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Introduction to Financial Mathematics Concepts and Computational Methods

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Introduction to Financial Mathematics Concepts and Computational Methods Introduction to Financial Mathematics Concepts and Computational Methods Arash Fahim Introduction to Financial Mathematics Concepts and Computational Methods Arash Fahim Florida State University July 8, 2019 Copyright © 2019, Arash Fahim This work is licensed under a Creative Commons Attribution-NonCommercial- ShareAlike 4.0 International License. You are free to: Share—copy and redistribute the material in any medium or format Adapt—remix, transform, and build upon the material Under the following terms: Attribution—You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. NonCommercial—You may not use the material for commercial purposes. ShareAlike—If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. This publication was made possible by an Alternative Textbook Grant issued by Florida State University Libraries. This work is published by Florida State University Libraries, 116 Honors Way, Tallahassee, FL 32306. DOI: 10.33009/financialmath1 ii Dedicated to Nasrin, Sevda and Idin. iv Notations v Table 1: Table of Notation A Y¨ B fi union of two disjoint set A and B Rd fi d-dimensional Euclidean space Rd Rd ` fi All points in with nonnegative coordinates x P Rd fi d-dimensional column vector minpx, yq “ fi component-wise minimum of two vectors pminpx1, y1q, ..., minpxn, ynqq x` “ minpx, 0q fi component-wise minimum of a vector with zero vector AT fi transpose of matrix A x ¨ y “ xTy fi inner (dot) product in Euclidean space A “ 0 fi all the entities of the matrix (vector) A are zero A ‰ 0 fi at least of the entities of the matrix (vector) A is nonzero A ¥ 0 (A ¡ 0) fi all the entities of the matrix (vector) A are non- negative (positive) ∇f fi column vector of gradient for a differentiable function f : Rd Ñ R ∇2f fi Hessian matrix for a twice differentiable function f : Rd Ñ R P or Pˆ fi Probability PpAq (resp. PˆpAq) fi Probability of an event A under probability P (resp. Pˆ) ErXs (resp. EˆrXs) fi Expected value of a random variable X under probability P (resp. Pˆ) vi Preface The story of this book started when I was assigned to teach an introductory financial math- ematics course at Florida State University. Originally, this course was all measure theory, integration and stochastic analysis. Then, it evolved to cover theory of measures, some probability theory, and option pricing in the binomial model. When I took over this course, I was not sure what I was going to do. However, I had a vision to educate students about some new topics in financial mathematics, while keeping the classical risk management material. My vision was to include some fundamental ideas that are shared between all models in financial mathematics, such as martingale property, Makrovian property, time- homogeneity, and the like, rather than studying a comprehensive list of models. To start, I decided to seek advise from a colleague to use a textbook by two authors, a quantitative financial analysts and a mathematician. The textbook was a little different and covered various models that quants utilize in practice. The semester started, and as I was going through the first couple of sections from the textbook, I realized that the book was unus- able; many grave mistakes and wrong theorems, sloppy format, and coherency issues made it impossible to learn from this textbook. It was my fault that I only skimmed the book before the start of semester. A few months later, I learned that another school had had the same experience with the book as they invited one of the authors to teach a similar course. Therefore, I urgently needed a plan to save my course. So, I decided to write my own lecture notes based on my vision, and, over the past three years, these lecture notes grew and grew to include topics that I consider useful for students to learn. In 2018, the Florida State University libraries awarded me the “Alternative Textbook Grant” to help me make my lecture notes into an open access free textbook. This current first edition is the result of many hours of effort by my library colleagues and myself. Many successful textbooks on financial mathematics have been developed in the recent decades. My favorite ones are the two volumes by Steven Shreve, Stochastic Calculus for Finance I and II ;[27, 28]. They cover a large variety of topics in financial mathematics with emphasis on the option pricing, the classical practice of quantitative financial analysts (quants). It also covers a great deal of stochastic calculus which is a basis for modeling almost all financial assets. Option pricing remains a must-know for every quant and stochas- tic calculus is the language of the quantitative finance. However over time, a variety of other subjects have been added to the list of what quants need to learn, including efficient vii computer programming, machine learning, data mining, big data, and so on. Many of these topics were irrelevant in 70s, when the quantitative finance was initially introduced. Since then, financial markets has changed in the tools that the traders use, and the speed of transactions. This is a common feature of many disciplines that the amount of data that can be used to make business decisions is too large to be handled by classical statistical techniques. Also, financial regulations has been adjusted to the new market environment. They now require financial institutions to provide structured measurements of some of their risks that is not included in the classical risk management theory. For instance, after the financial meltdown in 2007, systemic risk and central clearing became important research areas for the regulator. In addition, a demand for more robust evaluation of risks led to researches in the robust risk management and model risk evaluations. As the financial mathematics career grows to cover the above-mentioned topics, the prospect of the financial mathematics master’s programs must also become broader in topics. In the current book, I tried to include some new topics in an introductory level. Since this is an open access book, it has the ability to include more of the new topics in financial mathematics. One of the major challenges in teaching financial mathematics is the diverse background of students, at least in some institutions such as Florida State University. For example, some students whom I observed during the last five years, have broad finance background but lack the necessary mathematical background. They very much want to learn the mathemat- ical aspects, but with fewer details and stepping more quickly into the implementational aspects. Other students have majors in mathematics, engineering or computer science who need more basic knowledge in finance. One thing that both groups need is to develop their problem-solving abilities. Current job market favors employees who can work inde- pendently and solve hard problems, rather than those who simply take instructions and implement them. Therefore, I designed this book to serve as an introductory course in financial mathematics with focus on conceptual understanding of the models and problem solving, in contrast to textbooks that include more details of the specific models. It includes the mathematical background needed for risk management, such as probability theory, op- timization, and the like. The goal of the book is to expose the reader to a wide range of basic problems, some of which emphasize analytic ability, some requiring programming techniques and others focusing on statistical data analysis. In addition, it covers some areas which are outside the scope of mainstream financial mathematics textbooks. For example, it presents marginal account setting by the CCP and systemic risk, and a brief overview of the model risk. One of the main drawbacks of commercial textbooks in financial mathematics is the lack of flexibility to keep up with changes of the discipline. New editions often come far apart and with few changes. Also, it is not possible to modify them into the course needed for a specific program. The current book is a free, open textbook under a creative common license with attribution. This allows instructors to use parts of this book to design their own course in their own program, while adding new parts to keep up with the changes and viii the institutional goals of their program. The first two chapters of this book only require calculus and introductory probability and can be taught to senior undergraduate students. There is also a brief review of these topics in Sections A.1 and B of the appendix. I tried to be as brief as possible in the appendix; many books, including Stochastic Calculus for Finance I ([27, 28]) and Convex Optimization ([8]), cover these topics extensively. My goal to include these topics is only to make the current book self-sufficient. The main goal of Chapter 1 is to familiarize the reader with the basic concepts of risk management in financial mathematics. All these concepts are first introduced in a relatively nontechnical framework of one-period such as Markowitz portfolio diversification or the Arrow-Debreu market model. Chapter 2 generalizes the crucial results of the Arrow-Debreu market model to the multiperiod case and introduces the multiperiod binomial model and the numerical methods based on it. Chapter 3 discusses more advanced subjects in probability, which are presented in the remainder of Section B and Section C of the appendix. This chapter is more appropriate for graduate students. In Section 3.2, we first build important concepts and computational methods in continuous-time through the Bachelier model.
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