The Workshop on Probability and Statistics in Finance, sponsored by the IMS special interest group on “Finance: Probability and Statistics” (FPS) and the Statistics Department of Columbia University, took place on June 23, 2011 at Columbia University. After a continental breakfast and welcoming remarks, four speakers gave talks, punctuated by coffee breaks and lunch. A panel discussion followed their well attended talks. The workshop was organized by Xin Guo of U.C. Berkeley, Tze Leung Lai of Stanford University, and Philip Protter of Columbia University.
For more information on the recently formed special interest group FPS, including the forthcoming workshop at Berkeley/Stanford in June 2012 and the free membership sign-up, visit its URL http://lists.imstat.org/mailman/listinfo/fgfps
The Four Invited Talks
Robert Jarrow of the Cornell Business School in Ithaca, NY, kicked off the talks by returning to the classic topic of market eﬃciency, deﬁned long ago by Fama in 1970. In joint work with Martin Larsson he gave it a modern interpretation involving nested families of ﬁltrations. In this way, he provided a connection between the classic idea of market eﬃciency dependent on equilibrium theory and asset pricing models, with that of the more modern concept of the absence of arbitrage. In doing this, he was able to circumvent the classical conundrum of the “curse of the joint hypothesis.”
The second speaker was Arturo Kohatsu-Higa of Kyoto, Japan. His talk concerns the use of Lévy processes in the modeling of risky asset price processes; in particular, one often replaces Brownian motion as the driving noise in a stochastic diﬀerential equation with a Lévy process, and in particular a Lévy process that has “inﬁnite activity,” which is to say enough small jumps to present serious challenges to an eﬀective analysis. Financial quantities of interest often involve the expectation of functions, or functionals, of terminal values or paths of risky asset prices. When formulas are (almost always) not available for these quantities, one typically resorts to Monte Carlo methods, and this involves simulation issues. How one performs these simulations in general is a diﬃcult open problem, and Professor Kohatsu-Higa presented a summary of recent work in the area, focusing on his own signiﬁcant contributions, in particular those joint with S. Ortiz and P. Tankov.
After lunch the third speaker was Yingying Fan of USC. She spoke on a topic of much recent interest, involving the statistical analysis of path properties of risky asset prices observed in the market. Building on the celebrated test of Ait-Sahalia and Jacod for the determination of whether of not a risky asset price process is continuous or has jumps, she proposed a new test that has the advantage of reducing the variance, and in addition makes a contribution to the location of a jump.
The fourth and ﬁnal talk before the panel discussion was by Nizar Touzi of the Ecole Polytechnique, of France. Motivated by a desire to approximate no-arbitrage bounds on the prices of exotic options, given the implied volatility curve of a given maturity, Professor Touzi presented an extension of the Monge-Kantorovitch optimal transportation problem. In his framework the mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and diﬀusion coeﬃcients of the continuous semimartingale. The optimal transportation problem then minimizes the cost among all continuous semimartingales with given initial and terminal distributions.
The Panel Discussion
After a break following the last talk, a one hour panel discussion took place, with panelists Peter Carr of Morgan Stanley, Ronnie Sircar of Princeton, Steve Kou of Columbia, and Bala Rajaratnan of Stanford. Dr. Carr spoke of the possibilities of more collaboration between industry and academia and gave some concrete ideas; Professor Sircar spoke primarily about the role the new IMS group might play, and gave an illuminating presentation of the current state of the analogous group in SIAM; Professor Kou spoke of the recent mortgage scandals and how spatial statistics have a role to play in their study; and Professor Rajaratnam spoke of statistical issues related to the recent hot topic of high frequency trading data.
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