time. When there are borrowing cost and interest rebate, the optimal stopping strategy depends on a delicate balance between the loan fee (cost) and interest rate (benefit). When the loan fee is charged too high, the short-seller's optimal strategy is to stop immediately; in contrast, when the interest rate is sufficiently high, the short-seller forfeits the optionality to stop and wait-and-see until the recall time. As more interesting results we observe that in other cases, the optimal stopping rule is a down-and-out type (put-type problem) or an up-and-out type (call-type problem), depending on the levels of loan fee and interest rate relative to the short-seller's discount rate and expected return of the stock. The solution to the optimal stopping problems is obtained in closed-form and we show its optimality by the verification theorem. Given the closed-form solution, we are able to characterize explicitly the regimes (versus loan fee and interest rate) in which the short-seller is active or inactive. We can also determine the optimal loan fee charged by the broker-dealer given the short-seller's optimal strategy. The analysis in this paper can be readily extended to incorporate a stop-loss limit and regime-switching in the stock price process.