# An Expanded Local Variance Gamma model

## TL;DR

This paper introduces an expanded Local Variance Gamma model with added drift, enabling a closed-form solution for option pricing and a fast, non-optimization calibration method for the entire local volatility surface.

## Contribution

It extends the Local Variance Gamma model by incorporating drift and derives a closed-form solution, simplifying calibration of the local volatility surface.

## Key findings

- Closed-form solution for option prices using hypergeometric functions.
- Calibration can be performed term-by-term without optimization.
- Model effectively fits market smiles with fast computation.

## Abstract

The paper proposes an expanded version of the Local Variance Gamma model of Carr and Nadtochiy by adding drift to the governing underlying process. Still in this new model it is possible to derive an ordinary differential equation for the option price which plays a role of Dupire's equation for the standard local volatility model. It is shown how calibration of multiple smiles (the whole local volatility surface) can be done in such a case. Further, assuming the local variance to be a piecewise linear function of strike and piecewise constant function of time this ODE is solved in closed form in terms of Confluent hypergeometric functions. Calibration of the model to market smiles does not require solving any optimization problem and, in contrast, can be done term-by-term by solving a system of non-linear algebraic equations for each maturity, which is fast.