arXiv subjects

J. G. López-Salas

Publications and source records attributed to J. G. López-Salas.

14 recordsLinked to original sources

arXiv2024

Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA.

math.AP↗

arXiv2024

A second order finite volume IMEX Runge-Kutta scheme for two dimensional PDEs in finance

In this article we present a novel and general methodology for building second order finite volume implicit-explicit (IMEX) numerical schemes for solving two dimensional financial parabolic PDEs with mixed derivatives. In particular, applications to basket and Heston models are presented. The obtained numerical schemes have excellent properties and are able to overcome the well-documented difficulties related with numerical approximations in the financial literature. The methods achieve true second order convergence with non-regular initial conditions. Besides, the IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing very accurate and non-oscillatory approximations of the Greeks. Finally, in order to assess all the aforementioned good properties of the developed numerical schemes, we compute extremely accurate semi-analytic solutions using multi-dimensional Fourier cosine expansions. A novel technique to truncate the Fourier series for basket options is presented and it is efficiently implemented using multi-GPUs.

math.NA↗

arXiv2024

Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs

In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di\-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes.

math.NA↗

arXiv2024

Second order finite volume IMEX Runge-Kutta schemes for two dimensional parabolic PDEs in finance

We present a novel and general methodology for building second-order finite volume implicit-explicit Runge-Kutta numerical schemes for solving two-dimensional financial parabolic PDEs with mixed derivatives. The methods achieve second-order convergence even in the presence of non-regular initial conditions. The IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing accurate and non-oscillatory approximations of the Greeks.

math.NA↗

arXiv2024

IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing

The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order.

math.NA↗

arXiv2024

PDEs for pricing interest rate derivatives under the new generalized Forward Market Model (FMM)

In this article we derive partial differential equations (PDEs) for pricing interest rate derivatives under the generalized Forward Market Model (FMM) recently presented by A. Lyashenko and F. Mercurio in \cite{lyashenkoMercurio:Mar2019} to model the dynamics of the Risk Free Rates (RFRs) that are replacing the traditional IBOR rates in the financial industry. Moreover, for the numerical solution of the proposed PDEs formulation, we develop some adaptations of the finite differences methods developed in \cite{LopezPerezVazquez:sisc} that are very suitable to treat the presence of spatial mixed derivatives. This work is the first article in the literature where PDE methods are used to value RFR derivatives. Additionally, Monte Carlo-based methods will be designed and the results are compared with those obtained by the numerical solution of PDEs.

q-fin.PR↗

arXiv2024

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

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.

q-fin.PR↗

arXiv2024

PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan \cite{haganSABRLIBOR}, Mercurio \& Morini \cite{mercurioMorini} and Rebonato \cite{rebonatoWhite}. Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.

math.AP↗

arXiv2024

AMFR-W numerical methods for solving high dimensional SABR/LIBOR PDE models

In this work we mainly develop a new numerical methodology to solve a PDE model recently proposed in the literature for pricing interest rate derivatives. More precisely, we use high order in time AMFR-W methods, which belong to a class of W-methods based on Approximate Matrix Factorization (AMF) and are especially suitable in the presence of mixed spatial derivatives. High-order convergence in time allows larger time steps which combined with the splitting of the involved operators, highly reduces the computational time for a given accuracy. Moreover, the consideration of a large number of underlying forward rates makes the PDE problem high dimensional in space, so the use of AMFR-W methods with a sparse grids combination technique represents another innovative aspect, making AMFR-W more efficient than with full grids and opening the possibility of parallelization. Also the consideration of new homogeneous Neumann boundary conditions provides another original feature to avoid the difficulties associated to the presence of boundary layers when using Dirichlet ones, especially in advection-dominated regimes. These Neumann boundary conditions motivate the introduction of a modified combination technique to overcome a decrease in the accuracy of the standard combination technique.

math.NA↗

arXiv2024

An efficient implementation of parallel simulated annealing algorithm in GPUs

In this work we propose a highly optimized version of a simulated annealing (SA) algorithm adapted to the more recently developed Graphic Processor Units (GPUs). The programming has been carried out with CUDA toolkit, specially designed for Nvidia GPUs. For this purpose, efficient versions of SA have been first analyzed and adapted to GPUs. Thus, an appropriate sequential SA algorithm has been developed as a starting point. Next, a straightforward asynchronous parallel version has been implemented and then a specific and more efficient synchronous version has been developed. A wide appropriate benchmark to illustrate the performance properties of the implementation has been considered. Among all tests, a classical sample problem provided by the minimization of the normalized Schwefel function has been selected to compare the behavior of the sequential, asynchronous, and synchronous versions, the last one being more advantageous in terms of balance between convergence, accuracy, and computational cost. Also, the implementation of a hybrid method combining SA with a local minimizer method has been developed. Note that the generic feature of the SA algorithm allows its application in a wide set of real problems arising in a large variety of fields, such as biology, physics, engineering, finance, and industrial processes.

cs.DC↗

arXiv2024

Quasi-Regression Monte-Carlo scheme for semi-linear PDEs and BSDEs with large scale parallelization on GPUs

In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial differential equation (PDE) obtained through the well-known Feynman-Kac representation. For the sake of enriching the algorithm with high-order convergence a weighted approximation of the solution is computed and appropriate conditions on the parameters of the method are inferred. With the challenge of tackling problems in high dimensions we propose suitable projections of the solution and efficient parallelizations of the algorithm taking advantage of powerful many-core processors such as graphics processing units (GPUs).

math.NA↗

arXiv2024

Global optimization for data assimilation in landslide tsunamis models

The goal of this article is to make automatic data assimilation for a landslide tsunami model, given by the coupling between a non-hydrostatic multi-layer shallow-water and a Savage-Hutter granular landslide model for submarine avalanches. The coupled model is discretized using a positivity-preserving second-order path-conservative finite volume scheme. The data assimilation problem is posed in a global optimization framework and we develop and compare parallel metaheuristic stochastic global optimization algorithms, more precisely multi-path versions of the Simulated Annealing algorithm, with hybrid global optimization algorithms based on hybridizing Simulated Annealing with gradient local searchers, like L-BGFS-B.

physics.geo-ph↗

arXiv2024

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.

math.OC↗

arXiv2024

Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs

In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on many core processors such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization. In this way, we minimize the exposure to the memory requirements due to the storage of simulations. Indeed, we note the lower memory overhead of the method compared with previous works.

math.NA↗