Near-Maturity Asymptotics of Critical Prices of American Put Options under Exponential Lévy Models

Paper: arXiv 2512.17791

Authors: José E. Figueroa-López, Ruoting Gong

Abstract

In the present paper, we study the near-maturity ($t\rightarrow T^{-}$) convergence rate of the optimal early-exercise price $b(t)$ of an American put under an exponential Lévy model with a {\it nonzero} Brownian component. Two important settings, not previous covered in the literature, are considered. In the case that the optimal exercise price converges to the strike price ($b(T^{-})=K$), we contemplate models with negative jumps of unbounded variation (i.e., processes that exhibit high activity of negative jumps or sudden falls in asset prices). In the second case, when the optimal exercise price tend to a value lower than $K$, we consider infinite activity jumps (though still of bounded variations), extending existing results for models with finite jump activity (finitely many jumps in any finite interval). In both cases, we show that $b(T^{-})-b(t)$ is of order $\sqrt{T-t}$ with explicit constants proportionality. Furthermore, we also derive the second-order near-maturity expansion of the American put price around the critical price along a certain parabolic branch.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is a highly sophisticated theoretical mathematical finance study establishing near-maturity asymptotic properties for American options under general exponential Lévy processes. While it contains dense proofs, explicit formulas, and rigorous derivations, it is entirely analytical with no empirical backtests or market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Near-Maturity $t \to T^-$ Asymptotics of American Put Critical Price $b(t)$ under Exponential Lévy Models] --> B[Inputs: Nonzero Brownian Component & Infinite Activity Jumps]
    
    B --> C{Model Settings / Boundary Behavior}
    
    C -->|Case 1: $b(T^-) = K$| D[Analyze Unbounded Variation Negative Jumps High Activity]
    C -->|Case 2: $b(T^-) < K$| E[Analyze Infinite Activity Jumps of Bounded Variation]
    
    D --> F[Rigorous Asymptotic & Free Boundary Analysis]
    E --> F
    
    F --> G[Key Outcomes: <br/>1. $K - b(t) = O(\sqrt{T-t})$ with explicit constants<br/>2. Second-order price expansion along parabolic branch]