ELEMENTARY INTRODUCTION TO MATHEMATICS OF FINANCE W 4071

INTRODUCTION TO MATHEMATICS OF FINANCE W 4071.

Instructor: Professor Mikhail Smirnov

Time: Monday, Wednesday 7.40-8.55 PM

email smirnov@math.columbia.edu

web site www.math.columbia.edu/~smirnov

phone (212) 854-4303

Office 425 Mathematics

Office hours: Monday, Wednesday 9pm-10.30pm and by appointment

Prerequisites: working knowledge of calculus, knowledge of elementary probability theory, it is desirable but not required that students have some familiarity with partial differential equations

Teaching Assistants:

Helena Kauppila, Head Helena.kauppila@gmail.com, hmk2105@columbia.edu

Stephane Benoist          sbenoist@math.columbia.edu
Beomjun Choi                cbj521@math.columbia.edu,       bc2491@columbia.edu
Zhengyu Zong                zz2197@math.columbia.edu,      zz2197@columbia.edu

Grading: Homework grades (30%), Midterm exam (15%), Final exam both parts: take-home and in class (12.5% + 12.5%=25%), Group project (25%), Class participation and attendance(5%).

COURSE INFORMATION

This course focuses on mathematical methods in pricing of derivative securities, risk management, portfolio management and on other related questions of mathematical finance. The emphasis is on the basic mathematical ideas and practical aspects. All the necessary definitions and concepts from the probability theory: random variables, normal and log-normal distributions, Brownian motion etc, will be explained in the course.

Students will learn to use Bloomberg terminals with Excel (these terminals are located in the Business School Library), will learn and do basic models in VBA and Matlab. Matlab is available through CUIT.

Homeworks will be assigned on Mondays every 2 weeks, they are due on Mondays 2 weeks later. Homeworks will be distributed in class. Summary of lectures will also be distributed in class. Homeworks may be challenging but they will contain many questions often asked at interviews and will teach helpful practical skills. There will be weekly recitation sessions by Helena Kauppila addressing and helping with homeworks. Time of these sessions will be announced later. These sessions are optional.

Each student will be given a project. The groups of 2-5 students should be formed according to student’s preferences. As a rare exception projects can be individual. Topic should be discussed with professor Smirnov (appointment should be made preferably during office hours). Students should form groups by 10/3. After the group is formed its representatives e-mail project description proposal to professor Smirnov before October 15. Students are welcome and encouraged to discuss project plans with professor Smirnov at office hours and Helena Kauppila at her office hours.

Some of the topics of past student projects will be given in class. Students prepare projects and do 10 minute group powerpoint presentations in the last 4 classes (including extra class on a study day 12/12). These classes may be 30 minutes longer until 9.30pm to accommodate all the presenters. Attendance of these presentations is compulsory and attendance will be taken.

Class Attendance. Students are highly encouraged to attend and not skip classes. So class materials and homeworks will be given in class and not through courseworks to encourage attendance.

Required Main Texts: 1. J.Hull, Options Futures and other derivatives, Prentice Hall, NJ, 8 th Edition

(previous editions as well as international editions are acceptable)

2. R.Grinold, R.Kahn, Active Portfolio Management, McGraw-Hill, 1999

Not required but highly recommended

3. Paul Wilmott, Paul Wilmott on Quantitative Finance, 3 Volume Set, John Wiley & Sons; ISBN:0470018704

Also not required but highly recommended

4. N.Taleb, Dynamic Hedging, Wiley NY, 1996.

Readings will be assigned periodically.

Additional finance articles will be distributed and assigned in class.

Midterm exam: Take-home midterm will be handed on September 26. It is due on October 17.

We assume based on current information that final will be on December 19.

Final exam will have 2 parts. The take-home part will be handed on November 14, it is due December 19. In-class 1.5 hour final exam will be given on Wednesday, December 19, 7.40-9.10pm. Student project reports are due also December 19.

The final exam is compulsory and can not be rescheduled earlier or later. If there are conflicts with other exams please reschedule other exams.

There may be occasional guest speakers. They will be announced during the course.

SYLLABUS

9/5 Introductory lecture. Course Requirements. Overview. Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Forward contracts. Arbitrage.

9/10 Review of probabilistic models, random variables. Distribution of percentage returns and prices. Idealized assumptions of mathematical finance vs. market reality. Expectation, variance, standard deviation, skewness, kurtosis. Review of probability distributions and their properties. Normal random variables. Log-normal distribution and its properties. Examples. Distribution of the rate of return for stocks. Empirical evidence for the distribution of the rate of return for stocks and other assets. A model of the behavior of stock prices.

9/12 Futures, different types of futures. Mechanics of the futures markets. Margins, margin calls. Contango and backwardation, futures curves.

9/17 Futures trading. CTA’s, their strategies. Margin to equity, leverage, drawdown. Sharpe and other ratios.

9/19 Options and options combinations. Straddles, strangles, spreads etc.

9/24 The Black-Scholes model. Parameters of the model. Historical volatility, implied volatility, volatility smile. Put-Call parity. More complex option strategies. Use of derivatives for investment management.

9/26 Analogy between the behavior of the stock prices and Brownian motion. Ideas of L. Bachelier. Elementary description of Brownian motion. Further properties of Brownian motion. Geometric Brownian Motion and its properties. Other models.

Take-home midterm handed. Midterm is due 10/17.

10/1 Log-Normal distribution as a resulting price distribution from Geometric Brownian Motions. Black-Scholes formula through expected payoff. American options. Early exercise. Options on dividend paying stocks, currencies and futures.

10/3 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one step binomial model. Examples.

The last day to form a group for an individual project. After the group is formed its representatives e-mail project description proposal to professor Smirnov before October 15.

10/8 Trading and hedging of options. Greeks (sensitivities with respect to the inputs of the Black-Scholes): Delta, Gamma, Theta, Vega, Rho. Trading Gamma. Hedging of other Greeks. Dynamic option replication.

10/9 No class scheduled for that day but it is the LAST DAY TO DROP A CLASS for Barnard, Columbia College, General Studies, GSAS, and Continuing Education.

10/10 Ito lemma and its use. Examples. Martingales.

10/15 Derivation of the Black-Scholes equation using risk-free portfolio. Black-Scholes price as a solution of that equation using appropriate boundary conditions.

10/17 Take-home midterm due.

10/17 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function. Examples from physics. Complex derivatives.

10/22 Kolmogorov and Fokker-Planck equations and relation to Black-Scholes equation. Application to exotic options.

(These topics are optional and will not be asked in homeworks and exams)

10/24 Further topics in derivatives.

10/29 Risk measurement and risk management. Value-At-Risk, CVAR. Calculation and usage of Value-At-Risk. Methods of calculation Value-At-Risk (covariance matrix, historical, simulation). Examples. Alternative risk measures. Factor based risk models.

10/31 Further topics in risk management. Self-study topic given for 10/31-11/7: Elements of bond math. Duration and Convexity.

11/5 University Holiday before election day. No Lecture.

11/7 Portfolio theory. Portfolio optimization with volatility and with drawdown constraints. Examples.

11/12 Examples of portfolio models. Use of portfolio theories in investment management.

Portfolio construction in practice. Long/Short investing. Traditional and alternative investments and Hedge Funds.

11/14 Portfolio insurance. Constant proportion portfolio insurance of Black-Jones-Perold. Time invariant and other portfolio insurance.

Take-home final exam handed. In-class practice final handed.

11/19 Additional topics in portfolio management in alternative investments and Hedge Funds.

11/21 No lecture. Thanksgiving 11/22.

11/26 Additional topics in portfolio management.

11/28 Additional topics in portfolio management.

12/3 Student projects presentations 7.40-10pm (Attendance compulsory)

12/5 Student projects presentations 7.40-10pm (Attendance compulsory)

12/10 Student projects presentations 7.40-10pm (Attendance compulsory)

12/12 Student projects presentations 7.40-10pm (Attendance compulsory) That is on a study day.

12/19 Wednesday. Final exam. In Class part 7.40-9.10pm. Take-Home final due.

Some books for further reading and reference:

1. Albert N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory.

2. Christina I. Ray , The Bond Market: Trading and Risk Management.

3. B. Oksendal, Stochastic Differential Equations.

4. D.Cox, H.Miller The theory of stochastic processes.

5. E. G. Haug, The complete guide to option pricing formulas, McGraw-Hill , 2006 Book+Excel Disc

6. S.Natenberg, Option Volatility and Pricing.

7. F.Fabozzi, H.Markowitz, The Theory and Practice of Investment Management, 2004

8. W.Sharpe, G. Alexander, J.Bailey, Investments, 1999.

9. Taggart, Robert A., Quantitative Analysis for Investment Management.

10. A.Damordan, Investment Valuation.

11. C. Luca, Trading in the Global Currency Markets.

12. F.Fabozzi ed, Handbook of Fixed Income Instruments.

13. H. Hothakker and P. Williamson, The Economics of Financial Markets, Oxford