Optimal Stopping; Approximate Dynamic Programming; Interpretability; Decision Trees; Option Pricing
Optimal stopping is the problem of deciding when to stop a stochastic system to obtain the greatest reward, arising in numerous application areas such as finance, healthcare and marketing. State-of-theart methods for high-dimensional optimal stopping involve approximating the value function or the continuation value, and then using that approximation within a greedy policy.Although such policies can perform very well, they are generally not guaranteed to be interpretable; that is, a decision maker may not be able to easily see the link between the current system state and the policy's action. In this paper, the authors propose a new approach to optimal stopping, wherein the policy is represented as a binary tree, in the spirit of naturally interpretable tree models commonly used in machine learning. They show that the class of tree policies is rich enough to approximate the optimal policy. They formulate the problem of learning such policies from observed trajectories of the stochastic system as a sample average approximation (SAA) problem. The authors prove that the SAA problem converges under mild conditions as the sample size increases, but that computationally even immediate simplifications of the SAA problem are theoretically intractable. They thus propose a tractable heuristic for approximately solving the SAA problem, by greedily constructing the tree from the top down. They demonstrate the value of their approach by applying it to the canonical problem of option pricing, using both synthetic instances and instances using real S&P-500 data. Their method obtains policies that (1) outperform state-of-the-art noninterpretable methods, based on simulation-regression and martingale duality, and (2) possess a remarkably simple and intuitive structure.