Time Changes of Markov Processes: Applications in Financial Mathematics

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This project is concerned with stochastic modeling of financial variables, such as equity prices, foreign exchange rates, interest rates, and commodity and energy prices, and the pricing of financial derivatives, financial instruments that serve as tools to mitigate financial market and credit risk. The goal is to develop empirically realistic and analytically tractable models describing stochastic dynamics of financial variables and analytical and computational methods to implement these models. The project focuses in the following areas. (i) Develop a rich toolbox of financial models with state-dependent jumps, stochastic volatility, and default. This will be based on the methodology of time changes of Markov processes. The proposed model architecture is to pair Markov processes with analytically tractable semigroups with time changes with analytically tractable Laplace transforms to design analytically tractable processes with desired properties. (ii) Develop efficient pricing tools for derivative securities in these models based on Laplace transform inversion and, for symmetric Markov processes, on the spectral expansion method. (iii) Develop the next generation of unified credit-equity models with jumps, stochastic volatility, and default that provide a unified treatment for all financial obligations related to a given firm, including stock, debt, stock options, and credit derivatives in the framework of time changes of Markov processes with killing. Study options pricing and corporate debt valuation in this unified framework. (iv) Develop a novel class of models with mean-reverting jumps for interest rates, commodities, and energy markets by subjecting mean-reverting diffusions to time changes with jumps. The aim of this project is to develop sophisticated mathematical tools for financial practice. This research is expected to have significant practical impact. According to the Bank for International Settlements (2006), the size of the global derivatives markets is $343 trillion in notional amounts. Major market segments include interest rate, currency, equity, commodity, energy, and credit derivatives used to manage financial risks. The proposed research is expected to find applications in all of these financial market sectors. Empirically realistic and, at the same time, analytically and computationally tractable models developed in this project will facilitate consistent, fast, and accurate modeling and pricing of financial instruments used to manage market and credit risk. This will help improve efficiency and stability of financial markets. The project is also expected to have a broader mathematical impact on stochastic analysis and applied probability. Analytically tractable time changes of Markov processes developed in this project will provide a useful laboratory for theoretical developments in stochastic analysis, as well as find applications in a variety of areas that employ Markov processes for the modeling of physical, biological, engineering, and economic phenomena. The project will have an impact on education and human resources development. It is part of the long-term development effort at Northwestern University in financial engineering, including the recently established Ph.D. concentration. It will train highly qualified researchers for academia and industry.