Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing

[EN] The present PhD thesis is focused on numerical analysis and computing of finite difference schemes for several relevant option pricing models that generalize the Black-Scholes model. A careful analysis of desirable properties for the numerical solutions of option pricing models as the positivity, stability and consistency, is provided. In order to handle the free boundary that arises in American option pricing problems, various transformation techniques based on front-fixing method are applied and studied. Special attention is paid to multi-asset option pricing, such as exchange or spread option. Appropriate transformation allows eliminating of the cross derivative term. Transformation techniques of partial differential equations to remove convection and reaction terms are studied in order to simplify the models and avoid possible troubles of stability. This thesis consists of six chapters. The first chapter is an introduction containing definitions of option and related terms and derivation of the Black-Scholes equation as well as general aspects of theory of finite difference schemes, including preliminaries on numerical analysis. Chapter 2 is devoted to solve linear Black-Scholes model for American put and call options. A Landau transformation and a new front-fixing transformation are applied to the free boundary value problem. It leads to non-linear partial differential equation (PDE) in a fixed domain. Stable and consistent explicit numerical schemes are proposed preserving positivity and monotonicity of the solution in accordance with the behaviour of the exact solution. Efficiency of the front-fixing method demonstrated in Chapter 2 has motivated us to apply the method to some more complicated nonlinear models. A new change of variables resulting in a time dependent boundary instead of fixed one, is applied to nonlinear Black-Scholes model for American options, such as Barles and Soner and Risk Adjusted Pricing models. Chapter 4 provides a new alternative approach for solving American option pricing problem based on rationality of investor. There exists an intensity function that can be reduced in the simplest case to penalty approach. Chapter 5 deals with multi-asset option pricing. Appropriate transformation allows eliminating of the cross derivative term avoiding computational drawbacks and possible troubles of stability. Concluding remarks are given in Chapter 6. All the considered models and numerical methods are accompanied by several examples and simulations. The convergence rate is computed confirming the theoretical study of consistency. Stability conditions are tested by numerical examples. Results are compared with known relevant methods in the literature showing efficiency of the proposed methods.[ES] La presente tesis doctoral se centra en la construcción de esquemas en diferencias finitas y el análisis numérico de relevantes modelos de valoración de opciones que generalizan el modelo de Black-Scholes. Se proporciona un análisis cuidadoso de las propiedades de las soluciones numéricas tales como la positividad, la estabilidad y la consistencia. Con el fin de manejar la frontera libre que surge en los problemas de valoración de opciones Americanas, se aplican y se estudian diversas técnicas de transformación basadas en el método de fijación de las fronteras (front-fixing). Se presta especial atención a la valoración de opciones de múltiples activos, como son las opciones ''exchange'' y ''spread''. Esta tesis se compone de seis capítulos. El primer capítulo es una introducción que contiene las definiciones de opción y términos relacionados y la derivación de la ecuación de Black-Scholes, así como aspectos generales de la teoría de los esquemas en diferencias finitas, incluyendo preliminares de análisis numérico. El capítulo 2 está dedicado a resolver el modelo lineal de Black-Scholes para opciones Americanas put y call. Para fijar las fronteras del problema de frontera libre se aplican transformaciones como la de Landau y un nuevo cambio de variable propuesto. La eficiencia del método front-fixing mostrada en el capítulo 2 ha motivado el estudio de su aplicación a algunos modelos no lineales más complicados. En particular, se propone un cambio de variables que lleva a una nueva frontera dependiente del tiempo en lugar de una fija. Este cambio se aplica a modelos no lineales de Black-Scholes para opciones Americanas, como son el de Barles y Soner y el modelo RAPM (Risk Adjusted Pricing Methodology). El capítulo 4 ofrece una nueva técnica para la resolución de problemas de valoración de opciones Americanas basada en la racionalidad de los inversores. Aparece una función de la intensidad que se puede reducir en el caso más simple a la técnica de penalización (penalty method). Este enfoque tiene en cuenta el posible comportamiento irracional de los inversores. En la sección 4.2 se aplica esta técnica al modelo de cambio de regímenes lo que lleva a un nuevo modelo que tiene en cuenta el posible ejercicio irracional, así como varios estados del mercado. El enfoque del parámetro de racionalidad junto con una transformación logarítmica permiten construir un esquema numérico eficiente sin aplicar el método front-fixing o la conocida formulación de LCP (Linear Complementarity Problem). El capítulo 5 se dedica a la valoración de opciones de activos múltiples. Una transformación apropiada permite la eliminación del término de derivadas cruzadas evitando inconvenientes computacionales y posibles problemas de estabilidad. Las conclusiones se muestran en el capítulo 6. Se pone en relieve varios aspectos de la presente tesis. Todos los modelos considerados y los métodos numéricos van acompañados de varios ejemplos y simulaciones. Se estudia la convergencia numérica que confirma el estudio teórico de la consistencia. Las condiciones de estabilidad son corroboradas con ejemplos numéricos. Los resultados se comparan con métodos relevantes de la bibliografía mostrando la eficiencia de los métodos propuestos.[CA] La present tesi doctoral se centra en la construcció d'esquemes en diferències finites i l'anàlisi numèrica de rellevants models de valoració d'opcions que generalitzen el model de Black-Scholes. Es proporciona una anàlisi cuidadosa de les propietats de les solucions numèri-ques com ara la positivitat, l'estabilitat i la consistència. A fi de manejar la frontera lliure que sorgix en els problemes de valoració d'opcions Americanes, s'apliquen i s'estudien diverses tècniques de transformació basades en el mètode de fixació de les fronteres (front-fixing). Es presta especial atenció a la valoració d'opcions de múltiples actius, com són les opcions ''exchange'' i ''spread''. Esta tesi es compon de sis capítols. El primer capítol és una introducció que conté les definicions d'opció i termes relacionats i la derivació de l'equació de Black-Scholes, així com aspectes generals de la teoria dels esquemes en diferències finites, incloent aspectes preliminars d'anàlisi numèrica. El 2n capítol està dedicat a resoldre el model lineal de Black-Scholes per a opcions Americanes ''put'' i ''call''. Per a fixar les fronteres del problema de frontera lliure s'apliquen transformacions com la de Landau i s'ha proposat un nou canvi de variable proposat. Açò porta a una equació diferencial en derivades parcials no lineal en un domini fix. L'eficiència del mètode front-fixing mostrada en el 2n capítol ha motivat l'estudi de la seua aplicació a alguns models no lineals més complicats. En particular, es proposa un canvi de variables que porta a una nova frontera dependent del temps en compte d'una fixa. Este canvi s'aplica a models no lineals de Black-Scholes per a opcions Americanes, com són el de Barles i Soner i el model RAPM (Risk Adjusted Pricing Methodology). El 4t capítol oferix una nova tècnica per a la resolució de problemes de valoració d'opcions Americanes basada en la racionalitat dels inversors. Apareix una funció de la intensitat que es pot reduir en el cas més simple a la tècnica de penalització (penal method) . Este enfocament té en compte el possible comportament irracional dels inversors. En la secció 4.2 s'aplica esta tècnica al model de canvi de règims el que porta a un nou model que té en compte el possible exercici irracional, així com diversos estats del mercat. L'enfocament del paràmetre de racionalitat junt amb una transformació logarítmica permeten construir un esquema numèric eficient sense aplicar el mètode front-fixing o la coneguda formulació de LCP (Linear Complementarity Problem). El 5é capítol es dedica a la valoració d'opcions d'actius múltiples. Una transformació apropiada permet l'eliminació del terme de derivades mixtes evitant inconvenients computacionals i possibles problemes d' estabilitat. Les conclusions es mostren al 6é capítol. Es posa en relleu diversos aspectes de la present tesi. Tots els models considerats i els mètodes numèrics van acompanyats de diversos exemples i simulacions. S'estu-dia la convergència numèrica que confirma l'estudi teòric de la consistència. Les condicions d'estabilitat són corroborades amb exemples numèrics. Els resultats es comparen amb mètodes rellevants de la bibliografia mostrant l'eficiència dels mètodes proposats.Egorova, V. (2016). Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/68501TESISPremios Extraordinarios de tesis doctorale

Fire-spotting generated fires. Part II: the role of flame geometry and slope

This is the second part of a series of two papers concerning fire-spotting generated fires. While, in the first part, we focus on the impact of macro-scale factors on the growth of the burning area by considering the atmospheric stability conditions, in the present study we focus on the impact of meso-scale factors by considering the effects of the flame geometry and terrain slope. First, we discuss the phenomenological power law that relates flame length and fireline intensity by reporting literature data, analysing a formula originally proposed by Albini, and deriving an alternative formula based on the energy conservation principle. Subsequently, we extend the physical fire-spotting parametrisation RandomFrontadopted in the first part by including flame geometry and slope. Numerical examples show that fire-spotting is affected by flame geometry and, therefore, cannot be neglected in simplified fire-spread models used in operational software codes for wildfire propagation. Meanwhile, we observe that terrain slope enhances the spread of a fire at a higher rate than the augmentation of fire-spotting generated fires, such that a rapid merging occurs among independent fires.This research is supported by the Basque Government through the BERC 2014–2017 and the BERC 2018–2021 programs, the Spanish Ministry of Economy and Competitiveness MINECO through the BCAM Severo Ochoa excellence accreditations SEV-2013-0323 and SEV-2017-0718 and through the projects MTM2016-76016-R and PID2019-107685RB-I00, and by the PhD grant “La Caixa 2014”

Quadrature integration techniques for random hyperbolic PDE problems

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss?Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.This research has been funded by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-

An ETD method for multi-asset American option pricing under jump-diffusion model

In this paper, we propose a numerical method for American multi-asset options under jump-diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss-Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.Ministerio de Ciencia, Innovación y Universidades, Grant/Award Number: MTM2017- 89664-P; Ministerio de Economía y Competitividad, Grant/Award Number: PID2019-107685RB-I0

Solving American option pricing models by the front fixing method: numerical analysis and computing

This paper presents an explicit finite-difference method for nonlinear partial differential equation appearing as a transformed Black-Scholes equation for American put option under logarithmic front fixing transformation. Numerical analysis of the method is provided. The method preserves positivity and monotonicity of the numerical solution. Consistency and stability properties of the scheme are studied. Explicit calculations avoid iterative algorithms for solving nonlinear systems. Theoretical results are confirmed by numerical experiments. Comparison with other approaches shows that the proposed method is accurate and competitive

On the merits of sparse surrogates for global sensitivity analysis of multi-scale nonlinear problems: application to turbulence and fire-spotting model in wildland fire simulators

Many nonlinear phenomena, whose numerical simulation is not straightforward, depend on a set of parameters in a way which is not easy to predict beforehand. Wildland fires in presence of strong winds fall into this category, also due to the occurrence of firespotting. We present a global sensitivity analysis of a new sub-model for turbulence and fire-spotting included in a wildfire spread model based on a stochastic representation of the fireline. To limit the number of model evaluations, fast surrogate models based on generalized Polynomial Chaos (gPC) and Gaussian Process are used to identify the key parameters affecting topology and size of burnt area. This study investigates the application of these surrogates to compute Sobol' sensitivity indices in an idealized test case. The performances of the surrogates for varying size and type of training sets as well as for varying parameterization and choice of algorithms have been compared. In particular, different types of truncation and projection strategies are tested for gPC surrogates. The best performance was achieved using a gPC strategy based on a sparse least-angle regression (LAR) and a low-discrepancy Halton's sequence. Still, the LAR-based gPC surrogate tends to filter out the information coming from parameters with large length-scale, which is not the case of the cleaning-based gPC surrogate. The wind is known to drive the fire propagation. The results show that it is a more general leading factor that governs the generation of secondary fires. Using a sparse surrogate is thus a promising strategy to analyze new models and its dependency on input parameters in wildfire applications.This research is supported by the Basque Government through the BERC 2014–2017 and BERC 2018–2021 programs, by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa accreditations SEV-2013-0323 and SEV-2017-0718 and through project MTM2016-76016-R “MIP”, and by the PhD grant “La Caixa2014”. The authors acknowledge EDF R&D for their support on the OpenTURNS library. They also acknowledge Pamphile Roy and Matthias De Lozzo at CERFACS for helpful discussions on batman and scikit-learn tools

Dinámicas de aprendizaje colaborativo para el aula

RESUMEN: Algunos estudiantes del Grado en Economía perciben las asignaturas del contenido relacionado con la política económica como algo complicado y difícil de entender. Muchas de las causas se encuentran en las estrategias utilizadas por los profesores, quienes han promovido un sistema de enseñanza netamente conductista y monótono en el que se ha sembrado y cosechado indiferencia y rechazo hacia la asignatura. En este contexto, el aprendizaje colaborativo puede ser una solución en la que los estudiantes aprenden a gestionar sus tareas de manera cooperativa, reduciendo así la carga que recaía exclusivamente en la figura del docente. Por el otro lado, hoy en día el mundo laboral y profesional exige cada vez más una labor y dinámicas de equipo, grupos de trabajo y estructuras más horizontales que jerárquicas. Por lo que enfatizando los valores del aprendizaje colaborativo se puede asegurar, entre otros muchos fines, que nuestros estudiantes estén preparados para el nuevo mercado laboral.ABSTRACT: Some students of the Degree in Economics perceive the subjects of content related to economic policy as something complicated and difficult to understand. Many of the causes are found in the strategies used by teachers, who have promoted a monotonous teaching system in which indifference and rejection of the subject have been sown and harvested. In this context, collaborative learning can be a solution in which students learn to manage their tasks cooperatively, thus reducing the burden that fell exclusively on the figure of the teacher. On the other hand, today the work and professional world increasingly demands work and team dynamics, work groups and structures that are more horizontal than hierarchical. Therefore, emphasizing the values of collaborative learning can ensure, among many other purposes, that our students will be prepared for the new job market

A New Efficient Numerical Method for Solving American Option under Regime Switching Model

[EN] A system of coupled free boundary problems describing American put option pricing under regime switching is considered. In order to build numerical solution firstly a front-fixing transformation is applied. Transformed problem is posed on multidimensional fixed domain and is solved by explicit finite difference method. The numerical scheme is conditionally stable and is consistent with the first order in time and second order in space. The proposed approach allows the computation not only of the option price but also of the optimal stopping boundary. Numerical examples demonstrate efficiency and accuracy of the proposed method. The results are compared with other known approaches to show its competitiveness.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.Egorova, V.; Company Rossi, R.; Jódar Sánchez, LA. (2016). A New Efficient Numerical Method for Solving American Option under Regime Switching Model. Computers and Mathematics with Applications. 71:224-237. https://doi.org/10.1016/j.camwa.2015.11.019S2242377

An efficient method for solving spread option pricing problem: numerical analysis and computing

This paper deals with numerical analysis and computing of spread option pricing problem described by a two-spatial variables partial differential equation. Both European and American cases are treated. Taking advantage of a cross derivative removing technique, an explicit difference scheme is developed retaining the benefits of the one-dimensional finite difference method, preserving positivity, accuracy, and computational time efficiency. Numerical results illustrate the interest of the approach.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 the Ministerio de Economía y Competitividad Spanish Grant MTM2013-41765-P