英文标题：
《Geometric Local Variance Gamma model》
---
作者：
Peter Carr, Andrey Itkin
---
最新提交年份：
2018
---
英文摘要：
This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded Local Variance Gamma model), here we provide two innovations. First, in all previous papers the model was constructed based on a Gamma time-changed {\\it arithmetic} Brownian motion: with no drift in CI2017, and with drift in CI2018, and the local variance to be a function of the spot level only. In contrast, here we develop a {\\it geometric} version of this model with drift. Second, in CN2017 the model was calibrated to option smiles assuming the local variance is a piecewise constant function of strike, while in CI2018 the local variance is a piecewise linear} function of strike. In this paper we consider 3 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). We show that for all these new constructions it is still 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, and, moreover, it can be solved in closed form. Finally, similar to CI2018, we show that given multiple smiles the whole local variance/volatility surface can be recovered which does not require solving any optimization problem. Instead, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity which is fast.
---
中文摘要：
本文描述了P.Carr在2008年最初提出的局部方差Gamma模型的另一个扩展，随后Carr和Nadtochiy，2017（CN2017）和Carr和Itkin，2018（CI2018）进一步阐述了该模型。与CI2018中开发的最新版本的模型（称为ELVG（扩展的局部方差伽马模型））相比，这里我们提供了两项创新。首先，在之前的所有论文中，该模型是基于伽马时变{\\it算术}布朗运动构建的：CI2017年无漂移，CI2018年有漂移，局部方差仅为现货水平的函数。相反，这里我们开发了这个模型的{\\it geometric}版本，带有漂移。其次，在CN2017年，假设局部方差是罢工的分段常数函数，则将模型校准为期权微笑，而在CI2018年，局部方差是罢工的分段线性}函数。本文考虑3个分段线性模型：局部方差作为罢工函数，局部方差作为对数罢工函数，局部波动作为罢工函数（因此，局部方差是罢工的分段二次函数）。我们表明，对于所有这些新的构造，仍然可以推导出期权价格的常微分方程，该方程在标准局部波动率模型中起到Dupire方程的作用，而且可以以闭合形式求解。最后，与CI2018类似，我们表明，给定多个微笑，可以恢复整个局部方差/波动率曲面，而不需要解决任何优化问题。相反，它可以通过为每个成熟度快速求解非线性代数方程组来逐项完成。
---
分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
--
一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
--
一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
--

---
PDF下载：
-->