In the first project, it is proposed to establish new results in the theory of asymptotic analysis and homogenization of nonlinear partial partial differential equations with direct applications to several important and practical problems faced by practitioners in the financial industry. These problems include risk management under uncertain volatility, behavior of implied volatilities at short maturities, and portfolio optimization under stochastic volatility. In the second project, it is proposed to develop new models based on systems of interacting diffusions where the coupling is modeled in the drifts through lending preferences. Combined with sophisticated Monte Carlo methods, including interacting particle system methods, these models will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network.
Challenging nonlinear problems arising naturally in the context of risk management under uncertain or randomly fluctuating volatility are addressed in the first project. This research has direct applications to practical problems faced by practitioners in the financial industry. The second project is a new direction of research on systemic risk and mathematical analysis of the stability (or instability) of our banking system. The recent financial crisis has revealed a lack of understanding of the risk of cascade of defaults in banking networks. It is proposed to develop new models which will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network. This research, including its training component, is expected to contribute to the effort started by the regulators in the recent creation of the Office of Financial Research.
On the problem of uncertain volatility, the PI with his student B. Ren, has obtained an approximation of option prices when the volatility band (uncertainty) is small. The leading order term is simply the classical Black-Scholes price (with constant volatility) and the correction term is solution to a linear problem with source. The computation is therefore significantly reduced and it is shown that the approximation is efficient for not so small volatility band. On the problem of portfolio optimization under multiscale stochastic volatility and for general utility functions, expansions have been obtained around the constant volatility case (complete market) using the properties of the risk-tolerance function (PI and collaborators R. Sircar and T. Zariphopoulou). This work has opened several directions of research on optimization in finance which are now part of the PI's ongoing research. On systemic risk, the PI has explored several models of coupled diffusions in order to model and study the effect of inter-bank borrowing and lending on the stability of the system. The PI and his post doctoral fellow T. Ichiba studied a system squared-root processes coupled through their drifts parametrized by lending and borrowing preferences. The PI and his graduate student L.-H. Sun were able to identify a systemic event in a mean field model and compute its probability using large deviation type estimates. In the work with R. Carmona and L.-H. Sun, a game aspect has been introduced and Nash equilibria have been computed explicitly for a finite players game. Banks are borrowing and lending to a central bank and they try to minimize their cost along with an incentive from the regualtor to increase liquidity. The model shows that under equilibrium the central bank is acting as a clearing house and liquidity has been increased. In terms of broader impact, the PI has co-edited the Handbook on Systemic Risk published in 2013 by Cambridge University Press. It contains 35 articles looking at systemic risk from many points of view and will serve as a tool for regulators (at the newly established Office of Financial Research for instance) and researchers on stability of our banking system. The reseach has been published in journals, and talks have been delivered in conferences, seminars, and summer schools. Graduate students have been trained and a postdoctoral fellow was mentored.