Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach

In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regression by a Wiener chaos expansion. Not only does it allow us to deal with a non Markovian setting, but it also breaks the bottleneck induced by the least square regression as the coefficients of the chaos expansion are given by scalar products on the L^2 space and can therefore be approximated by independent Monte Carlo computations. This key feature enables us to provide an embarrassingly parallel algorithm.Comment: The Journal of Computational Finance, Incisive Media, In pres

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationComment: 25 page

Local Indicators for Plurisubharmonic Functions

A notion of local indicator for a plurisubharmonic function is introduced. The indicator is a certain plurisubharmonic function in the unit polydisc, which controls the behavior of the considered function near a fixed point of its singularity, as well as the residual Monge-Ampere mass of the function at this point. A pluricomplex Green function with respect to given indicators is constructed and applied to the Dirichlet problem for the Monge- Ampere operator in the class of plurisubharmonic functions with isolated singularities.Comment: 18 pages, LaTeX; to appear in J. Math. Pures App

Long time behaviour of a stochastic nano particle

In this article, we are interested in the behaviour of a single ferromagnetic mono-domain particle submitted to an external field with a stochastic perturbation. This model is the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model. The main results presented in this article are the rate of convergence of the magnetization toward the unique stable equilibrium of the deterministic model. The second result is a sharp estimate of the hysteresis phenomenon induced by the stochastic perturbation (remember that with no perturbation, the magnetic moment remains constant)

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationbackward stochastic differential equations, parallel computing, Monte- Carlo methods, non linear PDE, American options, local volatility model.

Green vs. Lempert functions: a minimal example

The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.Comment: v3: proof of the upper estimate for the Green function added; accepted in Pacific Journal of Mathematic