I will review results concerning free boundary problems arising from optimal stopping of one-dimensional diffusions in time-inhomogeneous setups. I will illustrate probabilistic methods that allow to determine peculiar features of free boundaries when the stopping payoff is not smooth. If time allows I will also present a fully probabilistic proof of continuous differentiability of the free boundary.

The talk is based on:

• De Angelis, T. (2022) Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon. Electronic Journal of Probability 27, pp. 1-41. doi:10.1214/21-EJP733
• De Angelis, T., Lamberton, D. (2024) A probabilistic approach to continuous differentiability of optimal stopping boundaries. arXiv:2405.16636

We study an optimal stopping problem (OSP) for Gauss–Markov processes conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we establish that this OSP is equivalent to that of a Brownian bridge with a random pinning point. These problems are scarcely addressed in the literature, not only due to the time-inhomogeneity of the underlying process and the non-Lipschitz, explosive behavior of its drift coefficient near the terminal time, but also because they deviate from the conventional monotonic optimal stopping boundary (OSB) framework. Furthermore, the OSB cannot even be regarded as the graph of a unique function in general, and its form depends on the process initial state at the time of conditioning.

This is a joint work with B. D'Auria.

In this talk we discuss continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time--specifically, a bounded, non-decreasing, càdlàg control process. To encourage exploration and facilitate learning, we introduce a regularized version of the problem by penalizing it with the cumulative residual entropy of the randomized stopping time. The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel. We address this through the dynamic programming principle, which enables us to identify the unique optimal exploratory strategy. For the specific case of a real option problem, we derive a semi-explicit solution to the regularized problem, allowing us to assess the impact of entropy regularization and analyze the vanishing entropy limit. Finally, we propose a reinforcement learning algorithm based on policy iteration. We show both policy improvement and policy convergence results for our proposed algorithm.

This talk is based on a joint project together with Giorgio Ferrari and Renyuan Xu.

We analyze the impact of liquidation costs on the pricing of American equity options within both discrete- and continuous-time models. In an otherwise arbitrage-free and frictionless market, the introduction of liquidation penalties changes the comparison between immediate exercise payoff and continuation value for American option holders. Specifically, when the sale proceeds achievable upon liquidation are lower due to penalties, immediate exercise becomes more advantageous, leading to a wider optimal early exercise region.

We start studying the impact of liquidation penalties in discrete time, and provide closed-form solutions for perpetual American call options in the binomial model. In the continuous-time lognormal and jump-diffusion models, we provide derive explicit pricing formulas for perpetual American call options with liquidation penalties and closed-form asymptotic solutions near maturity for the critical price that triggers optimal early exercise. Moreover, when the immediate payoff is a non-negative smooth function of the current underlying asset, we show that as the liquidation penalty rate approaches infinity, effectively prohibiting liquidation, the cashflow based value function converges to the immediate payoff.

Our results are relevant for executive stock options (ESOs), which typically exhibit liquidation penalties, and for the American equity options for which there is evidence of liquidation costs.

Variable annuities are life-insurance contracts designed to meet long-term investment goals. Such contracts provide several financial guarantees to the policyholder. A minimum rate is guaranteed by the insurer in order to protect the policyholder`s capital against market downturns. Moreover, the policyholder has the right to early terminate the contract (early surrender) and to receive the account value. In general, a penalty, which decreases in time, is applied by the insurer in case of early surrender. We provide a theoretical analysis of variable annuities with a focus on the holder`s right to an early termination of the contract. We obtain a rigorous pricing formula and the optimal exercise boundary for the surrender option. We also illustrate our theoretical results with extensive numerical experiments. The pricing problem is formulated as an optimal stopping problem with a time-dependent payoff, which is discontinuous at the maturity of the contract. This structure leads to non-monotone optimal stopping boundaries, which we prove nevertheless to be continuous. Because of this lack of monotonicity, we cannot use classical methods from optimal stopping theory and, thus, we contribute a new methodology for non-monotone stopping boundaries.

This talk is based on a joint work with T. De Angelis and G. Stabile.

We study smoothness of the value function of zero-sum games between a stopper and a singular-controller. The games are set on a finite-time horizon and the underlying dynamics is a one-dimensional, time-homogeneous, singularly controlled diffusion taking values on either the real line or the positive half-line. The saddle point of those games is characterised in terms of two moving boundaries: an optimal stopping boundary and an optimal control boundary. We show that the value function of the game belongs to the class of continuously differentiable functions on the whole domain and that the second-order spatial derivative and the second-order mixed derivative are continuous everywhere except for a (potential) jump across the stopper's optimal boundary. Our method relies on a new link between the value function of the game and the value function of an auxiliary optimal stopping problem with absorption, and on the convergence of the hitting times at those boundaries.

This is a combination of two joint works, one with T. De Angelis and one with A. Milazzo.