CAREER: An Approach to Pricing, Hedging, Stability, and Asymptotic Analysis in Financial Markets

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As financial markets have not only grown but have also become very complex, understanding both their qualitative and quantitative aspects are among highly active areas of research in mathematics. This award provides methods for pricing, hedging, stability and asymptotic analysis in financial markets. These projects will further our understanding of so-called incomplete markets (i.e., the markets which have a limited capability to offset risks) and the sensitivity of such markets to different types of perturbations and trading restrictions. The topics will lead to new developments in stochastic control, convex and stochastic analysis, to novel interdisciplinary research, and results applicable in the financial industry. Graduate students and post doctoral researchers are included in the work of the project.The first research topic is stability and asymptotic analysis of financial markets with respect to perturbations. Mathematically this topic leads to investigations of the responses of the underlying stochastic control problems to distortions of the input data. An appropriate form of parametrization for the perturbations and the corresponding value functions will allow for analysis involving only partial convexity (in one variable) of the underlying value function. Both dynamic and static formulations of the underlying stochastic control problem will be considered. The second topic is the pricing and hedging of financial instruments when additional trading constraints are imposed. This topic is connected to an investigation of the optimal investment problem with labor income, where extra trading constraints make the problem hard to analyze, and special forms of parametrization of the labor income and the value function are needed. The mathematical work relies on classical and modern results in stochastic analysis, stochastic control, finite and infinite-dimensional convex analysis, and new results to be established by the Principal Investigator.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.