# 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.

## Key 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.