October 15-16, at the University of Marne-la-Vallée

Auditorium, ground floor of Copernic building, cf. access map
(see also
Plans d'accès à l'UMLV ,
Plan de la Cité Descartes )

We organize a two days workwhop on the applications of partial
differential equations to finance.

The registration is
free but mandatory. The lunch will be offered if you register before
October 8

(see the list of
participants).

The program is the following (see also the abstracts at the end of the text):

Monday 15, October

9h30-10h00: Welcoming, coffee

10h00-10h55: Henri
Berestycki (EHESS),

...................."The implied volatility of long options"

11h00-11h55: Erik
Ekström (Uppsala University),

...................."Convexity Theory for the Term Structure Equation" (slides)

12h00-13h30: lunch at the ENPC

13h30-14h25: Huyen Pham (Paris 7
University),

...................."Impulse control problem on finite horizon with execution delay"(slides)

14h30-15h25: Cyril
Imbert (Paris 9 University),

...................."Dirichlet problem and Hölder regularity for non-local fully non-linear
elliptic equations" (slides)

15h30-16h00: Break

16h00-16h55: Espen
R. Jakobsen (NTNU),

...................."Monotone approximation schemes for integro-PDEs arising in optimal control theory"(slides)

17h00-17h55: Christoph
Schwab (ETH Zurich),

...................."Numerical Derivative Pricing in Non-BS Markets" (slides)

Tuesday 16, October

9h30-10h00: Welcoming, coffee

10h00-10h55: Johan
Tysk (Uppsala University),

...................."Feynman-Kac formulas for Black-Scholes type
operators" (reference paper)

11h00-11h55: Yves
Achdou (Paris 7 University),

...................."Remarks on the calibration of Lévy process with
American options" (slides)

12h00-13h30: lunch at the ENPC

13h30-14h25: Sergio
Polidoro (Bologna University),

...................."Kolmogorov equations related to Asian options" (paper) (slides)

14h30-15h25: Andrea
Pascucci (Bologna University),

...................."Obstacle and optimal stopping problems for
American Asian options" (slides)

15h30-16h00: Break

16h00-16h55:
Nizar
Touzi (CMAP, Ecole Polytechnique),

...................."Probabilistic numerical methods: from American options to nonlinear PDEs"(slides)

17h00-17h25: Teitur Arnarson (KTH),

...................."Free boundary regularity close to initial state and applications to finance"(slides)

17h30-17h55: Moeiz
Rouis (CMAP, Ecole Polytechnique),

...................."Estimating exponential Lévy models from option prices via Tikhonov regularization"

Organizers: Henri Berestycki (EHESS), Damien Lamberton (UMLV), Régis Monneau (ENPC).

This meeting is supported by the "Université de Marne-la-Vallée",
the "Ecole Nationale des Ponts et Chaussées",

the "Ecole des Hautes Etudes en
Sciences Sociales" and the ACI NIM 2003-83 "EDP et finance".

Address : | Université de Marne-la-Vallée | |

Département de mathématiques
5, boulevard Descartes Champs-sur-Marne F-77454 MARNE LA VALLEE CEDEX 2 |
||

Phone : | secretary: Mireille Morvan 33 (0)1 60 95 75 20 | |

Email : | damien.lamberton@univ-mlv.fr | monneau@cermics.enpc.fr |

List of abstracts

Henri Berestycki (Ecole des Hautes Etudes
en Sciences Sociales (EHESS), Centre d'Analyse et de Mathématiques Sociales
(CAMS))

Titre : The implied volatility of long options

Abstract : TBA

Erik Ekström (Uppsala University)

Title : Convexity Theory for the Term Structure Equation

Abstract :

In this joint work with Johan Tysk, we study convexity
and monotonicity properties for prices of
bonds and bond options when the short rate is modeled by a diffusion
process. We provide conditions under which convexity of the price in
the current short rate is guaranteed. Under these conditions the
price is decreasing in the drift and increasing in the volatility
of the short rate. We also study convexity properties of the
logarithm of the price.

Huyen Pham(Paris 7 University)

Title : Impulse control problem on finite horizon with execution delay

Abstract :

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one.
This is motivated by financial applications in the trading of illiquid assets such as hedge funds.
We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This easily implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn
as original arguments in the proofs for the boundary conditions and uniqueness results. Finally, we give some numerical experiments illustrating the impact of execution delay on trading strategies and on option pricing. This is based on a joint work with Benjamin Bruder.

Cyril Imbert (Paris 9 University)

Title : Dirichlet problem and Hölder regularity for non-local fully non-linear
elliptic equations

Abstract :

This is a joint work with Guy Barles (Tours) and Emmanuel Chasseigne
(Tours). We consider second order elliptic equations that can be
degenerate, non-linear and can contain singular integral terms.
The most important example is the Hamilton-Jacobi-Bellman equation
satisfied by the value function associated with a stochastic control
of jump processes. We investigate the Dirichlet problem and the
Hölder regularity of the solution. In particular, we exhibit structure condition on the singular measure that ensures that the
solution coincides with the Dirichlet datum at the boundary. We also explain how to get existence and uniqueness of such a solution.

Espen R. Jakobsen (NTNU)

Title : Monotone approximation schemes for integro-PDEs arising in optimal control theory.

Abstract :

I will discuss approximations of the dynamic programming equations (here integro-PDEs) in optimal control of jump-diffusion-processes. An example of such a control problem is the optimal portfolio problem in Finance.
Two classes of monotone approximations will be considered: Finite difference methods and finite element like control schemes. These schemes are of low order but robust, since as opposed to high order methods, they "always" converge to the correct solution. The main focus in this talk will be to provide rigorous error bounds.
The talk is based on joint papers with Biswas, Camilli, Karlsen, and La Chioma.

Christoph Schwab (ETH Zurich)

Title : Numerical Derivative Pricing in Non-BS Markets

Abstract :

We report on deterministic solution methods for Kolmogoroff equations.
Admissible processes are strong Markoff processes,
possibly nonstationary, of jump-diffusion and pure jump type,
including in particular Levy and additive processes.
Multivariate models with copula models for the dependence in the marginals' jump structure are allowed.
Our approach is based on stabilized Galerkin discretization of the process'
infinitesimal generator resp. its Dirichlet form in a wavelet basis.
The methods allow to analyze single period and multiperiod contracts of
european, american or exotic style, in single or multiple
periods and on single underlyings or on baskets in a unified fashion.
Numerical analysis in the domains of Dirichlet forms of the price processes is briefly addressed.
Examples include American and exotic contracts on Levy copula dependence
models, single or multiscale stochastic volatility models of BNS and coGARCH type.
Joint work of the CMQF group in the Seminar for Applied Mathematics, ETH Zurich documented in the

References:

E.W. Farkas, N. Reich and C. Schwab:
Anisotropic stable Lévy copula processes -- analytical and numerical
aspects, (to appear in Mathematical Models and Methods in the Applied Sciences 2007),
Report 2006-08 Seminar for Applied Mathematics, ETH Zürich,
http://www.sam.math.ethz.ch/reports/2006/08

M. Wilhelm and Ch. Winter:
Finite Element Valuation of American Style Swing Options
Report 2006-07 Seminar for Applied Mathematics, ETH Zürich,
http://www.sam.math.ethz.ch/reports/2006/07
(in review).

N. Hilber, C. Schwab and Ch. Winter
Variational Sensitivity Analysis of Parametric Markovian Market Models
(in preparation).

Johan Tysk (Uppsala Unisersity)

Title : Feynman-Kac formulas for Black-Scholes type operators

Abstract :

There are many references showing that a classical solution
to the Black-Scholes equation is a stochastic solution. However,
it is the converse of this theorem that is most relevant in
applications, and the converse is also more mathematically
interesting. In this talk we establish such a converse. We find
a Feynman-Kac-type theorem showing that the stochastic representation
yields a classical solution to the corresponding Black-Scholes
equation with appropriate boundary conditions under very general
conditions on the coefficients. We also study the pricing equation
in the presence of bubbles, ie when the price process is a strict
local martingale. In this case there is an infinite dimensional space of
classical solutions. These results are obtained jointly
with Svante Janson and Erik Ekström, respectively.

Yves Achdou (University of Paris VII)

Title : Remarks on the calibration of Lévy process with American options.

Abstract :

We consider the calibration of a Lévy process with American options. The
price of an American vanilla option as a function of
the maturity and the strike satisfies a forward in time linear
complementarity problem involving a partial integro-differential operator.
It leads to a variational inequality in a suitable
weighted Sobolev space. Calibrating the Lévy process amounts to solving an
inverse problem where the state variable satisfies the previously mentioned
variational inequality. We consider a regularized least square method.
After studying the variational inequality carefully, we find necessary
optimality conditions for the least square problem.
In the second part of the talk, we discuss a completely different topic, also
related to American options, i.e. mesh adaptivity for parabolic obstacle
problems.

Sergio Polidoro (Bologna University)

Title : Kolmogorov equations related to Asian options

Abstract :

I present a survey of results concerning an evolution hypoelliptic equation related to path-dependent options. Specifically, I will discuss about local regularity, Harnack inequality and Cauchy problem.

Andrea Pascucci (Bologna University)

Title : Obstacle and optimal stopping problems for American Asian options

Abstract :

In this talk I consider a quite general diffusion model, possibly correspondent to a
degenerate PDE, that includes Asian options and path dependent volatility models as particular
cases. Recent results are presented concerning the existence
and uniqueness of strong solutions to the free boundary and optimal stopping problems
related to the pricing of American contingent claims.

Nizar Touzi (CMAP, Ecole Polytechnique)

Title : Probabilistic numerical methods: from American options to for nonlinear PDEs

Abstract : TBA

Teitur Arnarson (KTH)

Title : Free boundary regularity close to initial state and applications to finance

Abstract:

The choice whether to hold or exercise an American option in finance is determined by a free boundary occurring in the obstacle problem solved by the option pricing function. It is, however, hard to calculate this free boundary numerically close to the option expiry (or initial state for the time reversed problem). The problem is well studied and good results are known in the basic one-dimensional Black-Scholes setting. We present a method for determining the free boundary regularity with less precision but in a more general, higher-dimensional, non-linear setting.

Moeiz
Rouis (CMAP, Ecole Polytechnique)

Title : Estimating exponential Lévy models from option prices via Tikhonov regularization

Abstract :

In the class of exponential Lévy models, the
dynamics of an asset is described by its diffusion coefficient
$\sigma\geq 0$ and its Lévy measure $\nu$, a positive (possibly
infinite) measure on $R\setminus\{0\}$ verifying
$$\int\min(1,x^2) \nu(dx)<\infty,\qquad \qquad\int
\nu(dx)e^x<\infty.$$
Option prices can then be shown to solve a partial
integro-differential equation (PIDE). The model
calibration problem in this context concerns the identification of
the parameters $(\sigma,\nu)$ of the PIDE from a finite number of
option prices. We propose a method which yields a stable solution
for this ill posed inverse problem, using a Tikhonov
regularization method plus a suitable parametrization of Lévy
measure. We solve the resulting optimization problem by a
gradient-based method where the gradient is computed by solving an
adjoint PIDE with an explicit-implicit finite difference method.
We discuss theoretical aspects of the regularization
procedure and test the performance of the algorithm on simulated
data and market prices of options.
This is a joint work with Rama CONT.

References:

Cont, R. and Tankov, P., Financial Modelling wih Jump Processes,
Chapman & all, CRC Press, (2004).

Cont, R. and Voltchkova , E., Integro-differential equations for
option prices in exponential Lévy models,
Finance and Stochastics, Volume 9, Number 3 (2005), pp. 299--325.

Cont, R. and Voltchkova , E., Finite difference methods for option
pricing in jump diffusion and exponential L\'evy models,
SIAM Journal on Numerical Analysis, Volume 43, Number 4 (2005), pp. 1596--1626.

A-M. Matache, T. von Petersdorff and C. Schwab:
Fast deterministic pricing of options on Lévy driven assets,
Mathematical Modelling and Numerical Analysis Vol. 38 No. 1 (2004) 37--72.

web address:

http://cermics.enpc.fr/~monneau/edpmethodsinfinance.html