Satellite meeting to AMAM 03

**Friday 14 and Saturday 15 February, 2003**

**University of Nice-Sophia
Antipolis**

**Venue: Laboratory of Mathematics**
**Jean-Alexandre
Dieudonné, UMR CNRS 6621**
**Parc
Valrose**

**Organizers : Francine and Marc Diener**

**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**

**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

Abstract:

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.

Abstract::

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.

Bruno DUPIRE
(consultant):

"Stochastic Volatility Models"

Abstract:

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":

Abstract:

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"

Abstract:

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"

Abstract:

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

Abstract:

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)

Abstract:

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

Abstract:

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.

**Registration form:**

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