Joint AMS-IMS-SIAM Conference Targets the Mathematics of Finance

November 1, 2003


Participants in the first AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences to be devoted to financial mathematics. Held in Snowbird in June, the conference presented advances in the rapidly evolving field and provided a forum for discussion of the state of financial mathematics in academia and in industry.

René Carmona, George Yin, and Qing Zhang

The field of financial mathematics continues to expand. Today's researchers make use of a wide spectrum of techniques, extending far beyond those of traditional applied mathematics. Stochastic calculus, dynamic programming, and partial differential equations have become indispensable tools in a field that not so long ago relied on ". . . a collection of anecdotes, rules of thumb, and shuffling of accounting data" (see [1]). Some of the research results have had a major impact on the global economy. At the same time, new applications are requiring and stimulating exciting new theoretical discoveries. The development of the field has created a high level of demand in the finance industry for mathematics graduates at both the master's and doctoral levels, resulting in the introduction of this topic into the curriculum of mathematical sciences departments at many universities.

Such rapid evolution in a field creates the need for new forms of communication and networking among researchers. Accordingly, financial mathematics---for the first time---was the theme of one of this year's AMS-IMS-SIAM Joint Summer Research Conferences in the Mathematical Sciences. Held in Snowbird, Utah, June 22-27, the conference attracted more than sixty participants---from Australia, Canada, England, France, Germany, Hong Kong, Italy, Japan, Poland, and the United States.

Beyond Black-Scholes Models

A distinctive feature of the conference was its interdisciplinarity, with disciplines represented including stochastic control, probability and stochastic processes, finance, economics, scientific computing, and statistics. A number of prominent figures and leading experts in the mathematics of finance and financial engineering were in attendance; along with established researchers, new PhDs and graduate students were well represented.

The members of the organizing committee were Wendell H. Fleming (Brown University), Jean-Pierre Fouque (North Carolina State University), George Papanicolaou (Stanford University), Bozenna Pasik-Duncan (University of Kansas), Stanley R. Pliska (University of Illinois at Chicago), Ronnie Sircar (Princeton University), and co-chairs George Yin (Wayne State University) and Qing Zhang (University of Georgia). The conference was supported in part by the National Science Foundation.

The scientific program was built around eight plenary talks, by Robert Elliott (University of Calgary), Wendell Fleming, Thaleia Zariphopoulou (University of Texas, Austin), Steve Shreve (Carnegie Mellon University), Stanley Pliska, Chris Rogers (Cambridge University), Wolfgang Runggaldier (University of Padova), and Bozenna Pasik-Duncan. Rounding out the program were 34 invited talks, a poster session, and a panel discussion on research and education.

Speakers presented widely ranging problems, models, and results involving modeling techniques, estimation, optimization, control, risk assessment and management, contingent claim pricing, dynamic hedging, and financial derivative design. The valuation of contingent claims remains the centerpiece of modern financial theory; its key components are financial market modeling and dynamic hedging. Black-Scholes models have been used for decades to characterize movements of asset prices; researchers now recognize the limited utility of these models, which assume deterministic rates of return and volatility, and ignore many aspects of financial markets, such as transaction costs, liquidity, and tax issues.

In the past few years, various approaches, including stochastic volatility, jump diffusions, and hybrid market models, have emerged to modify and generalize the Black-Scholes models. Optimal portfolio management, which originated in Robert Merton's pioneering work and continues to play an important role in the theory of finance, uses a stochastic control approach. The objective is to allocate financial assets dynamically among risky and fixed-income investments, with the goal of maximizing expected terminal utility. A closed-form solution is possible only for the simplest models. Typically, optimal investment and consumption control policies must be found by solving a partial differential equation of the
Hamilton-Jacobi-Bellman type. Nonlinearities make numerical implementation difficult, however, and efficient schemes are sorely needed. Adding to the difficulty of this effort are the issues of mathematical model selection and choice of utility function.

Financial risk management has attracted increasing attention in recent years. Devastating events like the Long-Term Capital Management default and the Enron bankruptcy are a clear indication of an urgent need for further research on corporate debt instruments, and on financial derivatives as a possible way to hedge these risks. As a result, one of the emerging research topics is the study of credit risk.

Because of the large number of exceptionally high-quality talks at the Snowbird conference, the program was quite dense. Even so, participants had ample time for discussion and exchange of ideas. A number of people used this opportunity to further existing collaborations; some new and potential collaborations also got under way.

Focus on the Job Market

One of the highlights of the meeting was a panel discussion on the state of financial mathematics in academia and in industry. The structure of the panel was standard: Each panelist gave a short presentation before the chair turned to the audience for questions.

Discussions emphasized the need for academic researchers to understand practical issues when modeling financial markets under uncertainty. To serve current and future needs, it became clear that effort must be directed to bridging the gap between theory and applications, and to facilitating communication between academic and industry researchers.

Nicole El Karoui (Ecole Polytechnique, Paris) and Wolfgang Runggaldier reported on the French and Italian experiences, respectively. Chris Rogers challenged participants to seek interactions with colleagues in the economics departments at their universities. Expressing concerns that may not be shared by some members of the financial community, either in academia or in industry, Marek Musiela (director of fixed-income research, BNP/Paribas, London) reminded the audience of the divergence of research directions in academia and the needs of the financial industry.

A good part of the panel discussion focused on the growing appetite among students for practical training in financial applications, and the steady growth of the number of professional programs in financial mathematics and financial engineering. René Carmona, who chaired the panel, emphasized the dangers of allowing the goal of training students for a particular career to overshadow the important responsibility of educating and preparing students for a wide variety of research challenges, both in industry and in academia. Our responsibility, he said, is to see beyond the short-term fluctuations of marketplace demands. Carmona also identified the unrealistic expectations of students seeking these professional degrees as one of the major issues in the dismal job market experienced by recent graduates. Demand for PhD-level quants continues to be strong, but a delicate question remains: Have we already saturated the financial job market with narrowly trained graduates at the master's level?

Musiela echoed this concern, stressing the need for a broad and solid education. Addressing the difficulties created by the current state of the economy, Tyrone Duncan (University of Kansas) suggested the insurance industry as a possible alternative to investment banking, which is failing to absorb current master's-level graduates of financial mathematics programs.

The extremely lively discussion and debate went on for two hours, stopped only by the dinner bell; clearly, a lot of open questions remain.

The Chinese describe an exceptionally rare event as a "June snow." As we learned, this description loses its meaning in Snowbird, where heavy snowfall occurred on two days during the meeting---not at all uncommon, we were told. Despite the unseasonably cool weather (even for Snowbird), participants enjoyed the snow in June and the natural beauty of the region. Along with stimulating talks and discussions, many found time to do some hiking. Participants also found time for the Wednesday evening banquet, where after-dinner speaker Stanley Pliska provided historical notes on the mathematics of finance from Bachelier to the 1970s.

Overall, the conference provided excellent and timely opportunities for researchers and practitioners alike. Today's financial research predicts tomorrow's financial practice (see [1]). In building on our inheritance from earlier researchers, and in ushering in the future, we see unprecedented challenges and opportunities. It is conceivable that this conference will have significant impacts on new directions of research in the mathematics of finance. In any case, it can be anticipated that it will continue to stimulate research in many areas of the mathematical sciences for years to come.

References
[1] J. Case, Robert Merton delivers Block community lecture, SIAM News, 31-7 (1998).

The proceedings of the conference will be published as part of the American Mathematical Society's Contemporary Mathematics series.

René Carmona (rcarmona@princeton.edu) is a professor in the Department of Operations Research and Financial Engineering at Princeton University. George Yin (gyin@math.wayne.edu) is a professor of mathematics at Wayne State University, and Qing Zhang (qingz@math.uga.edu) is a professor of mathematics at the University of Georgia, Athens.


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