Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

A numerical study of radial basis function based methods for option pricing under one dimension jump-diffusion model

The aim of this paper is to show how option prices in the Jump-diffusion model can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integro-differential equation (PIDE) in one-dimension for the Ameri- can put and the European vanilla call/put options on dividend-paying stocks in the Merton and Kou Jump-diffusion models. The radial basis function we select is the Cubic Spline. We also propose a simple numerical algorithm for finding a finite computational range of a global integral term in the PIDE so that the accuracy of approximation of the integral can be improved. Moreover, the solution functions of the PIDE are approximated explicitly by RBFs which have exact forms so we can easily compute the global intergal by any kind of numerical quadrature. Finally, we will also show numerically that our scheme is second order accurate in spatial variables in both American and European cases

Numerical algorithms for the valuation of installment options

Mestrado em Matemática FinanceiraInstallment options are financial derivatives in which part of the initial premium is paid up-front and the other part is paid discretely or continuously in installments during the option’s lifetime. This work deals with the numerical valuation of European installment options. Trough the study of the continuous case, we can show that numerical inversion of Laplace transform works well for computing the option value. In particular, we will investigate the De Hoog algorithm and compare it to other methods for the inverse Laplace transformation, namely Euler summation, Gaver-Stehfest and Kryzhnyi methods.Installment options são derivados financeiros cuja parte inicial do prémio é paga antecipadamente e a outra parte é dividida, discretamente ou continuamente, em parcelas durante o “tempo de vida” do contrato. Este trabalho estuda a valorização numérica de installment options do tipo Europeu. Estudando principalmente o caso contínuo podemos mostrar que a inversão numérica da transformada de Laplace é um bom método para calcular o valor da opção. Em particular, vamos investigar o algoritmo conhecido por De Hoog e compará-lo a outros métodos numéricos, sendo eles conhecidos por Euler summation, Gaver-Stehfest e método de Kryzhnyi

Fuzzy Transfer Pricing World: On the Analysis of Transfer Pricing with Fuzzy Logic Techniques

The arm’s length analysis of international transfer prices of multinational firms lacks sound methodological approach of the so-called function and risk analysis. In practice, such analyses are descriptive. Derived from Zadeh’s mathematical theory of fuzzy sets, this paper investigates a quantitative approach to identify the function and risk pattern of related parties of multinational companies. We illustrate our fuzzy logic approach with a simple case.

On the robustness of least-squares Monte Carlo (LSM) for pricing American derivatives

This paper analyses the robustness of Least-Squares Monte Carlo, a technique recently proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options provide evidence that a) this approach is very robust to the choice of different alternative polynomials and b) few basis functions are required. However, these conclusions are not reached when analyzing more complex derivatives.Least-Squares Monte Carlo, option pricing, American options

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model