Asymptotics and dynamics of forward implied volatility

渐近方法代表了一套工具(来自概率论、偏微分方程理论、几何学)，允许在某些参数变小或变大时研究系统。比方说，当一个方程没有显式解时，它特别有用，但当某个参数很小时，后者可以写成级数展开。因此，这产生了对解决方案的行为的近似而准确的理解(直到一些小误差)。在数学金融学中，在过去的40年里，为了反映资产价格和金融市场的动态，提出并使用了许多随机模型。在这些过程的基础上，可以写出定价方程并进行数值求解。这可以从概率的角度来执行，在概率的角度上，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，在分析的角度上，问题的解解决一些偏微分方程(积分)。尽管存在强大的数值方法，但它们往往需要大量使用计算机，而且不能轻松(和直观地)理解解决方案的行为。这类模型的基石是所谓的布莱克-斯科尔斯模型，对于该模型，欧式看涨期权价格有一个微不足道的封闭表达式。然而，在大多数模型中，期权价格没有闭合形式的表示，必须通过数值计算。对于相应的隐含波动率更是如此，它只是一个标准化的期权价格(目前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，从而有效地用一些简单的近似解取代了要求很高的数值计算。Lee是这一趋势的先驱之一，他在隐含波动率的行为和股票价格的尾部分布之间提供了精确的联系。此后，这一结果得到了几位作者的推广和改进，包括BenaimFriz，Gulisashvili-Stein，De Marco-Hillairet-Jquier。在这个方向上的其他重要结果由Henry-Labordere(使用微分几何)，Jquier，Keller-Ressel和Mijatovic(使用概率工具)以及Deuschel，Friz，Jquier和Violante(使用几何和概率方法)获得。然而，所有这些结果都没有提供任何关于隐含波动率动态行为的信息，这对于准确地模拟金融市场的时间演变性质是必不可少的。本项目的目标是了解一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于处理的公式。这在静态情况下已经部分实现了，但在动态情况下这个问题仍然悬而未决。为此，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；-将现有的无套利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的进展都将立即产生对当前实践中使用的模型的更好理解：它们是否足够准确？它们是否具备模拟金融市场行为的现实属性？它还将更深入地了解所谓的模型风险，即与将静态测试的模型用于动态目的有关的风险。最终，这可能会根据模型的实际用途对模型进行分类。

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.