The Mathematics of the New Financial Systems

The workshop will address the recent developments in the mathematics and the practical management of risk emanating from recent trends in the commodity markets, portfolio investment, and shocks to the financial system. Many problems arising from the analysis of commodities and energy markets, systemic risk of financial networks, modern portfolio management demand the development of new mathematical tools. These lead to challenges at the intersection of stochastic control and dynamic games, stochastic analysis, convex analysis as well as computational methods.

Each day of the three-day workshop will be devoted to one of these three themes and comprise five talks, three by senior researchers and two by junior researchers. Evening/late afternoon panel discussions will also be organized to discuss timely issues and challenges.

Day 1. Risk management in the commodity markets
Day 2. Systemic risk
Day 3. Investment and portfolio theory

A goal of the workshop will be to foster interactions between academia and industry. To this end we propose to end each day by a panel moderated by one of the organizers and addressing with academics and practitioners some of the themes of the day.

Mathematical Challenges in the New Financial Systems

Commodities and energy markets have gained increase importance both in terms of regulatory concerns such as a way to control emissions, and as a growing arena for investment to diversify from flat or declining traditional assets. Understanding the increased financialization of commodities markets can lead to feedback models and challenges in nonlinear PDEs. Discerning the relative interplay between growing demand due to industrialization and demand due to financial speculation is a delicate inverse problem that requires sophisticated numerical schemes. Many of these markets are oligopolies, or governed by a few energy providers, and this leads to problems in dynamic game theory that require analysis of BSDEs or systems of PDE. There is also a strong connection to the theory of mean-field games.

The financial crisis of 2007–09 highlights the need to better understand the behavior of risk in interconnected financial systems. The crisis has brought into focus the networked nature of the financial world. Interconnections often make a system robust, but they can also act as conduits for risk. The analysis of interactions and the quantification of risk involves the study of typical and atypical behavior of large interacting stochastic systems via laws of large numbers and large deviations principles. These can lead to challenging PDE and stochastic PDE problems. The solution of these requires intricate numerical schemes, including Monte Carlo algorithms. The Monte Carlo treatment of rare events leads to questions of efficiency. Quantitative investment theory raises new problems of understanding the interplay between investor preferences and markets, for example increased risk-aversion in down markets. At the same time traditional theory predicates the existence of an artificial time horizon at which performance is measured, and mathematicians are working on removing this and constructing forward performance measures. Doing so involves study of nonlinear PDE such as the fast-reaction equation, and some challenging SPDEs. The workshop will also cover latest developments in Stochastic Portfolio Theory that analyzes the collective behavior of large markets.

The overall mathematical theme is that whether we look at commodities markets, or systemic risk, or investment theory, we are dealing with complex stochastic systems whose control impacts the dynamics in non-trivial ways. The applications with the finance context are diverse, but the underlying mathematical challenges have a lot in common and we expect participants will be interested in all three days whatever their particular area of application may be.