Analytic techniques for option pricing under a hyperexponential L\'{e}vy model

TL;DR

This paper introduces series expansion techniques for option pricing under hyperexponential Lévy models, providing explicit formulas for prices, Greeks, and implied volatility, and enhancing numerical methods for exotic options.

Contribution

It develops novel series expansion methods for solving key equations in hyperexponential Lévy models, enabling explicit pricing formulas and improved computational efficiency.

Findings

Derived analytic formulas for European option prices and Greeks.

Provided asymptotic expansion for short-time implied volatility.

Showed how to accelerate numerical algorithms for exotic options.

Abstract

We develop series expansions in powers of $q^{-1}$ and $q^{-1/2}$ of solutions of the equation $\psi(z) = q$, where $\psi(z)$ is the Laplace exponent of a hyperexponential L\'{e}vy process. As a direct consequence we derive analytic expressions for the prices of European call and put options and their Greeks (Theta, Delta, and Gamma) and a full asymptotic expansion of the short-time Black-Scholes at-the-money implied volatility. Further we demonstrate how the speed of numerical algorithms for pricing exotic options, which are based on the Laplace transform, may be increased.

Full text

Analytic techniques for option pricing under a hyperexponential Lévy model

Daniel Hackmann 111 E-mail: dan@danhackmann.com. Web: www.danhackmann.com.