Solvable and Other Stochastic Models for Risk Modeling and Asset Pricing in Quantitative Finance

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Stochastic modelling is fundamental to financial mathematics, which is a core discipline in both finance and mathematics. Stochastic processes provide a natural setting for quantifying financial risk in all sectors of the economy. Financial derivatives are contingent claims that are heavily traded in all sectors of the economy. An important area of financial mathematics remains the development and application of realistic and tractable stochastic models for describing risk factors and the implementation of these models in the valuation and hedging of derivative contracts. Many financial derivatives are also inherently path-dependent. Such complex financial instruments, with built-in path dependencies, are useful for mitigating specific risks. Moreover, certain types of path dependencies of an asset can play a key role in linking option pricing with the credit risk associated to a given financial firm. For example, path-dependent quantities such as the first passage time of an asset to a critical level or the occupation time of the asset below a given critical level, as well as variations of such quantities, are useful for valuing financial contracts subject to default risk. My research will focus on further new developments of tractable stochastic models and implementing the models to various important problems in quantitative finance involving risk modelling, derivative pricing and hedging, and model calibration to historical market data. A main part of my research will continue to exploit and build upon my previous advancements on so-called solvable models with current and new areas of application that will also link option pricing and credit risk. This proposal will lead to new models and efficient numerical algorithms for financial applications employing such models. Another component of my research will exploit Monte Carlo simulation algorithms. Different approaches will be used to develop tractable realistic models. One approach will attempt to extend the mathematical methodology that we have already successfully developed for single-asset solvable models into the multi-asset domain. The other methodology will exploit the use of copulas in constructing tractable multi-asset models that will take into account some real market observed effects. The actual implementation of these models will lead to efficient algorithms for model calibration and pricing and hedging multi-asset financial products. Some of the algorithms will also exploit the use of high-performance computing software.