This 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.

中文翻译：

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几何局部方差伽玛模型

本文介绍了最初由Carr在2008年提出，然后由Carr和Nadtochiy在2017年以及在Carr和Itkin在2018年进一步阐述的局部方差伽马模型的另一种扩展。与由Carr和Itkin开发并称为在“扩展局部方差伽玛”（ELVG）模型中，本文提供了两项创新。首先，在之前的所有文章中，该模型都是基于伽马时变算术布朗运动构建的：Carr和Nadtochiy中没有漂移，Carr和Itkin中有漂移，并且局部方差仅是现货水平的函数。相反，本文开发了带有漂移的该模型的几何版本。第二，在Carr和Nadtochiy中，假设局部方差是罢工的分段常数函数，则模型对选项微笑进行了校准，而在Carr和Itkin中，当地方差被假定为罢工的分段线性函数。在本文中，作者考虑了三个分段线性模型：局部方差作为行使价的函数，局部方差作为对数行使价的函数以及局部波动率作为行使价的函数（因此，局部方差是分段的罢工的二次函数）。作者表明，对于所有这些新结构，仍然有可能导出期权价格的普通微分方程，该方程对于标准局部波动率模型起着Dupire方程的作用，而且可以封闭形式求解。最后，类似于Carr和Itkin，作者表明，给予多个微笑，可以恢复整个局部方差/波动表面，而无需解决任何优化问题。取而代之的是，可以通过求解每个成熟度的非线性代数方程组来逐项完成，这是一个明显更快的过程。主题：导数，统计方法，选项主要发现•基于带漂移的几何布朗运动，提出了局部方差伽马模型的扩展。•三种分段线性模型：考虑了作为方差的函数的局部方差，作为对数方差的函数的局部方差和作为方差的函数的局部波动率（因此，局部方差是由方差的分段二次函数） 。