American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f(St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine

European Options Sensitivity with Respect to the Correlation for Multidimensional Heston Models

International audienceThis 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

Calcul parallèle pour les problèmes linéaires, non-linéaires et linéaires inverses en finance

Handling multidimensional parabolic linear, nonlinear and linear inverse problems is the main objective of this work. It is the multidimensional word that makes virtually inevitable the use of simulation methods based on Monte Carlo. This word also makes necessary the use of parallel architectures. Indeed, the problems dealing with a large number of assets are major resources consumers, and only parallelization is able to reduce their execution times. Consequently, the first goal of our work is to propose "appropriate" random number generators to parallel and massively parallel architecture implemented on CPUs/GPUs cluster. We quantify the speedup and the energy consumption of the parallel execution of a European pricing. The second objective is to reformulate the nonlinear problem of pricing American options in order to get the same parallelization gains as those obtained for linear problems. In addition to its parallelization suitability, the proposed method based on Malliavin calculus has other practical advantages. Continuing with parallel algorithms, the last point of this work is dedicated to the uniqueness of the solution of some linear inverse problems in finance. This theoretical study enables the use of simple methods based on Monte CarloDe ce fait, le premier objectif de notre travail consiste à proposer des générateurs de nombres aléatoires appropriés pour des architectures parallèles et massivement parallèles de clusters de CPUs/GPUs. Nous testerons le gain en temps de calcul et l'énergie consommée lors de l'implémentation du cas linéaire du pricing européen. Le deuxième objectif est de reformuler le problème non-linéaire du pricing américain pour que l'on puisse avoir des gains de parallélisation semblables à ceux obtenus pour les problèmes linéaires. La méthode proposée fondée sur le calcul de Malliavin est aussi plus avantageuse du point de vue du praticien au delà même de l'intérêt intrinsèque lié à la possibilité d'une bonne parallélisation. Toujours dans l'objectif de proposer des algorithmes parallèles, le dernier point est l'étude de l'unicité de la solution de certains cas linéaires inverses en finance. Cette unicité aide en effet à avoir des algorithmes simples fondés sur Monte Carl

European Option Princing on a GPU Cluster

Presentation given at 2nd JTE "GPGPU", University Paris-6, 4 décembre 2008, Paris, France.The aim of this presentation is a comparison in terms of speed and energy consumption between CPU and GPU clusters using financial application as a benchmark. After a fast introduction on the field of application we will give details on the hardware and software architectures used. Then we will introduce a multiparadigm parallel algorithm, mixing coarse and fine grained parallelism, and its implementations using MPI+OpenMP on CPU cluster and MPI+CUDA on GPU cluster. Finally, some computing and energetic performances on different clusters will be compared

Evaluation of Resilience of randomized RNS implementation

Randomized moduli in Residue Number System (RNS) generate effectively large noise and make quite difficult to attack a secret key $K$ from only few observations of Hamming distances $H=(H_0, ..., H_{d-1})$ that result from the changes on the state variable. Since Hamming distances have gaussian distribution and most of the statistic tests, like NIST\u27s ones, evaluate discrete and uniform distribution, we choose to use side-channel attacks as a tool in order to evaluate randomisation of Hamming distances . This paper analyses the resilience against Correlation Power Analysis (CPA), Differential Power Analysis (DPA) when the cryptographic system is protected against Simple Power Analysis (SPA) by a Montgomery Powering Ladder (MPL). While both analysis use only information on the current state, DPA Square crosses the information of all the states. We emphasize that DPA Square performs better than DPA and CPA and we show that the number of observations $S$ needed to perform an attack increases with respect to the number of moduli $n$. For Elliptic Curves Cryptography (ECC) and using a Monte Carlo simulation, we conjecture that $S = O((2n)!/(n!)^2)$

Parallel computing for linear, nonlinear and linear inverse problems in finance

De ce fait, le premier objectif de notre travail consiste à proposer des générateurs de nombres aléatoires appropriés pour des architectures parallèles et massivement parallèles de clusters de CPUs/GPUs. Nous testerons le gain en temps de calcul et l'énergie consommée lors de l'implémentation du cas linéaire du pricing européen. Le deuxième objectif est de reformuler le problème non-linéaire du pricing américain pour que l'on puisse avoir des gains de parallélisation semblables à ceux obtenus pour les problèmes linéaires. La méthode proposée fondée sur le calcul de Malliavin est aussi plus avantageuse du point de vue du praticien au delà même de l'intérêt intrinsèque lié à la possibilité d'une bonne parallélisation. Toujours dans l'objectif de proposer des algorithmes parallèles, le dernier point est l'étude de l'unicité de la solution de certains cas linéaires inverses en finance. Cette unicité aide en effet à avoir des algorithmes simples fondés sur Monte CarloHandling multidimensional parabolic linear, nonlinear and linear inverse problems is the main objective of this work. It is the multidimensional word that makes virtually inevitable the use of simulation methods based on Monte Carlo. This word also makes necessary the use of parallel architectures. Indeed, the problems dealing with a large number of assets are major resources consumers, and only parallelization is able to reduce their execution times. Consequently, the first goal of our work is to propose "appropriate" random number generators to parallel and massively parallel architecture implemented on CPUs/GPUs cluster. We quantify the speedup and the energy consumption of the parallel execution of a European pricing. The second objective is to reformulate the nonlinear problem of pricing American options in order to get the same parallelization gains as those obtained for linear problems. In addition to its parallelization suitability, the proposed method based on Malliavin calculus has other practical advantages. Continuing with parallel algorithms, the last point of this work is dedicated to the uniqueness of the solution of some linear inverse problems in finance. This theoretical study enables the use of simple methods based on Monte Carl

AMERICAN OPTIONS BASED ON MALLIAVIN CALCULUS AND NONPARAMETRIC VARIANCE REDUCTION METHODS

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f(St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine

European Options Sensitivity with Respect to the Correlation for Multidimensional Heston Models

International audienceThis 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