PDE Methods in Finance 2007

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.