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

A new sequential algorithm for L2-approximation and application to Monte-Carlo integration

We design a new stochastic algorithm (called SALT) that sequentially approximates a given function in L2 w.r.t. a probability measure, using a finite sample of the distribution. By increasing the sets of approximating functions and the simulation effort, we compute a L2-approximation with higher and higher accuracy. The simulation effort is tuned in a robust way that ensures the convergence under rather general conditions. Then, we apply SALT to build efficient control variates for accurate numerical integration. Examples and numerical experiments support the mathematical analysis

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.

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

Deep Learning algorithms for solving high dimensional nonlinear Backward Stochastic Differential Equations

We study deep learning-based schemes for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). First we show how to improve the performances of the proposed scheme in [W. E and J. Han and A. Jentzen, Commun. Math. Stat., 5 (2017), pp.349-380] regarding computational time by using a single neural network architecture instead of the stacked deep neural networks. Furthermore, those schemes can be stuck in poor local minima or diverges, especially for a complex solution structure and longer terminal time. To solve this problem, we investigate to reformulate the problem by including local losses and exploit the Long Short Term Memory (LSTM) networks which are a type of recurrent neural networks (RNN). Finally, in order to study numerical convergence and thus illustrate the improved performances with the proposed methods, we provide numerical results for several 100-dimensional nonlinear BSDEs including nonlinear pricing problems in finance.Comment: 21 pages, 5 figures, 16 table