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

TL;DR

This work advances the understanding of near-maturity behavior for American put options in exponential Lévy models with a Brownian component. It establishes that the early-exercise boundary approaches the strike at a rate proportional to the square root of time to maturity in two regimes: with unbounded negative jumps and with infinite-activity jumps, providing explicit constants and a second-order price expansion along a parabolic trajectory. The authors derive precise rates for the European critical price, quantify the European–American boundary gap, and reveal a distinct rate when the drift d is negative with finite-variation jumps. These results extend classical Black–Scholes intuition to broad jump-diffusion settings and yield practically useful near-maturity approximations for pricing American puts under Lévy dynamics.

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.

Theorems & Definitions (23)

• Remark 2.2
• Proposition 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Theorem 3.5
• Theorem 4.1