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We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes $(X, Y )$. Here an exogenous component $Y$ (Markov by itself) is time-consuming to simulate, while the endogenous component $X$ (jointly Markov with $Y$) is quick to simulate given $Y$, but is responsible for most of the variance of the simulated payoff. To address the related variance issue, we introduce a conditionally independent, hierarchical simulation scheme, where several paths of $X$ are simulated for each simulated path of $Y$. We analyze the statistical convergence of the regression learning scheme based on such block-dependent data. We derive heuristics on the number of paths of $Y$ and, for each of them, of $X$, that should be simulated. The resulting algorithm is implemented on a graphics processing unit (GPU) combining Python/CUDA and learning with PyTorch. A CVA case study with a nested Monte Carlo benchmark shows that the hierarchical simulation technique is key to the success of the learning approach., Comment: This article has been accepted for publication in Mathematical Finance, published by Wiley

International audience; In this paper, we want to promote the influence of randomized arithmetic on the leaks during a code execution. When somebody wants to extract some specific information from these leaks, one can observe different emanations of the device like power consumption. These leaks mostly come from the variations of the Hamming distances of the successive states of the system. This phenomenon is particularly critical for cryptographic devices. Our work evaluates the resilience of randomized moduli in Residue Number System (RNS) against Correlation Power Analysis (CPA), Differential Power Analysis (DPA). Our analysis is illustrated through the evaluation of scalar multiplication on an elliptic curve using the Montgomery Powering Ladder (MPL) algorithm which protects from Simple Power Analysis (SPA). We also propose an evaluation based on the Maximum Likelihood Estimator (MLE), which crosses the information of the whole state vector, instead of analysing only the current state like with CPA or DPA. Furthermore, MLE gives better performance and smooths the results allowing a better evaluation of the behaviour of the leakage. Our experimental evaluation suggests that the number of observations, needed to perform exploitable information leakage, is proportional to the number of possible RNS bases.

This paper is about using the existing Monte Carlo approach for pricing European and American contracts on a state-of-the-art graphics processing unit GPU architecture. First, we adapt on a cluster of GPUs two different suitable paradigms of parallelizing random number generators, which were developed for CPU clusters. Because in financial applications, we request results within seconds of simulation, the sufficiently large computations should be implemented on a cluster of machines. Thus, we make the European contract comparison between CPUs and GPUs using from one up to 16 nodes of a CPU/GPU cluster. We show that using GPUs for European contracts reduces the execution time by~40 and diminishes the energy consumed by~50 during the simulation. In the second set of experiments, we investigate the benefits of using GPUs' parallelization for pricing American options that require solving an optimal stopping problem and which we implement using the Longstaff and Schwartz regression method. The speedup result obtained for American options varies between two and 10 according to the number of generated paths, the dimensions, and the time discretization. Copyright © 2012 John Wiley & Sons, Ltd.

This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially-observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be expressed in terms of the solutions and the current values of a Markov process adapted to the observation filtration. We shall illustrate the application of our results using the Longstaff-Schwartz algorithm for multiple optimal stopping times in a geometric Brownian motion stock price model with drift uncertainty., Comment: 29 pages; 3 figures

International audience; The aim of this paper is the efficient use of CPU and GPU clusters for a general path-dependent exotic European pricing, and their comparison in terms of speed and energy consumption. To reach our goal, we propose a parallel random number generator which is well suited to the parallelization paradigm, then, we implement a multidimensional Asian contract as a benchmark using g++/OpenMP/OpenMPI on CPUs and CUDA-nvcc/OpenMPI on GPUs. Finally, we give the detailed results of the two architectures for different size problems using 1-16 GPUs and 1-256 dual-core CPUs.

This paper is devoted to the sensitivity study of the European option prices according to the correlation parameters when dealing with the multi-asset Heston model. When the Feller condition is not fulfilled, the CIR flow regularity is needed to prove the differentiability of the price according to the correlation. In the bidimensional case when the Feller condition is satisfied, the regularity of the volatility according to the correlation allows us to establish an asymptotic expression of the derivative of the price with respect to the correlation. This approximation provides the monotony for the exchange options then heuristically for spread option prices at short maturities. We also obtain this monotony for some restrictive choices of the products $\{ \eta_i \rho_i \}_{i=1,2}$ and $\{ \eta_i \sqrt{1-\rho_i^2} \}_{i=1,2}$ where $\eta_i $ is the volatility of the volatility and $\rho_i$ is the asset/volatility correlation coefficient. Then, we explain how to extend the overall study to options written on more than two assets and on models that are derived from Heston model, like the double Heston model. We conclude by a large number of simulations that comfort the theoretical results.