Asymptotic Methods in Financial Mathematics

职务名称：金融数学中的渐近方法（NSF-DMS-0071744，2000年5月23日修订）主要研究者/项目总监：Jean-Pierre Fouque技术描述：在具有快速均值回复随机波动率的金融市场中，定价和对冲衍生品会导致抛物型偏微分方程（如股票市场中的Black-Scholes方程）的奇异摄动。本研究包括在开发必要的数学工具，以实现这种新的方法在各种情况下，如美国和异国情调的选择权在股票市场或期限结构的利率在固定income markets.The第一步，在这种方法中需要识别的时间尺度存在于隐藏的随机波动过程。这已经通过使用变差函数和谱分析方法对S P500进行了分析。其他工具和市场将进行调查。第二步包括通过随机微分方程的波动系数驱动的额外的一般遍历过程运行在更快的时间尺度上的价格过程的演变建模。风险中性概率由波动风险的市场价格参数化。在马尔可夫的情况下，衍生品的价格得到的偏微分方程，这是相应的经典常数波动率方程的奇异摄动的解决方案。渐近展开进行。零阶项对应于经典的恒定波动率的情况。一阶项只取决于当前的价格，他们计算显式或通过决议的偏微分方程。美式期权的情况下，导致奇异摄动的自由边界值问题，将详细研究。套期保值问题将被处理的鞅分解方法也用于非马尔可夫的情况下或无限维的情况下出现在固定收益markets.The下一步包括在扩展中获得的第一个修正中揭示的几个参数的校准。这是通过使用在流动性市场中观察到的衍生品价格来完成的，并用于定价其他奇异的衍生品。这种处理不完全随机波动率市场的新方法产生了易于校准和计算的模型独立的定价和对冲公式。它的数学分析，这个建议的心脏，触及应用数学的各个领域，并导致深远的新工具和result.Non-Technical描述：这个项目解决的问题，在金融数学的定价和对冲衍生证券在不确定和不断变化的市场波动的环境。这些风险管理问题对于从大型交易机构到养老基金的投资者以及金融市场和经济活动的监管机构（如联邦储备）都很重要。人们普遍认为，著名的布莱克-斯科尔斯模型将衍生品价格与当前股票价格联系起来，并通过恒定的波动率参数量化风险，其简单性已不再足以捕捉现代市场现象，特别是自1987年股灾以来，本研究包括一个新的方法建模，分析和估计，利用随机性质和快速的内在时间尺度的波动性的调查。这些观察到的波动性的性质是用来获得经典的常数波动率公式的修正。这些修正揭示了市场参数的重要分组，否则并不明显，事实证明，从市场数据中估计这些组合是非常有效和稳定的。大量的注意力将集中在更涉及数学的美式期权的情况下，它可以在到期前的任何时间行使，或其他越来越受欢迎的异国情调的期权，基于概率论中鞅的基本概念，开发了新的和原始的方法来处理复杂的固定收益市场，对利率收益率曲线的时间演变的理解是一个真实的挑战。这项研究的目标是产生复杂的数学结果，可以有效地实施和使用的从业人员在定量金融。

Title: Asymptotic Methods in Financial Mathematics (NSF-DMS-0071744 revised 5/23/00)Principal Investigator/Project Director: Jean-Pierre FouqueTechnical Description:Pricing and hedging derivatives in financial markets with fast mean-reverting stochastic volatility lead to singular perturbations of parabolic partial differential equations such as Black-Scholes equation in the case of stock markets. This research consists in developing the mathematical tools necessary to implement this new methodology in various situations such as American and exotic options in equity markets or term-structures of interest rates in fixed income markets.The first step in this method requires an identification of the time scales present in the hidden stochastic volatility processes. This has been done for S&P500 by using variograms and spectral analysis methods. Other tools and markets will be investigated The second step consists in modeling the evolution of the price process by means of stochastic differential equations with volatility coefficients driven by additional general ergodic processes running on faster time-scales. Risk-neutral probabilities are parameterized by market prices of volatility risk. In the Markovian case, derivative prices are obtained as solutions of partial differential equations, which are singular perturbations of the corresponding classical constant volatility equations. Asymptotic expansions are performed. The zero-order terms correspond to classical constant volatility situations. The first-order terms are shown to depend only on current prices and they are computed either explicitly or through the resolution of a partial differential equation. The case of American options leads to singular perturbations of free-boundary value problems that will be studied in detail. The hedging problem will be treated by a martingale decomposition approach also used in non-Markovian cases or infinite dimensional situations arising in fixed income markets.The next step consists in the calibration of the few parameters revealed in the first correction obtained in the expansion. This is done by using observed derivative prices in liquid markets and used for pricing other exotic derivatives.This new methodology to handle incomplete stochastic volatility markets gives rise to model independent pricing and hedging formulas easy to calibrate and to compute. Its mathematical analysis, the heart of this proposal, touches various fields of applied mathematics and leads to far reaching new tools and results.Non-Technical Description:This project addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These risk management problems are important to investors from large trading institutions to pension funds and to regulators of financial markets and economic activity such as the federal reserve.It is widely recognized that the simplicity of the famous Black-Scholes model which relates derivative prices to current stock prices and quantifies risk through a constant volatility parameter is no longer sufficient to capture modern market phenomena, especially since the 1987 crash.This research consists in the investigation of a new method for modeling, analysis and estimation that exploits the random nature and fast intrinsic time scale of the volatility. These observed volatility properties are used to derive corrections to the classical constant volatility formulas. These corrections reveal important groupings of market parameters, which otherwise are not obvious, and it turns out that estimation of these composites from market data is extremely efficient and stable.Much attention will be devoted to the more mathematically involved cases of American options, which can be exercised at any time before maturity, or other increasingly popular exotic options, which depend on the history of the underlying asset.New and original methods, based on the fundamental concept of martingale in probability theory, are developed to handle complex fixed income markets, for which the understanding of the time evolution of yield curves of interest rates is a real challenge. The goal of this research is to produce sophisticated mathematical results, which can be efficiently implemented and used by practitioners in quantitative finance.