# Decomposition formula for jump diffusion models

## TL;DR

This paper derives a generic decomposition for option pricing in jump diffusion models with finite activity jumps, introduces explicit approximation formulas for popular SVJ models, and demonstrates improved computational efficiency and insight into volatility smile behavior.

## Contribution

It extends the decomposition formula to jump diffusion models, providing explicit approximations and reformulations that enhance computation and understanding of volatility smiles.

## Key findings

- Approximation formulas significantly improve computation efficiency.
- Reformulation in terms of implied volatilities offers better intuition.
- Numerical comparisons show competitive accuracy with Fourier methods.

## Abstract

In this paper we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alos (2012) for Heston (1993) SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models - models utilizing a variance process postulated by Heston (1993). In particular, we inspect in detail the approximation formula for the Bates (1996) model with log-normal jump sizes and we provide a numerical comparison with the industry standard - Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behaviour under a specific SVJ model.