Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts

Abstract

In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfully apply it to two types of multidimensional partial differential equations in finance: a two-asset European call basket option under the Black-Scholes-Merton model, and a European call option under the Heston model. We also show that the performance of the improved method is equally high when it comes to pricing American options. By studying convergence, computational performance, and conditioning of the discrete systems, we show the superiority of the introduced approaches over previously used versions of the RBF-FD method in financial applications.

A high order method for pricing of financial derivatives using Radial Basis Function generated Finite Differences

“…Moreover, our model problem has a fairly short time to maturity T = 0.2. For longer times to maturity, FD does not perform equally well compared to the RBF methods, see [14,13]. We also establish that the fourth order methods quickly become superior when it comes to CPU-time to reach a certain ∆u max .…”

“…In this paper, we follow [5], [1], and [13] and use PHSs as basis functions together with polynomials of degree p in the interpolation. With that approach, the polynomial degree (instead of the RBF) controls the rate of convergence, while the RBFs contribute to reduction of approximation errors and are necessary in order to have a stable approximation.…”

“…This time-stepping scheme requires the solution at two previous time steps, and we therefore employ Backward Euler (BDF1) for the first time step. It is convenient to have the same coefficient matrix in all time steps, so we use non-equidistant time steps as described in [10] and later used in e.g., [14,13]. This is accomplished by discretizing the time interval with M steps of length ∆t = t − t −1 , where = 1, .…”

“…which we use to set the size of the stencils to m = 5ν, following [5,1,13]. We are aiming for a fourth order scheme and use p = 4 and q = 5 which gives…”

“…Numerical methods for the PDE formulation include adaptive Finite Differences (FD) [15,12,11,16,25], high-order compact schemes [3,4], Alternating Direction Implicit (ADI) schemes [8,6], Radial Basis Function (RBF) approximation [17,10], Radial Basis Function Partition of Unity (RBF-PU) method [20,22,21], and Radial Basis Function generated Finite Differences (RBF-FD) [14,13]. In [24] and [26] several methods for pricing of options are implemented and evaluated.…”