Friday 14 and Saturday 15 February, 2003
University of Nice-Sophia Antipolis
Venue: Laboratory of Mathematics
Jean-Alexandre Dieudonné, UMR CNRS 6621
Organizers : Francine and Marc Diener
How to get to the meeting
Invited speakers: M. Avellaneda, A. d'Aspremont, R. Cont, B. Dupire, N. El Karoui, H. Föllmer, C. Martini, B. Oksendal, G. Pages, C. Rogers, M. Verleysen
Program (11/02 version)
Available titles and abstracts:
Marco AVELLANEDA ( Courant Institute of Mathematical Sciences, NYU ):
Weighted monte carlo and steepest descent methods for multi-asset equity derivatives: theory and practice
This talk describes the method of Weigted Monte Carlo (WMC), or Maximum-Entropy Monte Carlo, in the context of multi-dimensional models for
equity derivatives. We show that WMC represents a simple solution for calibrating Monte Carlo simulations to option markets in situations where
there are multiple underlying instruments. We prove rigorously that, under mild regularity assumptions, WMC gives rise to a numerically feasible
numerical algorithm for computing the minimal martingale measure (least-relative-entropy martingale measure) consistent with options markets
on the underlying stocks. Finally, we compare the predictions of WMC for pricing index options with the recently-derived Steepest-Descent
Approximation (RISK, Oct 2002), in which ensemble-averaging over large groups of assets is replaced by evaluation over the most likely price
configuration. The talk provides extensive evidence of the performance of the methods (both in- and out-of-sample) in real market conditions.
Alexandre d ASPREMONT (Stanford University):
Symmetric cone programming and applications to calibration of multivariate models.
Rama CONT (CNRS - Ecole Polytechnique):
Model calibration and model uncertainty.
Recent work on "model calibration" focuses on developing computational methods for "marking to market" option pricing models using observed market prices of
benchmark options. While these observations enable to identify a pricing model in some theoretical settings with complete observations, in practice the market
information is insufficient to identify a pricing model, leading to ill-posedness and model uncertainty.
In this talk we propose a approach to the model calibration problem which takes into account its ill-posed nature and describe a computational method which
enables to obtain from a set of market prices a family of pricing models and use them to quantify model risk and incompleteness of information. Our approach
has a natural link with recent theoretical work on model uncertainty and robust risk measures.
"Stochastic Volatility Models"
We present a review of existing stochastic volatility models, show different ways to obtain forward equations in various settings, look at the quantities we can/cannot lock,
derive arbitrage relationships for the implied volatility dynamics, study the behavior of several calibrated models and examine on the mplied volatility market model.
Nicole EL KAROUI (Ecole Polytechnique):
"Entropic choice in Finance":
Entropic selection of "optimal implied distribution" in calibration problems (Avellaneda, Rubinstein) or choice of the best portfolio with respect to exponential
utility are connected problems via convex duality. Foellmer and Schied used similar criterium as example of convex risk measure. In incomplete market, this framework
may be used to give new insights for pricing and hedging derivative in incomplete market. The new pricing rule is associated with an entropic convex risk measure; the
optimal hedging portfolio may be characterized (Fritelli, Rouge, NEK, Musiela-Zariphopoulo, Delbaen & alii). More recently, the problem of the optimal design of derivatives
and risk transfer in incomplete market was solved in this framework, when the agents have acces to the same financial market. If it is not the case, the optimal solution
may be characterized via quadratic BSDE in Brownian markets.
Hans FÖLLMER (Humbold University, Berlin):
"Preferences in the face of Uncertainty and the Quantification of Financial Risk"
We present joint work with Alexander Schied on the structure of convex risk measures and some connections to the theory of robust preferences on a space of uncertain pay-offs.
Claude MARTINI (Kore Business Software and INRIA):
"Minimum relative-entropy calibration: theory, algorithms and numerical experiments"
(joint work with Pierre Cohort and Steven Farcy, Kore Business Software)
Bernt ØKSENDAL (Oslo University):
"Some recent applications of Malliavin calculus to finance"
In this talk we will briefly discuss how Malliavin calculus can be used to study some insider trading problems and partial observation portfolio problems in finance:
(i) By an insider we mean a person who has access to more information than the information that can be obtained by observing the prices on
the market. For example, an insider may have information about the future values of a certain stock. Insider trading is illegal in most
countries and it is important to be able to detect it, if it should occur. Central questions are: How much extra can an insider gain compared to an honest trader? How much different is the optimal portfolio of an insider compared to the optimal portfolio of an honest trader? We will give partial answers to these questions by setting up a general stochastic analysis model for insider trading. The model involves the forward integral, the Skorohod integral and the Malliavin calculus.Then we apply a similar model to study
a partial observation optimal portfolio problem for an honest trader in a market influenced by insider traders.
Gilles PAGES (Université Pierre et Marie Curie, Paris):
Pricing and hedging multi-dimensional American options using optimal quantization
We present here the optimal quantization method for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number
of conditional expectations by projection of the diffusion on optimal grids designed to minimize the (square mean) projection error. An algorithm to compute such
grids is described. We provide results concerning the approximation orders with respect to the regularity of the pay-off function and the global size of the grids.
We show how to derive some higher order schemes based on these grids. Numerical tests are performed in dimensions 2, 4, 6, 10 with American style exchange
options. They show that theoretical orders are probably pessimistic.
Chris ROGERS (Cambridge University):
The squared-Ornstein-Uhlenbeck market (with John Aquilina)
Why does a share have value? How is that value determined? These are basic questions which get assumed away in the Black-Scholes world, but to which economists
have perfectly satisfactory answers, in terms of a general equilibrium of a dynamic market. General equilibrium is both easy (to explain and understand) and difficult
(to illustrate with explicit examples). This talk (which is mainly pedagogical in aim) analyses a simple example where many of the things we are interested in (such as the equilibrium prices of bonds, shares, and more general financial instruments) can be solved in closed form (in terms of generalised hypergeometric functions). Some
rudimentary fitting to yield curve data suggests that the resulting models could be of practical value.
Michel VERLEYSEN (Université
Catholique de Louvain):
Neural networks in financial applications
Neural networks are nonlinear models adapted to learning from examples; they are widely used in applications where few or no assumptions can be made about the process to model, so that "black-bow" modelling is the
only alternative. Neural networks include a wide variety of models, from regression to data analysis and representation.
In this talk, we will introduce two largely used neural network models, the Radial-Basis Function Networks (RBFN) and Kohonen's Self-Organizing Maps (SOM). We will show two examples of their application to financial data: the forecasting of financial time series with RBFN, and the classification of investment funds with SOM. We will insist on the potentialities of the neural network models, but also on their limitations and the precautions that must be taken to evaluate their performances in an objective way.