FINN 222 Introduction to Mathematics of Finance Spring Semester 2018 (Tentative Under review) Instructor Ferhana Ahmad Room No. 314 Office Hours TBA Email ferhana.ahmad@lums.edu.pk, ferhanaahmad@gmail.com Telephone 042 3560 8044 Secretary/TA Sec: Bilal Ahmad Alvi/TA: TBA TA Office Hours TBA Course URL (if any) http://suraj.lums.edu.pk/~ro/ COURSE BASICS Credit Hours 3 Lecture(s) Nbr of Lec(s) Per Week 2 Duration 1.25 hours Recitation/Lab (per week) Nbr of Lec(s) Per Week Duration Tutorial (per week) Nbr of Lec(s) Per Week Duration COURSE DISTRIBUTION Core Elective Open for Student Category Close for Student Category No Yes Open for all None COURSE DESCRIPTION With the recent developments in Finance over the past decade, the usefulness of Mathematical tools in Finance has become significant than ever. The course provides an introduction to mathematics of finance and is ideal for developing an understanding and knowledge of basic mathematical finance that a student requires throughout his or her academic and professional career. It introduces the vocabulary of mathematics of finance that helps developing the understanding of financial instrument at large. The course serves as the basis for higher studies in finance, quantitative finance, computational finance, financial engineering, financial economics, economics, insurance, actuarial sciences or any similar field. The course will broaden your horizon of finance and financial industry. Not only that the course is essential for the undergraduate and graduate studies in finance but also plays an important role in the student s performance in their professional careers. The course covers topics including basics of calculus, time value of money, theory of interest, probability, normal random variables and probability, arbitrage theorem, random walks and Brownian motion. COURSE PREREQUISITE(S) FINN 100 Principles of Finance COURSE LEARNING OBJECTIVES The course helps students to get over their fear of mathematics as a student enrolled in SDSB. The course will develop the vocabulary of mathematics of finance familiarize students with basic mathematical tools that are used in finance introduce students to probability theory required in finance introduce students with the concepts of randomness in finance prepare students to take advanced courses in finance and quantitative finance
LEARNING OUTCOMES Upon completion of the course, students will be able to; apply basic mathematical concepts used in finance implement the learnt theory of interest rates in other courses and in their professional lives apply the randomness concepts in finance and probability model randomness using Brownian motion simulate a simple random walk simulate stock prices as random processes UNDERGRADUATE PROGRAM LEARNING GOALS & OBJECTIVES General Learning Goals & Objectives Goal 1 Effective Written and Oral Communication Objective: Students will demonstrate effective writing and oral communication skills Goal 2 Ethical Understanding and Reasoning Objective: Students will demonstrate that they are able to identify and address ethical issues in an organizational context. Goal 3 Analytical Thinking and Problem Solving Skills Objective: Students will demonstrate that they are able to identify key problems and generate viable solutions. Goal 4 Application of Information Technology Objective: Students will demonstrate that they are able to use current technologies in business and management context. Goal 5 Teamwork in Diverse and Multicultural Environments Objective: Students will demonstrate that they are able to work effectively in diverse environments. Goal 6 Understanding Organizational Ecosystems Objective: Students will demonstrate that they have an understanding of Economic, Political, Regulatory, Legal, Technological, and Social environment of organizations. Major Specific Learning Goals & Objectives Goal 7 (a) Discipline Specific Knowledge and Understanding Objective: Students will demonstrate knowledge of key business disciplines and how they interact including application to real world situations (Including subject knowledge). Goal 7 (b) Understanding the science behind the decision making process (for MGS Majors) Objective: Students will demonstrate ability to analyze a business problem, design and apply appropriate decision support tools, interpret results and make meaningful recommendations to support the decision maker Indicate below how the course learning objectives specifically relate to any program learning goals and objectives. PROGRAM LEARNING GOALS AND OBJECTIVES Goal 1 Effective Written and Oral Communication Goal 2 Ethical Understanding and Reasoning Goal 3 Analytical Thinking and Problem Solving Skills COURSE LEARNING OBJECTIVES The course provides an opportunity to students to write and deliver effectively the mathematical nature problems arising in Finance. The course equips students with basic problem solving techniques in Finance using quantitative methods. It enables students to analytically think a problem and solve it using the problem solving techniques they're learning throughout the course COURSE ASSESSMENT ITEM Written: Assignments, Quizzes Oral: CP Assignments, Quizzes, Exams
Goal 4 Application of Information Technology Goal 5 Teamwork in Diverse and Multicultural Environments Goal 6 Understanding Organizational Ecosystems Goal 7 (a) Discipline Specific Knowledge and Understanding Goal 7 (b) Understanding the science behind the decision making process Students will simulate stock prices using Excel The course forces students to learn in teamwork. The discussion on assignments and lecture notes will help them in working in diverse environments NA Students will learn quantitative skills in finance that they can apply and model the real world financial situations/problems This is a basic course in mathematics of finance. Students will learn tools that may help them in future if they opt for Quantitative finance career in designing and solving a problem in finance using quantitative skills. Class work and Assignments Assignments and Projects Quizzes, Assignments, Project, and Exams Assignments, Quizzes, and Exams GRADING BREAKUP AND POLICY Assignments: 20% There will be around 8 10 assignments during the semester. Students are advised to submit the assignments before the deadline. Please don t ask for an extension in assignment submission deadlines. No make up assignments will be given! Quizzes: 10% There will be around 4 6 announced quizzes in class. The N 1 policy will be applicable only when the number of quizzes exceeds 4. No make up quizzes will be conducted. Class Participation & Attendance: 5% Attendance policy: You can have up to 3 absences during the without losing any attendance points. Late arrival by 5 minutes will mark you absent for the session Absent from class on medical leave/ tours on behalf of LUMS or any other personal or professional reasons will mark you absent for the session unless approved by OAS Use of mobile phones in the class or bringing food in class will mark you absent for the session. Marks will be deducted at a rate of 1% per class missed after 3 absences mentioned in the first point. Midterm Examination: 30% Final Examination: 35% The exams are closed book/closed notes. Cheat sheets are not allowed! EXAMINATION DETAIL Midterm Exam Final Exam Yes/No: YES Combine Separate: Duration: 1 hour 15 minutes Preferred Date: Exam Specifications: Yes/No: YES Combine Separate: Duration: 2 hours Exam Specifications:
WEEK/ LECTURE/ MODULE TOPICS RECOMMENDED READINGS SESSION OBJECTIVE(S) Basics of Calculus The module is based on the revision of the basic mathematics and calculus concepts that students have already learnt in Further mathematics or Calculus I. The module prepares students to look at finance from mathematical side. The module reviews the real number system, vectors and array, polynomial and series concepts along with functions, their derivatives and integration. 1 3 Calculus Review Thomas Calculus (Review) Ch: 2 (Limits and Continuity) (2.1 2.7) Ch: 3 (Differentiation) (3.1, 3.2, 3.5 (chain rule)) Ch: 4 (Optimization) (4.1, 4.3, 4.4) (First and Second derivative tests for maxima and minima Ch5: (Integration) 5.3 (integral definition Riemann sums), 5.4(Fundamental theorem of calculus) Students will revise the concepts of limits & continuity, derivatives and integration They ll apply the concepts on finding maxima and minima the first step towards optimization 4 5 Calculus Review Thomas Calculus Ch:11 (746 748, 761 765,794 795,800,805 808) Students will learn the sequences, series and sums Application of series to present value computations Taylor Series expansion 6 7 Partial Derivatives Thomas Calculus Ch:14 (965 966, 14.2,14.3,14.4) Mean Value optimization (Handouts given in the class) Students will learn functions of multiple variables how to take derivatives of a multivariate function how to interpret the derivatives the application of partial derivatives in Lagrangian optimization (Overview of Mean Variance optimization problem) Finance The module is based on the basic concepts in finance and interest rate theory. The students learn the core concepts in finance such as time value of money and rate of return. They will learn the mathematics behind the interest rate compounding and measuring the rates in discrete and continuous times.
8 9 10 Time value of money Theory of interest rates Arbitrage Principles of Managerial Finance 13 th Edition Chapter 5 (Finn 100 readings) Paul Wilmott introduces Quantitative Finance Second Edition Chapter 1 (1.6) Kavin J Hasting Ch: 1 (1.1,1.2,1.3 & 1.6) J Robert Buchanan Pages (81 84) Kevin J Hastings (5.1.2) Students are advised to bring examples of arbitrage for class discussion Time value of money concepts will be given and discussed with students to emphasize the need to mathematical concepts to model interest rates and other financial instruments Students will learn Rate of return and present value Compound interest Annuities Measuring rate of return Continuous time interest Students will be given the concept of arbitrage in finance. They will be provided with example of taking advantage of arbitrage and how pricing is mostly based on no arbitrage arguments Probability Theory Randomness is a core concept in finance, especially when we model the financial securities and instruments. It provides the analytical tools to solve practical problems in the complex and rapidly evolving world of today's financial industry. Due to the randomness involved in the financial industry, we work in a probabilistic way. The module provides the essential concepts that are required for defining and understanding the modeling of a Brownian motion. 11 12 13 14 15 Probability theory Random experiments, random variable, events, sample space Algebra, Sigma algebra, filtration, measure, probability measure, probability space Expectations, conditional expectations, independence Grimmett & Stirzekar Chapter 2 (2.1 (25 29), 32) Grimmett & Stirzekar Chapter 1: (Pages 1 14) Grimmett & Stirzekar Ch: 3 (3.1,3.2,3.3,3.7) Ch:4 (4.1,4.2,4.3,4.6) Students will learn the concepts in probability theory from a mathematical side. The will learn What constitutes a random experiment What is a random variable What we mean by events How do we define sample space In this session, they will learn The concepts required to define probability space Concepts such as algebra, sigma algebra, filtration will be introduced to students Measure will be defined as a function Probability measure and probability space will be defined Expectations, conditional expectations and independent events will defined using mathematical definitions
16 17 Normal distribution, mean variance and moment generating function Log normal distribution J Robert Buchanan Chapter 3 Normal distribution has an essential role in the theory of randomness in finance. The distribution function, mean, variance and moment generating function of a normal distribution will be discussed in the sessions. The usage of moment generating function in finding moments will be discussed. By the end of the sessions, students will be comfortable will the concepts that are required in stochastic calculus. Randomness concepts in Finance The module on randomness concepts in finance provides the basics of quantitative and computational finance. The module starts with discussion of a random walk, the pattern the stocks follow in reality are discussed for comparison. Brownian motion is defined and its properties are discussed as the basic ingredient of the mathematics of finance and stochastic calculus. The modeling of financial instruments is all based on the concept and modeling of randomness. The students will learn those basic concepts in the module 18 19 20 21 22 23 24 25 Random Walk Brownian motion Properties of Brownian motion Geometric Brownian motion and its properties Martingale and Markov process Stock prices as a lognormal process J Robert Buchman Chapter 5 J Robert Buchman Chapter 5 J Robert Buchman Chapter 5 Lecture Notes Ubbo F Wiersema Pages (31 37) Reading Provided Richard Ivey School of Business Modeling Stock Prices Students will learn the concept or randomness random walk and its properties how to formulate a random walk Brownian motion will be defined as as a continuous time limit of a random walk as a random variable from a normal distribution Alternative definition of a Brownian motion Students will learn different properties of Brownian motion such as scaling, time inversion, time reversal The will learn to show whether a process is a Brownian motion or not We model stock prices as a geometric Brownian motion, in this session students will learn the properties of a geometric Brownian motion Students will learn martingales and will show that Brownian motion is a martingale Students will learn the Markov property and will show that the BM is Markov Students will price stock prices as a lognormal process Simulation of stock prices will be done on Excel
26 27 Options, Pricing options using Monte Carlo Simulation Introduction to first order differential equations John Hull Chapter 8 Zill & Cullen Chapter 1.1 (pages 2 5) Chapter 2 (2.2,2.3,2.4) Students will learn the Applications of modeling randomness Monte Carlo simulation for pricing an option (Overview, application, Excel) (Overview) Students will learn first order differential equations and their solutions using integrating factors and separation of variables 28 Differential equations are an integral part of financial modeling. The session will prepare students to think about analytical solutions, if any, of the financial instruments and modeling of the financial instruments. TEXTBOOK(S)/SUPPLEMENTARY READINGS 1. J Robert Buchanan. An Undergraduate Introduction to Financial Mathematics, World Scientific Publishing Company, 3 rd Edition, [ISBN 9789814407441] (Finance and Randomness) 2. Kevin J. Hastings. Introduction to Financial Mathematics, Chapman and Hall, 2015. [ISBN 9781498723909] (Finance and Randomness) 3. Geoffrey R Grimmett and David R Stirzekar, Probability and Random Processes, Oxford University Press, 3 rd Edition, 2001. [ISBN 13: 978 0198572220] (Probability Theory) 4. George B Thomas, Maurice D Weir, Joel Hass and Franke R Giordano, Thomas Calculus, 11 th Edition, Addison Wesley, 2004. [ISBN 9780321185587] (Calculus review) 5. Dennis G Zill and Micheal R Cullen, Differential Equations with Boundary Conditions, 7 th Edition, Cengage Learning, 2009. [ASIN: B008UB1WJE] (Introduction to Differential Equations Overview)