Financial modelling and derivatives pricing under alternative Stochastic processes

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The development of realistic stochastic models for risky asset price processes and their implementation to the pricing of generally exotic derivatives is at the heart of modern day financial mathematics. A vast body of research is devoted to this important area which poses computational and theoretical challenges. Current models go well beyond the celebrated Black-Scholes model which does not support commonly observed features such as volatility clustering and fat tail distributions for asset price returns. The observed leverage effect and the market implied volatility smiles are also completely absent in this model and not well replicated in some mathematical extensions of the model. The resolution of these model discrepancies is a fundamental issue and this continues to fuel interest in financial modelling and option pricing with the use of more realistic models. The goal of my research is to make further progress in this and related critical areas of financial mathematics. This proposal focuses on further innovative developments of alternative stochastic processes in the single and multi-asset domain with several applications to derivatives pricing. An underlying component of my research involves the use of our newly developed families of analytically tractable models. Our research to date has shown that our models are rich in their ability to realistically describe option market volatility smiles and skews. My most current work also develops analytically exact spectral expansions for first passage time densities, barrier options and seasoned lookback options for these new models. We have also recently succeeded in developing efficient algorithms for pricing exotic options under subfamilies of our new volatility smile models. This proposal will continue to build on new developments and applications of such alternative stochastic models. The development of efficient numerical algorithms for describing such processes leads to various applications in finance as well as in many other areas of mathematical modelling that involve stochastic processes. The mathematical and computational results that will be generated by this research are expected to significantly impact the field of financial mathematics.