Financial Mathematics at ICIAM

October 21, 2007

Nicole El Karoui of the Ecole Polytechnique in Paris gave an invited talk in mathematical finance: �New Developments in Dynamic Pricing and Hedging and Measuring Financial Risk.�

Mike Giles and Ronnie Sircar

Financial mathematics was well represented at ICIAM '07, with one plenary lecture, by Nicole El Karoui, and nine well-attended minisymposia---sessions every day of the meeting, morning and afternoon. This followed the large inaugural meeting of the SIAM Activity Group on Financial Mathematics & Engineering in Boston last year, held in conjunction with the SIAM Annual Meeting. Financial mathematics was of course covered at ICIAM '99 and '03, in Edinburgh and Sydney. But the strong presence at ICIAM '07 reflects a field that is now coming of age. The sessions in Zurich had two main themes: the growing subfield of computational finance, and modeling issues in credit risk and stock market volatility, among other applications.

The major challenges in computational finance arise not from difficult geometries, as in many physical problems, but from the need for rapid calculation of an expectation, or the solution of its associated Kolmogorov partial differential equation. Efficiency is at the forefront, because models are re-estimated as new market data arrives and calibration (or "marking to market") embeds the expectation/PDE calculation in an iterative solution to an inverse problem. Practitioners require that this be done as the data moves, so that the models can be used to quote consistent prices on other instruments. Other twists include (i) the need for sensitivities (or "Greeks") of prices with respect to input parameters--in other words, partial derivatives of the expectation/PDE solution; and (ii) in some cases, the dimensionality of the problem. Lively debates often arose during the minisymposia: How high is high-dimensional? Can numerical PDE methods really tackle realistic problems in finance, such as valuation of options on baskets of stocks, where the dimension might be in the hundreds?

Nicole El Karoui of the Ecole Polytechnique in Paris gave an excellent overview lecture to an audience of upward of 200 people. As the co-founder of one of the first and most successful master's programs in financial mathematics, El Karoui has been responsible for the reputation and international demand for French quants in the finance industry. (She was the subject of a profile in The Wall Street Journal in March 2006.)

Another excellent plenary talk, by Andrew Stuart of Warwick University, though not explicitly on finance, was highly relevant to financial and, in particular, econometric applications. Stuart discussed MCMC (Markov Chain Monte Carlo) algorithms for parameter estimation with stochastic systems, and
new mathematical developments. Heinz Engl, who received the ICIAM Pioneer Prize in Zurich for his work on inverse problems, was a speaker in a finance session on inverse methods for parameter estimation.

Organizers of the nine minisymposia in financial mathematics were Abdul Khaliq and Qin Sheng ("Advances in Modeling" and "Advances in Computation"); Jorge Zubelli, Marco Avellaneda, and Bernd Hofmann ("Inverse Problems in Financial Modelling"); Olivier Pironneau and Jari Toivanen ("Numerical Methods in Option Pricing"); and Christoph Schwab and Rama Cont ("Financial Modeling with Jump Processes").

Numerous sessions on related topics complemented those explicitly devoted to financial mathematics. In a number of sessions on quasi-Monte Carlo methods for high-dimensional integrals, for example, researchers, among them outgoing ICIAM president Ian Sloan, were looking to finance for applications. Topics of other sessions with financial connections include stochastic differential equations, fractional calculus, robust optimization, large-scale optimization, game theory, environmental economics, wavelets, sparse grid methods, and, of particular interest to one of us, automatic differentiation tools, which can greatly simplify the efficient calculation of Greeks.

Finally, the industrial relevance of all this research was emphasised in an Industry Day session, "Risk Management in Financial and Energy Markets." Shortly after ICIAM, and in contrast to the tranquility of Zurich in the summer, financial markets were thrown into turmoil by the sub-prime mortgage crisis in the U.S. Some major commentators in the press were quick to pounce on mathematicians, referring to hedge funds that employ pattern recognition strategies and quickfire program trading to exploit "statistical arbitrage." Wizardry of this kind was not on display at ICIAM, but such crises in the past spurred responsible forward-looking pioneers in the industry to seek out advances in modeling and computation---for example, dynamic credit risk models (as discussed in the talks of Knut Solna and Philipp Schonbucher) and high-dimensional PDE methods (as discussed by Abdul Khaliq, Christoph Schwab, and others). There is every reason, then, to expect that the current crisis will again lead to the demand for more math, not less. ICIAM showcased an impressive range of advances in financial mathematics. What is left to be done? Everything.

Mike Giles is a professor in the Computing Laboratory at Oxford University. Ronnie Sircar is a professor in the Department of Operations Research & Financial Engineering at Princeton University.

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