Near-Maturity Asymptotics of Critical Prices of American Put Options under Exponential L\'{e}vy Models

TL;DR

This paper analyzes the near-maturity behavior of American put option prices and optimal exercise boundaries under exponential Lévy models with jumps, providing explicit asymptotics and second-order expansions.

Contribution

It extends existing results by deriving near-maturity asymptotics for models with negative jumps of unbounded variation and infinite activity jumps, including explicit constants.

Findings

The difference b(T−)−b(t) is of order √(T−t) with explicit constants.

Derived second-order near-maturity expansion of American put prices.

Extended asymptotic analysis to models with complex jump structures.

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 L\'{e}vy 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.