Asymptotics and dynamics of forward implied volatility

远期隐含波动率的渐近性和动态

• 批准号: EP/M008436/1
• 负责人: Antoine Jacquier
• 金额: $ 12.34万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM008436%2F1
• 关键词: Asymptotics; dynamics; forward; implied; volatility

Asymptotic methods represent a set of tools (from probability, PDE theory, geometry) allowing to study systems when some parameters become small or large. It is particularly useful when, say, an equation does not have an explicit solution, but the latter can be written as a series expansion when some parameter is small. This therefore yields approximate yet accurate understanding of the behaviour of the solution (up to some small error). In mathematical finance, many stochastic) models have been proposed and used in the past four decades in order to reflect the dynamics of asset prices and financial markets. Based on these processes, pricing equations can be written and solved numerically. This can be performed, either from a probabilistic point of view, where computing expectations boils down to (often complex) numerical integration, or from an analytic perspective, where the solution of the problem solves some partial (integro-) differential equation. Even though powerful numerical methods exist, they are often computer-intensive and do not provide easy (and intuitive) understanding of the behaviour of the solution.The cornerstone of such models is the so-called Black-Scholes model, for which European call option prices have a trivial closed-form expression. However, in most models, option prices do not have closed-form representations, and have to be computed numerically. This is even more so for the corresponding implied volatility, which is just a standardised option price (now universally used in practice as a quoting mechanism). Over the past fifteen years, active research has been carried out to obtain explicit analytical approximations for this implied volatility, thus effectively replacing the highly demanding numerical computations by some simple approximate) solution. Lee was one of the pioneers of this stream, providing a precise link between the behaviour of the implied volatility and the tail distribution of the stock price. This result has since been extended and improved by several authors, including Benaim-Friz, Gulisashvili-Stein, De Marco-Hillairet-Jacquier. Other important results in this direction were obtained by Henry-Labordere (using differential geometry), Jacquier, Keller-Ressel and Mijatovic (using probabilistic tools) and Deuschel, Friz, Jacquier and Violante (using both geometric and probabilistic methods). All these results however do not give any information on the dynamic behaviour of the implied volatility, which is essential in order to accurately model the time-evolving nature of financial markets.The goal of this project is to understand this dynamic behaviour of the implied volatility for a large class of models, and to propose a tractable formula describing it. This has been partially achieved in the static case, but the question remains wide open in the dynamic case. In order to do so, the PI intends to follow two main directions:- determine the asymptotic behaviour of the dynamic implied volatility for a large class of stochastic models;- extend to the dynamic case the existing arbitrage-free implied volatility parameterisation.Progress in either of these directions would immediately yields a better understanding of the models currently used in practice: are they accurate enough? Do they possess realistic properties to model the behaviour of financial markets? It would also provide deeper insight on so-called model risk, namely the risk associated to the use of a statically tested model for dynamic purposes. Ultimately this could yield a classification of models according to their actual usefulness.

渐近方法代表了一组工具（来自概率，PDE理论，几何），允许在某些参数变小或变大时研究系统。当一个方程没有显式解，但当某些参数很小时，后者可以写成级数展开式时，它特别有用。因此，这产生了对解的行为的近似但准确的理解（直到一些小的误差）。在过去的四十年中，许多随机模型被提出和使用，以反映资产价格和金融市场的动态。基于这些过程，定价方程可以用数字来写和求解。这可以从概率的角度来执行，其中计算期望归结为（通常是复杂的）数值积分，或者从分析的角度来执行，其中问题的解决方案解决了一些偏（积分）微分方程。尽管存在强大的数值方法，但它们通常是计算机密集型的，并且不能提供简单（和直观）的解的行为理解。这些模型的基石是所谓的布莱克-斯科尔斯模型，其中欧式看涨期权价格有一个平凡的封闭形式表达式。然而，在大多数模型中，期权价格没有封闭形式的表示，必须用数值计算。对于相应的隐含波动率来说更是如此，它只是一个标准化的期权价格（现在在实践中普遍用作报价机制）。在过去的15年里，人们进行了积极的研究，以获得这种隐含波动率的显式解析近似，从而有效地取代了一些简单的近似解的高要求的数值计算。李是这一理论的先驱之一，他在隐含波动率的行为和股票价格的尾部分布之间建立了精确的联系。这个结果后来被几个作者扩展和改进，包括Benaim-Friz，Gulisashvili-Stein，De Marco-Hillairet-Jacquier。其他重要的结果在这一方向上获得了亨利-阿卡德雷（使用微分几何），雅基耶，凯勒-Ressel和Mijatovic（使用概率工具）和Deuschel，Friz，雅基耶和Violante（使用几何和概率方法）。然而，所有这些结果并没有给出任何关于隐含波动率动态行为的信息，而这对于准确地模拟金融市场的时间演化性质是必不可少的。本项目的目标是理解大类模型的隐含波动率的动态行为，并提出一个描述它的易处理的公式。这在静态情况下已经部分实现，但在动态情况下，这个问题仍然很开放。为了做到这一点，PI打算遵循两个主要方向：-确定一大类随机模型的动态隐含波动率的渐近行为;-将现有的无仲裁隐含波动率参数化扩展到动态情况。在这两个方向中的任何一个方向上的进展都将立即产生对目前在实践中使用的模型的更好理解：它们是否足够准确？它们是否具有模拟金融市场行为的现实属性？它还将提供对所谓的模型风险的更深入的了解，即为动态目的使用经过静态测试的模型所带来的风险。最终，这可以根据模型的实际用途对其进行分类。

项目成果

The implied volatility of Forward-Start options: ATM short-time level, skew and curvature

远期启动期权的隐含波动率：ATM 短期水平、偏斜和曲率

• DOI: 10.1080/17442508.2018.1499105
• 发表时间: 2018
• 期刊: Stochastics
• 影响因子: 0.9
• 作者: Alòs E

Asymptotic Behavior of the Fractional Heston Model

分数赫斯顿模型的渐近行为

• DOI: 10.1137/17m1142892
• 发表时间: 2018
• 期刊: SIAM Journal on Financial Mathematics
• 影响因子: 1
• 作者: Guennoun H

No-arbitrage bounds for the forward smile given marginals

给定边际前向微笑的无套利界限

• DOI: 10.1080/14697688.2016.1267392
• 发表时间: 2017
• 期刊: Quantitative Finance
• 影响因子: 1.3
• 作者: Badikov S

An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients

• DOI: 10.1137/15m1017788
• 发表时间: 2014-05
• 期刊: SIAM J. Financial Math.
• 作者: J. Chassagneux;A. Jacquier;I. Mihaylov

Asymptotic Behaviour of the Fractional Heston Model

分数赫斯顿模型的渐近行为

• DOI: 10.2139/ssrn.2531468
• 发表时间: 2014
• 期刊: SSRN Electronic Journal
• 作者: Guennoun H