On an Optimal Stopping Problem with a Discontinuous Reward

ArXiv ID: 2311.03538 “View on arXiv”

Authors: Unknown

Abstract

We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity contract with guaranteed minimum maturity benefit, under the assumption that the policyholder’s surrender behaviour maximizes the risk-neutral value of the contract. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool to understand their interrelation and how it affects early surrenders and the optimal surrender boundary. Under certain conditions on these two functions, we develop three representations for the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature.

Keywords: Optimal stopping, Variable annuity, Surrender behavior, American option pricing, Value function

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical analysis of an optimal stopping problem with discontinuous reward functions, using concepts like free-boundary problems and Itô’s formula, but lacks any empirical validation, backtesting, or numerical implementation details.

  flowchart TD
    A["Research Goal<br>Optimal Stopping with Discontinuous Reward"] --> B{"Methodology"}
    B --> C["Alternative Value Function<br>Representation"]
    B --> D["General Fee & Charge<br>Functions"]
    C --> E["Analytical Properties<br>& Surrender Region Analysis"]
    D --> E
    E --> F["Computational Process<br>Numerical Solution via PDE"]
    F --> G{"Outcomes"}
    G --> H["Condition for<br>Optimal Maturity Surrender"]
    G --> I["Characterization of<br>Surrender Region"]