Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied.[ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates.[CA] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates.El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917TESI

The influence of mineralogical composition and alkali reactivity for utilization of some Egyptian crushed granites as a concrete aggregate

Egyptian Eastern Desert is rich in many areas that contain granites masses throughout the geological era; some of them show good characteristics of the rock hardness, durability, density and mineralogy. This current research aims to utilize three main types of granite aggregates based on their mineralogical composition and Alkali reactivity with cement during concrete production. The studied granite aggregate can be also classified into red younger granite aggregate, white older granite aggregate and grey older granite aggregate. Evaluating these granite rocks as aggregate used in concrete mixture is interesting by produced three mixes using the three studied granite aggregate symbolized (Red GA), (White GA) and (Grey GA), tested mechanically to give a more detailed for the obtained results to be not restricted for only studied granite aggregate criteria but also to follow the actual reaction of this studied granite aggregate with cement.   It was obtained that all studied granite aggregates within acceptable limits of concrete aggregate by following Egyptian code (ECP-203) although their variation on its mineralogical composition. Some reflections produced from change in mineralogical composition between the three studied granite aggregates exhibited by relative regression in the average physico-mechanical values for both (White and Grey GA) than (Red GA). On the other hand, slight reactive for (Red GA) than others at the age of 28 day. In addition, all produced (Red GA), (White GA) and (Grey GA) mixes were acceptable mechanically with limits of (ECP-203) giving benefit for using all of the studied granite aggregate after their detailed study involving its mineralogical composition and alkali aggregate reactivity (AAR)

Numerical valuation of two-asset options under jump diffusion models using Gauss-Hermite quadrature

In this work a finite difference approach together with a bivariate Gauss–Hermite quadrature technique is developed for partial integro-differential equations related to option pricing problems on two underlying asset driven by jump-diffusion models. Firstly, the mixed derivative term is removed using a suitable transformation avoiding numerical drawbacks such as slow convergence and inaccuracy due to the appearance of spurious oscillations. Unlike the more traditional truncation approach we use 2D Gauss–Hermite quadrature with the additional advantage of saving computational cost. The explicit finite difference scheme becomes consistent, conditionally stable and positive. European and American option cases are treated. Numerical results are illustrated and analysed with experiments and comparisons with other well recognized methods.FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) Ministerio de Economía y Competitividad Spanish grant MTM2013-41765-

Target Detection in a Known Number of Intervals Based on Cooperative Search Technique

Finding hidden/lost targets in a broad region costs strenuous effort and takes a long time. From a practical view, it is convenient to analyze the available data to exclude some parts of the search region. This paper discusses the coordinated search technique of a one-dimensional problem with a search region consisting of several mutual intervals. In other words, if the lost target has a probability of existing in a bounded interval, then the successive bounded interval has a far-fetched probability. Moreover, the search domain is swept by two searchers moving in opposite directions, leading to three categories of target distribution truncations: commensurate, uneven, and symmetric. The truncated probability distributions are defined and applied based on the proposed classification to calculate the expected value of the elapsed time to find the hidden object. Furthermore, the optimization of the associated expected time values of various cases is investigated based on Newton's method. Several examples are presented to discuss the behavior of various distributions under each case of truncation. Also, the associated expected time values are calculated as their minimum values.Comment: 32 pages, 11 figure

Numerical valuation of two-asset options under jump diffusion models using Gauss-Hermite quadrature

In this work a finite difference approach together with a bivariate Gauss-Hermite quadrature technique is developed for partial-integro differential equations related to option pricing problems on two underlying asset driven by jump-diffusion models. Firstly, the mixed derivative term is removed using a suitable transformation avoiding numerical drawbacks such as slow convergence and inaccuracy due to the appearance of spurious oscillations. Unlike the more traditional truncation approach we use 2D Gauss-Hermite quadrature with the additional advantage of saving computational cost. The explicit finite difference scheme becomes consistent, conditionally stable and positive. European and American option cases are treated. Numerical results are illustrated and analyzed with experiments and comparisons with other well recognized methods.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economía y Competitividad Spanish grant MTM2013-41765-P

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

[EN] In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P.El-Fakharany, M.; Company Rossi, R.; Jódar Sánchez, LA. (2016). Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes. Journal of Computational and Applied Mathematics. 296:739-752. https://doi.org/10.1016/j.cam.2015.10.027S73975229

On the Geometry of Equiform Normal Curves in the Galilean Space G4

In our article, we establish the definition of the Equiform Normal curves in Galilean space G4. To obtain the position vector of an Equiform Normal curve in G4, we have to solve an integro-differential equation in μ2, where μ2 is the position function of a space curve γ (σ ) in the direction of third vector V3 of the Galilean space. Special cases of Equiform Normal curvatures are discussed. Finally, we prove that there is no equiform normal curve that is congruent to an Equiform Normal curve in G4

Effectiveness of human, camel, bovine and sheep lactoferrin on the hepatitis C virus cellular infectivity: Comparison study

Purpose. The prevalence of HCV infection has increased during recent years and the incidence reach 3% of the world''s population, and in some countries like Egypt, may around 20%. The developments of effective and preventive agents are critical to control the current public health burden imposed by HCV infection. Lactoferrin in general and camel lactoferrin specifically has been shown to have a compatitive anti-viral activity against hepatitis C virus (HCV). The purpose of this study was to examine and compare the anti-infectivity of native human, camel, bovine and sheep lactoferrin on continuous of HCV infection in HepG2 cells. Material and methods. Used Lfs were purified by Mono S 5/50 GL column and Superdex 200 5/150 column. The purified Lfs were evaluated in two ways; 1. the pre-infected cells were treated with the Lfs to inhibit intracellular replication at different concentrations and time intervals, 2. Lfs were directly incubated with the virus molecules then used to cells infection. The antiviral activity of the Lfs were determined using three techniques; 1. RT-nested PCR, 2. Real-time PCR and 3. Flowcytometric. Results: Human, camel, bovine and sheep lactoferrin could prevent the HCV entry into HepG2 cells by direct interaction with the virus instead of causing significant changes in the target cells. They were also able to inhibit virus amplification in HCV infected HepG2 cells. The highest anti-infectivity was demonstrated by the camel lactoferrin. Conclusion: cLf has inhibitory effect on HCV (genotype 4a) higher than human, bovine and sheep lactoferrin

Role of Assistive Devices on Gait in Patients with Incomplete Spinal Cord Injury: Systematic Review

Background: People with incomplete spinal cord injury disabilities can be able to live a healthy, productive, and dignified life by using Assistive devices as their role in improving gait. Facilitate locomotion rehabilitation. And enable people with incomplete SCI to ambulate in an upright position. Objective: This systematic review aimed to examine the effectiveness of the role of using assistive devices in gait rehabilitation in patients with incomplete SCI. Material and Methods: Studies were identified from 2000 to 2020 by electronic search using PubMed, Cochrane Database of Systematic Reviews, Google Scholar, and Physiotherapy Evidence Database (Pedro). They were reviewed if they were randomized control trials focused on the effectiveness of Assistive Devices on Gait in Patients in age more than 18 years with incomplete Spinal Cord Injury being published in English. Eight studies were selected according to inclusive and exclusive criteria and descriptive analysis was conducted due to heterogeneity. Results: Eight trials were identified with good quality methodology. Descriptive analysis was applied for three studies that supported the use of assistive devices for those patients and meta-analysis was applied for five studies. The mean difference across all the five studies is -0.69 (95% CI -0.93, -0.45). According to AACPDM, there is level II evidence that supports the use of the assistive device as a method to be able to live a healthy, productive, and dignified life. Conclusion: The current level of evidence supports the effectiveness of assistive devices in improving gait in patients with incomplete spinal cord injury