Accelerating the calibration of stochastic volatility models

This paper compares the performance of three methods for pricing vanilla options in models with known characteristic function: (1) Direct integration, (2) Fast Fourier Transform (FFT), (3) Fractional FFT. The most important application of this comparison is the choice of the fastest method for the calibration of stochastic volatility models, e.g. Heston, Bates, Barndorff-Nielsen-Shephard models or Levy models with stochastic time. We show that using additional cache technique makes the calibration with the direct integration method at least seven times faster than the calibration with the fractional FFT method. --Stochastic Volatility Models,Calibration,Numerical Integration,Fast Fourier Transform

Accelerating the calibration of stochastic volatility models

This paper compares the performance of three methods for pricing vanilla options in models with known characteristic function: (1) Direct integration, (2) Fast Fourier Transform (FFT), (3) Fractional FFT. The most important application of this comparison is the choice of the fastest method for the calibration of stochastic volatility models, e.g. Heston, Bates, Barndor®-Nielsen-Shephard models or Levy models with stochastic time. We show that using additional cache technique makes the calibration with the direct integration method at least seven times faster than the calibration with the fractional FFT method.Stochastic Volatility Models; Calibration; Numerical Integration; Fast Fourier Transform

Accelerating Reconfigurable Financial Computing

This thesis proposes novel approaches to the design, optimisation, and management of reconfigurable computer accelerators for financial computing. There are three contributions. First, we propose novel reconfigurable designs for derivative pricing using both Monte-Carlo and quadrature methods. Such designs involve exploring techniques such as control variate optimisation for Monte-Carlo, and multi-dimensional analysis for quadrature methods. Significant speedups and energy savings are achieved using our Field-Programmable Gate Array (FPGA) designs over both Central Processing Unit (CPU) and Graphical Processing Unit (GPU) designs. Second, we propose a framework for distributing computing tasks on multi-accelerator heterogeneous clusters. In this framework, different computational devices including FPGAs, GPUs and CPUs work collaboratively on the same financial problem based on a dynamic scheduling policy. The trade-off in speed and in energy consumption of different accelerator allocations is investigated. Third, we propose a mixed precision methodology for optimising Monte-Carlo designs, and a reduced precision methodology for optimising quadrature designs. These methodologies enable us to optimise throughput of reconfigurable designs by using datapaths with minimised precision, while maintaining the same accuracy of the results as in the original designs

Characteristic functions in the Cheyette Interest Rate Model

We investigate the characteristic functions of multi-factor Cheyette Models and the application to the valuation of interest rate derivatives. The model dynamic can be classiffied as an affine-diffusion process implying an exponential structure of the characteristic function. The characteristic function is determined by a model specific system of ODEs, that can be solved explicitly for arbitrary Cheyette Models. The necessary transform inversion turns out to be numerically stable as a singularity can be removed. Thus the pricing methodology is reliable and we use it for the calibration of multi-factor Cheyette Models to caps. --Cheyette Model,Characteristic Function,Fourier Transform,Calibration of Multi-Factor Models

Boundary-safe PINNs extension: Application to non-linear parabolic PDEs in counterparty credit risk

[Abstract]: The goal of this work is to develop a novel strategy for the treatment of the boundary conditions for multi-dimension nonlinear parabolic PDEs. The proposed methodology allows to get rid of the heuristic choice of the weights for the different addends that appear in the loss function related to the training process. It is based on defining the losses associated to the boundaries by means of the PDEs that arise from substituting the related conditions into the model equation itself. The approach is applied to challenging problems appearing in quantitative finance, namely, in counterparty credit risk management. Further, automatic differentiation is employed to obtain accurate approximation of the partial derivatives, the so called Greeks, that are very relevant quantities in the field.Xunta de Galicia; ED431C 2018/33Xunta de Galicia; ED431G 2019/01A.L and J.A.G.R. acknowledge the support received by the Spanish MINECO under research project number PDI2019-108584RB-I00, and by the Xunta de Galicia, Spain under grant ED431C 2018/33. All the authors thank to the support received from the CITIC research center, funded by Xunta de Galicia and the European Union (European Regional Development Fund - Galicia Program, Spain ), by grant ED431G 2019/01

Optimal damping with hierarchical adaptive quadrature for efficient Fourier pricing of multi-asset options in Lévy models

Efficient pricing of multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods become competitive compared to alternative techniques because the integrand in the frequency space has often higher regularity than in the physical space. However, when designing a numerical quadrature method for most of these Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of the damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of the high dimensionality. To address these challenges, we propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of damping parameters based on a proposed heuristic optimization rule. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. Our extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some L'evy models demonstrates the advantages of adaptivity and our damping rule on the numerical complexity of the quadrature methods. Moreover, our approach achieves substantial computational gains compared to the Monte Carlo method

Calibrating Option Pricing Models with Heuristics

Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

[EN] In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P.El-Fakharany, M.; Company Rossi, R.; Jódar Sánchez, LA. (2016). Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes. Journal of Computational and Applied Mathematics. 296:739-752. https://doi.org/10.1016/j.cam.2015.10.027S73975229