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

9803169波动率的渐近和统计分析及其对衍生品定价和风险管理的影响K.Ronnie Sircar在金融数学中，存在一种用于定价和对冲衍生品证券风险的方法(由于布莱克和斯科尔斯)。期权)，当股票的波动率不变时。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。这个项目的目的是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年股市崩盘以来，这种观点是否变得更加悲观。该分析将利用推动股票价格过程的布朗运动和波动随机过程在波动时间尺度上的差异。过去20年衍生品市场规模的惊人增长(目前它在美国的营业额高达数万亿美元)，加上最近臭名昭著(同样引人注目)的风险(Mis)管理灾难，如霸菱和奥兰治县的惨败，产生了对良好的数学和计算模型的迫切需求，以量化此类投资的各自风险和回报。这个项目旨在以布莱克、斯科尔斯和默顿的诺贝尔奖获得者的方法为基础，考虑到市场波动的波动性质。数学工具与对过去价格的统计分析相结合，以产生公式和软件，准确地捕捉当今巨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.