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.
