La determinazione del fair value di opzioni su valuta impiegando funzioni a base radiale: un’applicazione al framework di pricing di Garman-Kohlhagen

Abstract

Garman – Kohlhagen framework, which is an extension of the most popular Black-Scholes-Merton model, is often used by financial institutions in order to price options with a currency as underlying. These pricing techniques have in common the definition of a partial differential equation used for the definition of the future value of the derivative, called Fundamental PDE. The financial instruments characterization depends on the derivative pay-off and it is realized through the specification of the initial conditions (IC) and the Dirichlet’s boundary conditions (BC). For standard contracts, called plain-vanilla derivatives, and for a few class of non-standard instruments, called exotic derivatives, this problem can be solved analytically reaching a theoretical fair value through a closed formula (CF) valuation, otherwise a numerical method must be used. Classical numerical integration schemes, which have been implemented for this purpose, are Finite Difference Method (FDM) and Finite Elements Method (FEM). In the last ten years, financial engineering has focused on an innovative methodology for option pricing which has its foundations on Radial Basis Functions (RBF). This paper aims to examine how this technique works in the financial field and how this method can be applied to the fair-value determination of vanilla and exotic Forex options.

Keywords

• radial basis functions (rbf)
• meshless method in quantitative finance
• currency option pricing
• partial differential equation (pde)
• garman – kohlhagen pricing framework