Non-negativity preserving numerical algorithms for stochastic differential equations

Construction of splitting-step methods and properties of related non-negativity and boundary preserving numerical algorithms for solving stochastic differential equations (SDEs) of Ito-type are discussed. We present convergence proofs for a newly designed splitting-step algorithm and simulation studies for numerous numerical examples ranging from stochastic dynamics occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in biology and physics.Comment: 23 pages, 7 figures. Figures 6.2 and 6.3 in low resolution due to upload size restrictions. Original resolution at http://gisc.uc3m.es/~moro/profesional.htm

An ETD method for American options under the Heston model

A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed. A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs. Free boundary is treated by the penalty method. Transformed nonlinear partial differential equation is solved numerically by using the method of lines. For full discretization the exponential time differencing method is used. Numerical analysis establishes the stability and positivity of the proposed method. The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.This work has been supported 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-P

Positivity preserving solutions of partial integro-differential equations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2009."May 15th, 2009."Includes bibliographical references (leaves 246-249).Differential equations are one of the primary tools for modeling phenomena in chemical engineering. While solution methods for many of these types of problems are well-established, there is growing class of problems that lack standard solution methods: partial integro-differential equations. The primary challenges in solving these problems are due to several factors, such as large range of variables, non-local phenomena, multi-dimensionality, and physical constraints. All of these issues ultimately determine the accuracy and solution time for a given problem. Typical solution techniques are designed to handle every system using the same methods. And often the physical constraints of the problem are not addressed until after the solution is completed if at all. In the worst case this can lead to some problems being over-simplified and results that provide little physical insight. The general concept of exploiting solution domain knowledge can address these issues. Positivity and mass-conservation of certain quantities are two conditions that are difficult to achieve in standard numerical solution methods. However, careful design of the discretizations can achieve these properties with a negligible performance penalty. Another important consideration is the stability domain. The eigenvalues of the discretized problem put restrictions on the size of the time step. For "stiff' systems implicit methods are generally used but the necessary matrix inversions are costly, especially for equations with integral components. By better characterizing the system it is possible to use more efficient explicit methods.(cont.) This work improves upon and combines several methods to develop more efficient methods. There are a vast number of systems that be solved using the methods developed in this work. The examples considered include population balances, neural models, radiative heat transfer models, among others. For the capstone portion, financial option pricing models using "jump-diffusion" motion are considered. Overall, gains in accuracy and efficiency were demonstrated across many conditions.by Alexander M. Lewis.Ph.D

Some extensions of the Black-Scholes and Cox-Ingersoll-Ross models

In this thesis we will study some financial problems concerning the option pricing in complete and incomplete markets and the bond pricing in the short-term interest rates framework. We start from well known models in pricing options or zero-coupon bonds, as the Black-Scholes model and the Cox-Ingersoll-Ross model and study some their generalizations. In particular, in the first part of the thesis, we study a generalized Black-Scholes equation to derive explicit or approximate solutions of an option pricing problem in incomplete market where the incompleteness is generated by the presence of a non-traded asset. Our aim is to give a closed form representation of the indifference price by using the analytic tool of (C0) semigroup theory. The second part of the thesis deals with the problem of forecasting future interest rates from observed financial market data. We propose a new numerical methodology for the CIR framework, which we call the CIR# model, that well fits the term structure of short interest rates as observed in a real market

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 sten- cil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods

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

Mathematical control theory and Finance

Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio

Pathwise functional calculus and applications to continuous-time finance

This thesis develops a mathematical framework for the analysis of continuous- time trading strategies which, in contrast to the classical setting of continuous-time finance, does not rely on stochastic integrals or other probabilistic notions. Using the recently developed `non-anticipative functional calculus', we first develop a pathwise definition of the gain process for a large class of continuous-time trading strategies which include the important class of delta-hedging strategies, as well as a pathwise definition of the self-financing condition. Using these concepts, we propose a framework for analyzing the performance and robustness of delta-hedging strategies for path-dependent derivatives across a given set of scenarios. Our setting allows for general path-dependent payoffs and does not require any probabilistic assumption on the dynamics of the underlying asset, thereby extending previous results on robustness of hedging strategies in the setting of diffusion models. We obtain a pathwise formula for the hedging error for a general path-dependent derivative and provide sufficient conditions ensuring the robustness of the delta hedge. We show in particular that robust hedges may be obtained in a large class of continuous exponential martingale models under a vertical convexity condition on the payoffs functional. Under the same conditions, we show that discontinuities in the underlying asset always deteriorate the hedging performance. These results are applied to the case of Asian options and barrier options. The last chapter, independent of the rest of the thesis, proposes a novel method, jointly developed with Andrea Pascucci and Stefano Pagliarani, for analytical approximations in local volatility models with L\ue9vy jumps. The main result is an expansion of the characteristic function in a local L\ue9vy model, which is worked out in the Fourier space by considering the adjoint formulation of the pricing problem. Combined with standard Fourier methods, our result provides effcient and accurate pricing formulae. In the case of Gaussian jumps, we also derive an explicit approximation of the transition density of the underlying process by a heat kernel expansion; the approximation is obtained in two ways: using PIDE techniques and working in the Fourier space. Numerical tests confirm the effectiveness of the method