Collaborative Research: High-Performance Computational Methods for Continuous-Time Markov Processes in Financial Engineering

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This grant provides funding for the development of high-performance computational tools for financial engineering. The goal is to develop computational methods to evaluate complex financial products used to manage foreign exchange, interest rate, equity, commodity, energy, and credit risks, manage portfolios of assets, and evaluate financial contracts and equipment leases in manufacturing industries. The methodology is based on the extension to financial engineering of finite element methods used to solve numerically partial differential equations. Partial integro-differential equations arise in Markov jump-diffusion models and associated optimal stopping and stochastic control problems in financial engineering. Finite element methods will be applied to jump-diffusion processes to develop computational tools to be used by practitioners in the financial services industry, as well as researchers in financial engineering, applied probability and operations research. Applications include pricing algorithms for a wide range of financial contracts (equity and foreign exchange options, interest rate derivatives, commodity contracts, and equipment leases in manufacturing), as well as portfolio optimization with transaction costs and trading restrictions. If successful, methodologies developed in this project will help financial institutions, corporate treasuries of manufacturing and service firms, and energy companies accurately value complex financial instruments and efficiently manage financial risks. This project will also have a broader impact on research and application areas that use continuous-time Markov processes, such as heavy traffic limits in queueing theory, inventory control, scheduling, and manufacturing. The results on constructive approximations to the valuation, optimal stopping and stochastic control problems will also help simulation and stochastic optimization research by providing reliable benchmarks for simulation and stochastic optimization. This grant supports the Ph.D. concentration in financial engineering at Northwestern. This will result in training highly qualified researchers in financial engineering for academia and industry. The grant will also help the Department of Mathematical Sciences at the University of Nevada establish a research program in financial mathematics.