Forward deterministic pricing of options using Gaussian radial basis functions

Abstract

The price of a fixed-term option is the expected value of the payoff at the time of maturity. When not analytically available, the option price is computed using stochastic or deterministic numerical methods. The most common approach when using deterministic methods is to solve a backward partial differential equation (PDE) such as the Black-Scholes equation for the option value. The problem can alternatively be formulated based on a forward PDE for the probability of the asset value at the time of maturity. This enables simultaneous pricing of several contracts with different payoffs written on the same underlying asset. The main drawback is that the initial condition is a (non-smooth) Dirac function. We show that by using an analytical expansion of the solution for the first part of the time interval, and applying a high-order accurate radial basis function (RBF) approximation in space, we can derive a competitive forward pricing method. We evaluate the proposed method on European call options and barrier options, and show that even for just one payoff it is more efficient than solving the corresponding backward PDE.

An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

“…Nonetheless, limited by the efficiency and data processing ability of these algorithms, many challenges still arise in practical applications. In recent years, the radial basis function (RBF) method has attracted much attention because of its simple format and high accuracy [15][16][17][18][19], and it has gradually developed into a significant numerical method in the scientific computing domain [20][21][22][23]. However, for large-scale problems such as image processing, the RBF method incurs excessive computational costs due to the generation of a dense matrix [23][24][25][26], and the large condition number of this matrix can also causes calculation instability [27,28].…”

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

“…Hence, in order to approximate the cumulative first passage time distribution in an interval, we have to solve this equation several times. Although FPE is a well-known PDE which is widely used in different fields of science and engineering such as chemistry [75], physics [76], protein folding [77], and option pricing [78,79] and various numerical algorithms such as variational iteration method [80], Adomian decomposition method [81], local discontinuous Galerkin method [82], spectral and pseudo-spectral methods [83,84], and collocation methods based on orthogonal and non-orthogonal basis functions [85,86,87] have been developed for solving this class of PDEs. But the considered equation has two main challenges 1) time-dependent boundaries and 2) singularity at the initial condition.…”

Numerical Approximation of the First-Passage Time Distribution of Time-Varying Diffusion Decision Models: A Mesh-Free Approach

“…Hence, in order to approximate the cumulative first passage time distribution in an interval, we have to solve this equation several times. Although FPE is a well-known PDE which is widely used in different fields of science and engineering such as chemistry [75], physics [76], protein folding [77], and option pricing [78,79] and various numerical algorithms such as variational iteration method [80], Adomian decomposition method [81], local discontinuous Galerkin method [82], spectral and pseudo-spectral methods [83,84], and collocation methods based on orthogonal and non-orthogonal basis functions [85,86,87] have been developed for solving this class of PDEs. But the considered equation has two main challenges 1) time-dependent boundaries and 2) singularity at the initial condition.…”