Pricing foreign exchange options under stochastic volatility and interest rates using an RBF–FD method

Abstract

This paper proposes a numerical method for pricing foreign exchange (FX) options in a model which deals with stochastic interest rates and stochastic volatility of the FX rate. The model considers four stochastic drivers, each represented by an Itô's diffusion with time-dependent drift, and with a full matrix of correlations. It is known that prices of FX options in this model can be found by solving an associated backward partial differential equation (PDE). However, it contains non-affine terms, which makes its difficult to solve it analytically. Also, a standard approach of solving it numerically by using traditional finite-difference (FD) or finite elements (FE) methods suffers from the high computational burden. Therefore, in this paper a flavor of a localized radial basis functions (RBFs) method, RBF-FD, is developed which allows for a good accuracy at a relatively low computational cost. Results of numerical simulations are presented which demonstrate efficiency of such an approach in terms of both performance and accuracy for pricing FX options and computation of the associated Greeks.

Four-factor model of Quanto CDS with jumps-at-default and stochastic recovery

“…The proposed model together with the obtained results are new and represents the main contribution of this paper. Also, despite the numerical method has been already described in the previous papers of the authors (both joint and separate), its application to solving a system of the 4D PDEs derived in this paper is new (also new boundary conditions are used as compared with, e.g., Soleymani and Itkin (2019)). Finally, in this paper the method was parallelized to achieve the best efficiency.…”

“…Moreover, the RBF-FD scheme is able to tackle irregular geometries and scattered node layouts. A particular flavor of the RBF-FD method used in this paper is described in Soleymani and Ullah (2018); Soleymani and Itkin (2019). In this paper as the basis functions for the RBF method we use the Gaussian RBF φ(r i ) = e −ǫ 2 r 2 i , where r i = x − x i 2 denotes the Euclidean distance, and ǫ is the shape parameter.…”

“…( 22) is done first, along the spatial variables by using the RBF-FD methodology, and then the method-of-lines (MOL) is used to finally reach a system of Ordinary Differential Equations (ODEs). The RBF-FD weights for approximation of the first and second derivatives in the above PDEs are obtained as in Soleymani and Itkin (2019), and provide the second order of convergence.…”

“…Finally, for solving our systems of PDEs we use a different flavor of the Radial Basis Function (RBF) method 3 which is a combination of localized RBF and finite-difference methods (see Milovanovich and Shcherbakov (2017); Flyer et al (2016); Bayona et al (2017) and references therein), and is known in the literature as the RBF-FD method. More specifically, a flavor of the RBF-FD method used in this paper is described in Soleymani and Ullah (2018); Soleymani and Itkin (2019). Both the RBF-PUM and RBF-FD methods belong to a localized version of the classical RBF, and demonstrate similar features, such as high accuracy, sparsity of the differentiation matrices, meshfree nature and multi-dimensional extendability.…”

On the Statistical GARCH Model for Managing the Risk by Employing a Fat-Tailed Distribution in Finance

“…Accurate risk identification and the assignment of risk to the party make the stock-holders or financial managers to carefully step ahead and minimize the risk of losing the capital. This could improve quality, reduce delays and resolve disputes efficiently [11].…”