Optimal accelerated share repurchases

An accelerated share repurchase allows a firm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks or as much as several months. Some contracts allow the intermediary to specify when the repurchase ends, at which point the firm and the intermediary exchange the difference between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. Here, we address the intermediary’s optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds $$\zeta (t,{q_t}){\kern 1pt} {S_t}$$ζ(t,qt)St where $${S_t}$$St is the midprice of the asset and $$\zeta $$ζ is a deterministic function of time and inventory. Moreover, we develop a dimensional reduction of the stochastic control and stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies. We also provide bounds on the optimal strategy and characterize the convexity and monotonicity of the optimal strategies in addition to exploring its behaviour numerically and through simulation studies.