Optimal strategy and deep hedging for share repurchase programs

TL;DR

This work develops a unified, machine-learning-based framework to jointly optimize ASR execution and hedging under realistic trading constraints, using optimized certainty equivalent risk measures and indifference pricing. By parameterizing both the buyback policy and hedging strategy with neural networks and employing differentiable relaxation of stopping, the authors enable gradient-based optimization over a high-dimensional control space. Results show that joint optimization substantially improves risk metrics and increases sustainable discounts (δ^fair and δ^ind), with particularly large gains for the network policy. The paper also demonstrates the importance of explicitly modeling hedging constraints and of integrating hedging into the optimization, rather than treating it as a separate post-hoc step, yielding a more realistic and profitable ASR management framework.

Abstract

In recent decades, companies have frequently adopted share repurchase programs to return capital to shareholders or for other strategic purposes, instructing investment banks to rapidly buy back shares on their behalf. When the executing institution is allowed to hedge its exposure, it encounters several challenges due to the intrinsic features of the product. Moreover, contractual clauses or market regulations on trading activity may make it infeasible to rely on Greeks. In this work, we address the hedging of these products by developing a machine-learning framework that determines the optimal execution of the buyback while explicitly accounting for the bank's actual trading capabilities. This unified treatment of execution and hedging yields substantial performance improvements, resulting in an optimized policy that provides a feasible and realistic hedging approach. The pricing of these programs can be framed in terms of the discount that banks offer to the client on the price at which the shares are delivered. Since, in our framework, risk measures serve as objective functions, we exploit the concept of indifference pricing to compute this discount, thereby capturing the actual execution performance.

Figures (9)

• Figure 1: $b_n(S_n, \text{ } A_n)$ function: when the ratio between $S_n \text{ and } A_n$ falls in the interval $[1 + \epsilon_r - 0.5 \cdot \delta_r, \text{ } 1 + \epsilon_r + 0.5 \cdot \delta_r]$, the repurchase $b_n$ is determined by a linear interpolation over $[q_n^{min}, \text{ } q_n^{max}]$.
• Figure 2: Payoff distribution for the smooth bang-bang and network strategy. The payoff is normalized by $W_{Min}$ and expressed in basis points.
• Figure 3: Payoff distribution for the sequential model applied to the smooth bang-bang and network policies. The payoff $PnL = PnL^{ASR} + PnL^{Hedge}$ is normalized by $W_{Min}$ and expressed in basis points.
• Figure 4: Distribution of $PnL^{ASR}$: in orange it is represented the smooth bang-bang in absence of the hedging portfolio, as in Section \ref{['sec:buyback']}. In light-blue is represented the joint hedging model. The payoff is normalized by $W_{Min}$ and expressed in basis points.
• Figure 5: Distribution of $PnL^{ASR}$: in orange it is represented the network policy in absence of the hedging portfolio, as in Section \ref{['sec:buyback']}. In light-blue is represented the joint hedging model. The payoff is normalized by $W_{Min}$ and expressed in basis points.

Theorems & Definitions (1)

• Definition 1: Optimized Certainty Equivalent Risk Measure