Computational Finance

MATLAB® Oriented Modeling

Francesco Cesarone

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367493035
• eBook ISBN: 9781000169034
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

Computational finance is increasingly important in the financial industry, as a necessary instrument for applying theoretical models to real-world challenges. Indeed, many models used in practice involve complex mathematical problems, for which an exact or a closed-form solution is not available. Consequently, we need to rely on computational techniques and specific numerical algorithms. This book combines theoretical concepts with practical implementation. Furthermore, the numerical solution of models is exploited, both to enhance the understanding of some mathematical and statistical notions, and to acquire sound programming skills in MATLAB®, which is useful for several other programming languages also. The material assumes the reader has a relatively limited knowledge of mathematics, probability, and statistics. Hence, the book contains a short description of the fundamental tools needed to address the two main fields of quantitative finance: portfolio selection and derivatives pricing. Both fields are developed here, with a particular emphasis on portfolio selection, where the author includes an overview of recent approaches. The book gradually takes the reader from a basic to medium level of expertise by using examples and exercises to simplify the understanding of complex models in finance, giving them the ability to place financial models in a computational setting. The book is ideal for courses focusing on quantitative finance, asset management, mathematical methods for economics and finance, investment banking, and corporate finance.

Part II

Portfolio selection

Chapter 2

Preliminary elements in Probability Theory and Statistics

Since most financial models represent random phenomena, in this chapter we give a short introduction to the basic elements of Probability Theory and Statistics, which are the typical mathematical tools used in this framework. More precisely, in Section 2.1 we provide some preliminary elements in probability. Section 2.2 introduces the concept of discrete random variables, while Section 2.3 shows how to characterize a random variable by means of its cumulative distribution function. In Section 2.4 we introduce continuous random variables, describing how to characterize them through their probability density functions. In Section 2.5 we discuss some standard synthetic indices of a distribution. Finally, Section 2.6 is devoted to present some of the most frequently used probability distributions in finance.

2.1 Basic concepts in probability

Firstly, we seek to define the probability associated to a specific event. In the Classical (Bernoulli and Laplace) definition, the probability of an event is the ratio between the number of cases favorable to it and the number of possible cases, which should be considered equally likely. In the Frequentist definition, the probability of an event is the limit of the frequency of the occurrence of such an event when the number of trials is large (i.e., it tends to infinity). In the Subjective definition (Ramsey, De Finetti, Savage), the probability can be considered as the degree of confidence that an event will occur. It is determined by the information available to the individual and by his experience.

Remark 61 (Different approaches to define probability) The Classical definition can be applied only when all the events have the same probability of occurrence (a situation that is generally not the case in financial contexts). The Frequentist definition can be used when the experiment is repeated multiple times under the same exact conditions (this is seldom the case in finance). In addition, it is not clear how to treat the concept of the limit to infinity of the frequency of the occurrence of an event. Both of these limitations can be overcome by the Subjective approach, but nonetheless with this definition it is impossible to assign a unique probability to an event, since different individuals can provide different estimations of the probability.

Remark 62 In view of the nature of the events in finance, we us...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• I Programming techniques for financial calculus
• II Portfolio selection
• III Derivatives pricing
• References
• Suggested lesson plan