Solvable models in option pricing and credit risk

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The Science of Mathematical Finance deals with problems in financial markets, such as investing, hedging, pricing and risk management. Over the last 30 years ideas and methods of Math Finance have been adopted by the leading banks, insurance companies and pension funds around the globe and have become essential for the efficient operation of financial markets.At the heart of Mathematical Finance lies the modelling of financial data. A good model should be realistic and computationally efficient at the same time. Those models which have explicit formulas for important quantities (such as the prices of European options) are called "solvable" and are the most popular in the industry. The problem is that there are just a few solvable models and most of them are just too simple to capture the phenomena observed in market data. The work I propose to do is to construct new solvable stochastic models (and classify existing ones), which might be better suited for the particular problems in math finance, as well as other areas in applied science. This research lies at the intersection of several branches of science, such as applied mathematics, theory of stochastic processes and theory of special functions. The possible applications of these new solvable models in Mathematical Finance include pricing of American type options, exotic options, credit risk modelling, interest rate and stochastic volatility modelling.