Paper: arXiv 2610.30637

Authors: Katsunori Ano

Abstract

This article studies a finite-horizon American put option with up to $m$ exercise rights under the risk-neutral Black–Scholes model, where consecutive exercises must be separated by a prescribed refractory period $δ>0$ and unused rights may expire. For each number of remaining rights, we show that optimal exercise is characterized by a single exercise boundary and that these boundaries are ordered by the number of remaining rights. We prove optimality of the first boundary-hitting time and establish local Lipschitz continuity of the exercise boundaries within each open deadline strip. Away from the deadlines, we derive smooth fit, the associated quasi-variational inequality and free-boundary formulation, an early-exercise-premium representation, recursive Volterra-type boundary equations, and a verification theorem. We also extend the recursive formulation to an independent random termination time, thereby allowing for exogenous termination of the option contract.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on rigorous mathematical proofs for option pricing. While it introduces novel analytical solutions for multiple exercise rights, it lacks empirical validation or backtesting. The mathematical derivations are dense and advanced, characteristic of a theoretical quant finance paper.

Research Flowchart

  flowchart TD
    A[Research Goal: Model & Characterize American Put Option with Multiple Exercise Rights & Refractory Period] --> B{Methodology: Risk-Neutral Black-Scholes Model, Optimal Stopping Theory};
    B --> C[Inputs: Finite Horizon American Put Option, m Exercise Rights, Refractory Period δ > 0, Unused Rights Expiry];
    C --> D{Computational Process: Derivation of Exercise Boundaries, Quasi-Variational Inequality, Free-Boundary Formulation, Volterra-type Boundary Equations};
    D --> E[Outcomes: Optimal Exercise Characterized by Single Boundary for Each Right Count, Boundaries Ordered by Rights, Local Lipschitz Continuity, Smooth Fit, Early Exercise Premium];
    E --> F[Further Extension: Recursive Formulation to Independent Random Termination Time];