Asymptotic Methods in Financial Mathematics

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This project studies some problems in financial mathematics related to stochastic volatility models and portfolio optimization. The specific problems under consideration here are 1) identification and analysis of time-scales in market volatility; 2) analysis of "alternative" mechanisms for pricing and hedging derivative securities via stochastic control methods, in particular to model "crash-o-phobia"; 3) optimal investment decisions under stochastic stock price models incorporating asymmetry in returns distributions.The spectacular growth in the size of the financial derivatives market over the last thirty 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, Orange County and Long Term Capital Management fiascos, have created an urgent need for smart mathematical and computational models to quantify the respective risks and rewards of such investments. This continuing project aims to build on the methodology introduced by Black, Scholes and Merton, to take into account the uncertain nature of market volatility. Mathematical and computational tools are combined with statistical analysis of past prices to produce formulas and software that better understand the potentially serious consequences of changing volatility for portfolios. This issue is important for investors from large trading institutions to individuals with pension funds.