Asymptotic and Statistical Analysis of Volatility and its Implications for Derivative Pricing and Risk Management

DMS-9803169 波动率的渐近统计分析及其启示 衍生品定价和风险管理 K. 罗尼·瑟卡尔 在金融数学中，存在一种方法（由于布莱克和斯科尔斯）来定价和对冲衍生证券的风险（例如。当股票的波动率是常数时，然而，人们普遍认为波动率具有随机成分，必须用合适的随机模型来描述。本项目旨在利用历史价格数据对股票波动动态进行统计描述，并将其与衍生品价格偏微分方程的渐近分析相结合，以推断市场对未来可能的股票走势的看法，特别是自1987年崩盘以来，这种看法是否变得更加悲观。该分析将利用布朗运动驱动的股票价格过程和波动率随机过程之间的波动时间尺度的差异。 过去20年衍生品市场规模的惊人增长（目前美国的交易额已达数万亿美元），加上最近臭名昭著的（同样惊人的）风险管理灾难，如巴林银行和橙子县的惨败，迫切需要良好的数学和计算模型来量化此类投资的风险和回报。该项目旨在建立在布莱克，斯科尔斯和默顿的诺贝尔奖获得者的方法，考虑到市场波动的波动性质。数学工具与过去价格的统计分析相结合，产生公式和软件，准确地捕捉当今庞大的衍生品市场的潜在损失和收益。

DMS-9803169 Asymptotic and Statistical Analysis of Volatility and its Implications for Derivative Pricing and Risk Management K. Ronnie Sircar In financial mathematics, a methodology (due to Black and Scholes) exists for pricing and hedging against the risk of derivative securities (eg. options) on stocks when the volatility of the stock is constant. However, it is widely believed that volatility has a random component and must be described by a suitable stochastic model. This project aims to produce a statistical description of stock volatility dynamics using historical price data, and to combine it with an asymptotic analysis of the partial differential equation for derivative prices to infer the market's view of probable future stock movements, and in particular, whether this view has become more pessimistic since the 1987 crash. The analysis will exploit the discrepancy in time-scales of fluctuation between the Brownian motion driving the stock price process and the volatility stochastic process. The spectacular growth in the size of the derivatives market over the last twenty years (currently it has a turnover of trillions of dollars in the US) plus recent infamous (and equally spectacular) risk (mis)management disasters such as the Barings and Orange County fiascos, have created an urgent need for good mathematical and computational models to quantify the respective risks and rewards of such investments. This project aims to build on the Nobel Prize winning methodology of Black, Scholes and Merton, to take into account the fluctuating nature of market volatility. Mathematical tools are combined with statistical analysis of past prices to produce formulas and software that accurately capture the potential losses and gains in today's vast derivative market.