Grant: Discovery Grants Program - Individual

In modeling financial variables mathematically it is important to consider dynamics which lead to tractable solutions while at the same time capture qualitative features of empirical data. The proposed research program involves the study of various stochastic equations, their properties, and their application to modeling in finance. In particular we shall focus on the pricing and hedging of various term-structure derivatives, commodity contracts, default (credit) risk, and insurance products. Attention shall also be given to the foundations and analytic tools necessary to implement the models effectively. The main class of stochastic equations to be considered are forward-backward stochastic differential equations (FBSDEs, for short). A FBSDE is a coupled system of stochastic equations of which certain components evolve forward in time from an initial condition and other components evolve backward in time from a random terminal condition. Several problems in the literature have been solved by characterizing financial models in terms of FBSDEs. We plan to consider various new models for factors of the economy and derivative securities in terms of FBSDEs that are more general than those previously considered and better capture qualitative features of observed financial data. We shall first incorporate spikes or jumps in the price processes to model the unpredictable events that traditional financial models do not allow. A price process that allows parameters to switch between a finite number of values, modeling a change in financial or economic fundamentals over time, is called a regime-switching model. To better model changes in financial or economic conditions over time we plan to incorporate regime-switching in the dynamics for the factors of the economy and allow these to influence price processes of assets, interest rates, and commodities. Innovative new numerical methods for solving FBSDEs shall also be developed to allow our theoretical results to be applied by practitioners and industry professionals.