Simulation of BSDEs with jumps by Wiener Chaos Expansion

We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy

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

Error expansion for the discretization of Backward Stochastic Differential Equations

We study the error induced by the time discretization of a decoupled forward-backward stochastic differential equations $(X,Y,Z)$. The forward component $X$ is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme $X^N$ with $N$ time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors $(Y^N-Y,Z^N-Z)$ measured in the strong $L\_p$-sense ($p \geq 1$) are of order $N^{-1/2}$ (this generalizes the results by Zhang 2004). Secondly, an error expansion is derived: surprisingly, the first term is proportional to $X^N-X$ while residual terms are of order $N^{-1}$.Comment: 27 page

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.

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumitrescu-Labart (2014), which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples.Comment: arXiv admin note: text overlap with arXiv:1406.361

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps

Pricing Parisian options using Laplace transforms

International audienceIn this work, we propose to price Parisian options using Laplace transforms. Not only do we compute the Laplace transforms of all the different Parisian options, but we also explain how to invert them numerically. We prove the accuracy of the numerical inversion

A Parallel Algorithm for solving BSDEs

International audienceWe present a parallel algorithm for solving backward stochastic differential equations. We improve the algorithm proposed in Gobet Labart (2010), based on an adaptive Monte Carlo method with Picard's iterations, and propose a parallel version of it. We test our algorithm on linear and non linear drivers up to dimension 8 on a cluster of 312 CPUs. We obtained very encouraging speedups greater than 0.7

Simulation of BSDEs by Wiener Chaos Expansion

International audienceWe present an algorithm to solve BSDEs based on Wiener Chaos Expansion and Picard's iterations. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. We use the Malliavin derivative to compute $Z$. Concerning the error, we derive explicit bounds with respect to the number of chaos and the discretization time step. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy