Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes

[EN] This paper deals with the numerical analysis of PIDE option pricing models with CGMY process using double discretization schemes. This approach assumes weaker hypotheses of the solution on the numerical boundary domain than other relevant papers. Positivity, stability, and consistency are studied. An explicit scheme is proposed after a suitable change of variables. Advantages of the proposed schemes are illustrated with appropriate examples.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and by the Spanish M.E.Y.C. Grant DPI2010-20891-C02-01.Company Rossi, R.; Jódar Sánchez, LA.; El-Fakharany, M. (2013). Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes. Abstract and Applied Analysis. 2013:1-12. https://doi.org/10.1155/2013/517480S1122013Kou, S. 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Numerical methods for Lévy processes

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

An implementation of the Wiener-Hopf factorization into finite difference methods for option pricing under Lévy processes

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.On considère le problème d'évaluation d'options pour une large classe de processus de Lévy. On propose une approche numérique basée sur une approximation par différences finies pour l'équation de Black-Scholes généralisée. Le but est d'introduire la factorisation de Wiener-Hopf dans la méthode de différences finies pour l'évaluation d'options dans des modèles de Lévy avec sauts. La méthode s'applique au cas des options barrières et les options américaines. Le problème d'évaluation se réduit à une suite de systèmes linéaires algébriques avec matrice dense de Toeplitz, pour laquelle la méthode de factorisation de Wiener-Hopf est appliquée. Nous donnons une interprétation probabiliste basée sur la théorie des distributions infiniment divisibles des opérateurs de Laurent de l'identité de factorisation correspondante. Notre algorithme a la même complexité que le shéma explicite avec matrice tridiagonale, mais est plus précis. Nous illustrons l'avantage de cette méthode en termes de précision et convergence, sur des expériences numériques

Fractional diffusion models and option pricing in jump models

Mestrado em Mathematical FinanceO problema de valorização de derivados tem sido o foco da investigação em Matemática Financeira desde a sua conceção. Mais recentemente, a literatura tem-se focado por exemplo em modelos que assumem que as dinâmicas do preço do ativo subjacente são governadas por um processo de Lévy (por vezes chamado um processo com saltos). Este tipo de modelo admite a possibilidade de eventos extremos (saltos), que não são devidamente capturados por modelos clássicos do tipo Black-Scholes, alicerçados no movimento Browniano. Foi também demonstrado ao longo da última década que se as dinâmicas do preço do ativo subjacente seguem certos processos de Lévy, tais como o CGMY , o FMLS e o KoBoL, os preços das opções satisfazem uma equação diferencial parcial fracionária. Nesta dissertação, iremos mostrar que se as dinâmicas do ativo subjacente seguem o denominado Processo Estável Temperado Generalizado, que admite como caso particular os suprareferidos processos CGMY e KoBoL, então os preços das opções satisfazem igualmente uma equação diferencial parcial fracionária. Além disso, iremos implementar um método simples de diferenças finitas para resolver numericamente a equação deduzida, e valorizar opções do tipo europeu.The problem of pricing financial derivatives has been the focal point of research within the field of Mathematical Finance since its conception. In recent years, one of the main areas of focus within the literature has been on models which assume that the dynamics of the price of the underlying asset are governed by a Lévy process (sometimes referred to as a jump process). This type of model admits the possibility of extreme events (jumps), which are not captured by classical Black-Scholes type models based on the Brownian motion. Over the last decades, the literature has further shown that if the dynamics of the price of the underlying is governed by certain Lévy processes, such as the CGMY , the FMLS and the KoBoL, the price processes of European-style options satisfy a variety of fractional partial differential equations (FPDEs). In this dissertation, we will show that if the underlying price dynamic follows a Generalized Tempered Stable process, which admits as particular cases the aforementioned CGMY and KoBoL processes, prices of options satisfy an FPDE of the same type. Further, we will implement a simple finite difference scheme to solve the FPDE numerically to price European-type options.info:eu-repo/semantics/publishedVersio