Geometric Local Variance Gamma Model

Abstract

The article describes another extension of the local variance gamma model originally proposed by Carr in 2008 and then further elaborated by Carr and Nadtochiy in 2017 and Carr and Itkin in 2018. As compared with the latest version of the model developed by Carr and Itkin and called the “expanded local variance gamma” (ELVG) model, two innovations are provided in this article. First, in all previous articles the model was constructed on the basis of a gamma time-changed arithmetic Brownian motion: with no drift in Carr and Nadtochiy, with drift in Carr and Itkin, and with the local variance a function of the spot level only. In contrast, this article develops a geometric version of this model with drift. Second, in Carr and Nadtochiy the model was calibrated to option smiles assuming that the local variance is a piecewise constant function of strike, while in Carr and Itkin the local variance was assumed to be a piecewise linear function of strike. In this article, the authors consider three piecewise linear models: the local variance as a function of strike, the local variance as a function of log-strike, and the local volatility as a function of strike (so, the local variance is a piecewise quadratic function of strike). The authors show that for all these new constructions, it is still possible to derive an ordinary differential equation for the option price, which plays the role of Dupire’s equation for the standard local volatility model, and moreover, it can be solved in closed form. Finally, similar to in Carr and Itkin, the authors show that given multiple smiles the whole local variance/volatility surface can be recovered without requiring solving any optimization problem. Instead, it can be done term-by-term by solving a system of nonlinear algebraic equations for each maturity, which is a significantly faster process.

TOPICS: Derivatives, statistical methods, options

Key Findings

• • An extension of the Local Variance Gamma model is proposed on the basis of the Geometric Brownian motion with drift.

• • Three piecewise linear models: the local variance as a function of strike, the local variance as function of log-strike, and the local volatility as a function of strike (so, the local variance is a piecewise quadratic function of strike) are considered.

• • For all these new constructions an ODE is derived which replaces the Dupire equation and can be solved in closed form.