A framework for adaptive Monte-Carlo procedures

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives

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

Convenient Multiple Directions of Stratification

This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along non-orthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.Comment: 21 pages, 11 table

Using Premia and Nsp for Constructing a Risk Management Benchmark for Testing Parallel Architecture

International audienceFinancial institutions have massive computations to carry out overnight which are very demanding in terms of the consumed CPU. The challenge is to price many different products on a cluster-like architecture. We have used the Premia software to valuate the financial derivatives. In this work, we explain how Premia can be embedded into Nsp, a scientific software like Matlab, to provide a powerful tool to valuate a whole portfolio. Finally, we have integrated an MPI toolbox into Nsp to enable to use Premia to solve a bunch of pricing problems on a cluster. This unified framework can then be used to test different parallel architectures

Using Premia and Nsp for Constructing a Risk Management Benchmark for Testing Parallel Architecture

Financial institutions have massive computations to carry out overnight which are very demanding in terms of the consumed CPU. The challenge is to price many different products on a cluster-like architecture. We have used the Premia software to valuate the financial derivatives. In this work, we explain how Premia can be embedded into Nsp, a scientific software like Matlab, to provide a powerful tool to valuate a whole portfolio. Finally, we have integrated an MPI toolbox into Nsp to enable to use Premia to solve a bunch of pricing problems on a cluster. This unified framework can then be used to test different parallel architectures.Premia;Mpi;Nsp

El asiento de avería y el origen de la compañía privilegiada en España

El presente artículo analiza cómo la autonomía patrimonial del capital destinado a la organización de las flotas con América acabó adquiriendo la forma de una verdadera compañía privilegiada de comercio, antecedente directo de la actual sociedad anónima, cuando la Corona dejó por entero a los particulares la administración de la avería a través del sistema de asientos. Los paralelismos de los asientos con las compañías llegaron a ser tan grandes que los funcionarios a quienes el conde-duque de Olivares encomendó el estudio de las compañías extranjeras, con objeto de introducirlas en España, vieron en los asientos una verdadera compañía privilegiada.This paper analyses how the progressive autonomy of the capital used to arrange the treasure fleets from Seville to America derived in a real chartered company, a precedent of today’s joint-stock company, when the Crown finally privatised it through the asientos de avería. The parallelism among the asientos and the companies were so noticeable that the public officers appointed by the Count-Duke of Olivares to study the foreign companies couldn’t detect any difference between both institutions.Publicad

Une application de la théorie des excursions à une diffusion réfléchie dégénérée

International audienceWe consider the R²-valued reflected stochastic differential equation associated with the degenerated differential operator (1/2)∂²/∂x²+x∂/∂y. We study the excursion phenomenon around the point (0, 0). We are able to identify the law of the inverse of the local time as a stable subordinator with order 1/4 and to give for it an occupation time formula. These facts enable us to study the behavior of the law of the process near the point (0, 0) and to refine, in our case, results obtained by Krée (1985) by means of analytical methods.Nous considérons l'équation différentielle stochastique réfléchie à valeur dans R² associée à l'opérateur différentiel dégénéré (1/2)∂²/∂x²+x∂/∂y. On étudie alors le phénomène d'excursion autour du point origine. En particulier, on peut identifier la loi de l'inverse du temps local comme un subordinateur stable d'ordre 1/4 et en donner une formule du type temps d'occupation. Ceci permet alors d'étudier le comportement de la loi du processus au voisinage du point origine pour préciser, dans ce cas particulier, des résultats obtenus par Krée (1985) grâce à des méthodes analytiques