José A. García-Rodríguez

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.

Simulation of shallow-water systems using graphics processing units

This paper addresses the speedup of the numerical solution of shallow-water systems in 2D domains by using modern graphics processing units (GPUs). A first order well-balanced finite volume numerical scheme for 2D shallow-water systems is considered. The potential data parallelism of this method is identified and the scheme is efficiently implemented on GPUs for one-layer shallow-water systems. Numerical experiments performed on several GPUs show the high efficiency of the GPU solver in comparison with a highly optimized implementation of a CPU solver.

Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of Roes Q-scheme are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [Vazquez-Cendon ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Comp Phys 1999;148:497-526; Garcia-Navarro P, Vazquez-Cendon ME. On numerical treatment of the source terms in the shallow water equations. Comput Fluids 2000;29(8):17-45; Castro MJ, Macias J, Pares C. A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. Math Model Numer An 2001;35(1):107-27]. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data and some tests on friction sensitivity are also presented.

Original article: Static and dynamic SABR stochastic volatility models: Calibration and option pricing using GPUs

For the calibration of the parameters in static and dynamic SABR stochastic volatility models, we propose the application of the GPU technology to the Simulated Annealing global optimization algorithm and to the Monte Carlo simulation. This calibration has been performed for EURO STOXX 50 index and EUR/USD exchange rate with an asymptotic formula for volatility or Monte Carlo simulation. Moreover, in the dynamic model we propose an original more general expression for the functional parameters, specially well suited for the EUR/USD exchange rate case. Numerical results illustrate the expected behavior of both SABR models and the accuracy of the calibration. In terms of computational time, when the asymptotic formula for volatility is used the speedup with respect to CPU computation is around 200 with one GPU. Furthermore, GPU technology allows the use of Monte Carlo simulation for calibration purposes, the computational time with CPU being prohibitive.

SABR/LIBOR market models: Pricing and calibration for some interest rate derivatives

Abstract In order to overcome the drawbacks of assuming deterministic volatility coefficients in the standard LIBOR market models to capture volatility smiles and skews in real markets, several extensions of LIBOR models to incorporate stochastic volatilities have been proposed. The efficient calibration to market data of these more complex models becomes a relevant target in practice. The main objective of the present work is to efficiently calibrate some recent SABR/LIBOR market models to real market prices of caplets and swaptions. For the calibration we propose a parallelized version of the simulated annealing algorithm for multi-GPUs. The numerical results clearly illustrate the advantages of using the proposed multi-GPUs tools when applied to real market data and popular SABR/LIBOR models.

Numerical methods to solve PDE models for pricing business companies in different regimes and implementation in GPUs

In this paper we propose appropriate numerical methods for companies valuation models proposed in [2]. Moreover, small modifications of these models allow to price the debt of the company and obtain the credit spread. The models are formulated in terms of final-boundary value problems associated to Kolmogorov type equations, which in some cases include an additional unilateral constraint on the solution. We also analyze the required boundary conditions so that the final-boundary value problem is well posed. This allows us to remove one of the unnecessary boundary conditions proposed in [2]. The numerical methods are mainly based on the use of characteristics (also known as semilagrangian) schemes in the direction without diffusion combined with implicit second order finite differences schemes in the direction where diffusion is present in the equation. This particular choice of numerical methods allows to develop an original parallelization strategy, which results to be specially highly efficient when using GPUs technologies.

GPU parallel implementation for asset-liability management in insurance companies

Abstract In this work we present a stochastic asset liability management (ALM) model for a life insurance company together with its numerical simulation, based in a Monte Carlo balance sheet projection, and we carry out its efficient parallel computation using graphics processing units (GPUs) hardware. The liabilities of the company consist of a portfolio comprising with-profit life insurance policies, that evolve according to the policyholder saving account, surrender and biometric models. On the asset side, we mainly consider bonds, equity and cash, so that appropriate stochastic models are considered for their evolution. We consider some innovations with respect to literature in the modeling of the surrenders of the policyholders. Another important innovative aspect comes from the implementation of ALM in the new high performance computing architectures provided by GPUs technology. Numerical results illustrate the high speed up of the calculus by using GPUs and the coherence of the computations (asset evolution, default probabilities and so on).

Parallel two-phase methods for global optimization on GPU

Abstract Developing general global optimization algorithms is a difficult task, specially for functions with a huge number of local minima in high dimensions. Stochastic metaheuristic algorithms can provide the only alternative for the solution of such problems since they are aimed at guaranteeing global optimality. However, the main drawback of these algorithms is that they require a large number of function evaluations in order to skip/discard local optima, thus exhibiting a low convergence order and, as a result, a high computational cost. Furthermore, the situation can become even worse with the increase of dimension. Usually the number of local minima highly increases, as well as the computational cost of the function evaluation, thus increasing the difficulty for covering the whole search space. On the other hand, deterministic local optimization methods exhibit faster convergence rates, requiring a lower number of functions evaluations and therefore involving a lower computational cost, although they can get stuck into local minima. A way to obtain faster global optimization algorithms is to mix local and global methods in order to benefit from higher convergence rates of local ones, while retaining the global approximation properties. Another way to speedup global optimization algorithms comes from the use of efficient parallel hardware architectures. Nowadays, a good alternative is to take advantage of graphics processing units (GPUs), which are massively parallel processors and have become quite accessible cheap alternative for high performance computing. In this work a parallel implementation on GPUs of some hybrid two-phase optimization methods, that combine the metaheuristic Simulated Annealing algorithm for finding a global minimum, with different local optimization methods, namely a conjugate gradient algorithm and a version of Nelder–Mead method, is presented. The performance of parallelized versions of the above hybrid methods are analyzed for a set of well known test problems. Results show that GPUs represent an efficient alternative for the parallel implementation of two-phase global optimization methods.

Speedup of Calibration and Pricing with SABR Models: From Equities to Interest Rates Derivatives

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.

Chapter 25 – A parallel 2D finite volume scheme for solving the bilayer shallow-water system: Modellization of water exchange at the Strait of Gibraltar

Publisher Summary This chapter presents a parallel 2D finite volume scheme for solving the bilayer shallow-water system, with a focus on modellization of water exchange at the Strait of Gibraltar. In the Strait of Gibraltar, which connects the Atlantic Ocean with the Mediterranean Sea, two layers of different waters can be distinguished: (1) the colder and less saline Atlantic water flowing at surface and penetrating into the Mediterranean, and (2) the deeper, denser Mediterranean water flowing into the Atlantic. In this chapter, a 2D two-layer shallow water system is presented together with an explicit edge-based finite volume numerical scheme. Due to the size of the domain, a parallel implementation based on object oriented MPI (OOMPI) is performed. The domain decomposition is carried out by using Chaco as a mesh partitioner integrated in the code. A special treatment of the open boundaries is implemented to reduce the communication among processors. The numerical model is applied to the simulation of the flow trough the Strait of Gibraltar with real bathimetric and coastline data. The measurements of the performance of the parallel implementation are also discussed.