ELEMENTARY INTRODUCTION TO MATHEMATICS OF FINANCE W 4071.
This Course is divided into 2 subcourses ELEMENTARY INTRODUCTION TO MATHEMATICS OF FINANCE taught on Mondays 7.40-8.55 by professor Smirnov and QUANTITATIVE METHODS IN INVESTMENT MANAGEMENT taught on Wednesdays 7.40-8.55 by professor Greyserman, Visiting Professor at Columbia University and a partner at a hedge fund Mint Investment Management
Students taking class for credit must take both parts of the course for 3 credits. Their grade will be calculated as 50% for the Introduction to Mathematics of Finance part and 50% for Quantitative Methods in Investment Management part.
Grading For Introduction to Mathematics of Finance Part (Total 50%): Homework grades (15%), Midterm exam (10%), Final exam both parts (20%), Class participation (5%).
Grading For Quantitative Methods in Investment Management (Total 50%): Individual or Group project 50%.
Individual or group project for Quantitative Methods in Investment Management. Each student will be given a project. The groups of 2-5 students should be formed according to student’s preferences. Topic should be discussed with professor Greyserman (appointment should be made preferably during office hours).
SYLLABUS AND ADDITIONAL INFORMATION FOR INTRODUCTION TO MATHEMATICS OF FINANCE SECTION
This course focuses on mathematical methods in pricing of derivative securities, portfolio management and on other related questions of mathematical finance. The emphasis is on the basic mathematical ideas and practical aspects.
Basic financial instruments. The distribution of the rate of return of stocks. Random walk model of stock prices, ideas of L. Bachelier and B. Mandelbrot, Brownian motion. Historical data, normal and log-normal distributions. Derivative securities: options, futures, swaps, exotic derivatives.
Black-Scholes formula, its modifications. Applications. Trading strategies involving options, straddles, strangles, spreads etc.Trading and hedging of derivatives. Greeks: Delta, Gamma, Theta, Vega, Rho. Trading Gamma. Hedging of other greeks.
Elementary derivation of Black-Scholes formula, arbitrage, risk neutralvaluation, binomial models, modifications of binomial models. Exotic options, Asians, Barrier options, Binary options.
Fixed Income Market Overview. Duration and Convexity.
At the very end the course we will discuss more advanced topics related to partial differential equations and stochastic differential equations, these topics will not be included in the final exam for undergraduates taking this class.
All the necessary definitions and concepts from the probability theory: random variables, normal and log-normal distributions etc,will be explained in the course.
Main Text: J.Hull, Options Futures and other derivatives Prentice Hall NJ 1999
Optional Text: N.Taleb, Dynamic Hedging, Wiley NY, 1996. Additional mathematical articles will be distributed and assigned in class.
Software: Excel 97 or higher (better for PC). Mathematica 3.0 or higher optional.
Recommended Hardware: Hewlett Packard calculator HP 12C is useful for some bond calculations. (Fair street price $60-75)
Problem sets: Homework will be assigned on Mondays every 2 weeks, it is due on Mondays 2 weeks later. Problem sets will be distributed in class. Summary of lectures will be distributed in class every 2 weeks.
Midterm exam: Take-home midterm will be handed on October 8. It is due on October 22. Practice midterm exam will be available a week before the actual exam.
Final exam will have 2 parts. The take-home part will be handed on November 26,it is due December 19. In-class 1 hour final exam will be given on Wednesday, December 19
Reviews : Part of the class before each of the two exams will be devoted to review.
Guest speakers: there will be guest speakers. They will be announced during the course.
SYLLABUS
9/5 Introductory lecture. Overview. Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Forward contracts. Arbitrage.
9/10 Probabilistic models, random variables. Distribution of percentage returns and prices. Idealized assumptions of mathematical finance vs. market reality. Expectation, variance, standard deviation. 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 Quantitative Methods in Investment Management, Professor Greyserman.
9/15 Saturday (Highly recommended to pre-register and attend) Columbia Practitioners Conference on Mathematics of Finance. Havermeyer Hall 9am (See appendix for details).
9/17 Futures, options, other derivatives. Mechanics of the futures markets. Margins, margin calls. Contango and backwardation.
9/19 Quantitative Methods in Investment Management, Professor Greyserman
9/24 Options and options combinations. Straddles, strangles, spreads etc. The Black-Scholes model. Parameters of the model. Historical volatility, implied volatility, volatility smile. Put-Call parity. More complex option strategies.
9/26 Quantitative Methods in Investment Management, Professor Greyserman
10/1 The last day to form a group for an individual project. After the group is formed its representatives should discuss project with professor Greyserman before the middle of October.
10/1 Analogy between the behavior of the stock prices and Brownian motion. Ideas of L. Bachelier and B. Mandelbrot. Other models. Elementary description of Brownian motion. Further properties of Brownian motion. Geometric Brownian Motion and its properties. Log-Normal distribution as a resulting price distribution. Black-Scholes formula through expected payoff.
10/3 Quantitative Methods in Investment Management, Professor Greyserman
10/8 Take-home midterm handed.
10/8 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one step binomial model. Examples. Review of key concepts learned so far.
10/10 Quantitative Methods in Investment Management, Professor Greyserman
10/15 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.
10/17 Quantitative Methods in Investment Management, Professor Greyserman
10/22 Take-home midterm due.
10/22 Ito lemma and its use. Derivation of the Black-Scholes equation using risk-free portfolio. Black-Scholes price as a solution of that equation using appropriate boundary conditions.
10/24 Quantitative Methods in Investment Management, Professor Greyserman
10/29 American options. Early exercise. Options on dividend paying stocks, currencies and futures. Elements of bond math. Duration and Convexity. Bond options.
10/31 Quantitative Methods in Investment Management, Professor Greyserman
11/5-11/7 Election day holiday. No Lecture 11/5. No Lecture 11/7
11/12 Risk measurment and risk management. Value-At-Risk. Calculation and usage of Value-At-Risk. Examples. Modern portfolio theories. CAPM and APM. Examples. Portfolio insurance. Constant proportion portfolio insurance of Black-Jones-Perold. Time invariant (or drawdown control) portfolio insurance.
11/14 Quantitative Methods in Investment Management, Professor Greyserman
11/19 Guest speaker or lecture moved because of the guest speaker.
11/21 No lecture. Thanksgiving 11/22
11/26 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function. Examples from physics. Application to complex derivatives. Kolmogorov and Fokker-Planck equations and relation to Black-Scholes equation. Application to barrier options.
11/26 Take-home final exam handed. In-class practice final handed.
11/28 Quantitative Methods in Investment Management, Professor Greyserman
12/3 Special topics in derivatives. Review.
12/5 Quantitative Methods in Investment Management, Professor Greyserman
12/19 (Wednesday) Final exam. (In Class part) 7.40-9.10pm. Take-Home final due. Individual projects due.
Further reading:
1. B. Oksendal, Stochastic Differential Equations, Springer, 1995
2. D.Cox, H.Miller The theory of stochastic processes, L 1965
3. Christina Ray, Bond Markets, 1997
4. E. G. Haug, The complete guide to option pricing formulas, McGraw-Hill , 1997 Book+Excel Disc
5. S.Natenberg, Option Volatility and Pricing. Advanced Trading Strategies & Techniques, Probus,1994 or later
6. A.Damordan, Investment Valuation. Wiley 1996
7. C. Luca, Trading in the Global Currency Markets, New York Institute of Finance
8. F.Fabozzi ed, Handbook of Fixed Income Instruments, McGraw Hill
9. H. Hothakker and P. Williamson, The Economics of Financial Markets, Oxford
Highly recommended book:
Campbell, Lo and MacKinlay, The Econometrics of Financial Markets, Princeton University Press
Recommended articles:
F.Black, M.Scholes , The pricing of options and corporate liabilities, Journal of Political Economy , 81 (1973) 637-654
Quantitative Methods in Investment Management, Professor Greyserman
This course will survey the field of quantitative investment strategies from a "buy side" perspective, through the eyes of portfolio managers, analysts, and investors. Financial modeling often involves avoiding complexity in favor of simplicity and practical compromise. The "buy side" of the marketplace is dominated less by highly rigorous mathematics or miraculous discoveries, and more by a mix of analytical and financial understanding, computation, sensible risk management, and a general sense of humbleness in search for an "edge" in investing and performance at reasonable risk/reward levels.
The purpose of this course is to give students direct exposure to those problems facing "buy side" quantitative analysts on Wall Street. In this practically oriented course, we combine all necessary material scattered in finance, computer science and statistics into a project-based curriculum, which give students hands-on experience to solve real world problems in portfolio management.
Students will work with market and historical data to develop and test trading and risk management strategies with an eye towards the practical considerations of financial data modeling, as well as real-world considerations and limitations.
In this course, we will wear the hat of a buy-side quant (e.g. someone hired by a mutual fund or a hedge fund, etc…), whose responsibility is to develop two profitable quantitative investment strategies over the course of 3 months (length of the course). Programming projects are required to complete this course.
Topics Covered :
Data Mining
Discussion of appropriate topics in Data Aggregation, Data Analysis and Data Mining in search for the investment "edge". Emphasis on identification and mitigation of hidden biases (e.g. selection bias) in analyzing data. We will collect and analyze fair amounts of historical data.
Statistical Validation
We will employ a variety of tools, including back-tests, in-sample/out-sample comparisons and Monte Carlo analysis, to study a strategy’s robustness and sensitivity to a given choice of parameters.
Benchmarking, Attribution and Performance Analysis
Discussion of appropriate topics in style and performance analysis and general evaluation of strategy performance against objectives and benchmarks. We will also discuss appropriateness of various reward/risk measures (e.g. Sharpe Ratio and other statistics) as they pertain to specific strategies and objectives.
Transaction Cost Model
We will develop practical transaction cost models based on evaluation of the chosen markets. We will also discuss how to incorporate various practical limitations into a proper model.
Portfolio Construction and Optimization
Given a perceived edge and trading approach, we will study appropriate mechanisms for constructing and managing the portfolio over time.
Analysis of Leverage
With leverage (explicit or implicit) being a common source of actual (or hidden) risk, we will focus on analyzing hidden leverage due to potential non-linearities in portfolio relationships. This is especially relevant for any hedge-fund types of strategies or strategies employing futures or other "non-standard" instruments.
Risk Management
We will survey VAR-based and other approaches to risk management. As with the whole course, the emphasis will be from a "buy"-side perspective, i.e. we will not be concerned with risk from the point of view of fraud, business risks, etc... rather we will be concerned as investment managers to manage the risk of the portfolio on a daily basis.
Grading emphasis will be placed on the completeness and rigor of the student’s approach. During the length of the course, several industry speakers will be invited to speak on various issues in quantitative portfolio management.
Instructor : Dr. Alex Greyserman is currently principal in a private hedge-fund investment partnership. Over the past 5 years, he was Chief Investment Officer a 500mil+ hedge fund, with responsibility for research, development, and implementation of all trading strategies. Dr. Greyserman holds a MS in Electrical Engineering from Columbia University and a Ph.D. in Statistics from Rutgers University.
Prerequisites and Programming :
Students are expected to have completed an intro course to Mathematical Finance or its equivalent. Sufficient programming skills to carry out data-intensive analysis are required. Possible programming languages include C/C++/Perl/Matlab, and others. Given sufficient proficiency, it may be possible to conduct the analysis in Excel.
Reading Materials :
Samplings from the following books will be provided for valuable reading materials:
1. Schwager, J. Market Wizards (1990)
2. Schwager, J. New Market Wizards (1994)
3. Schwager, J. Stock Market Wizards (2000)
4. O'Shaughnessy, James P., What Works on Wall Street, (McGraw Hill, 1996)
5. Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk, (Wiley, 1996)
6. Bernstein, Peter L. Capital Ideas (Free Press Paperback, 1993)
7. Fabozzi, Frank Investment Management (Prentice Hall, 1995)
8. Haugen, Robert A. The Inefficient Stock Market (Prentice Hall, 1998)
9. Haugen, Robert A. Beast on Wall Street (Prentice Hall, 1999)
10. Sharpe, William F., Gordon J. Alexander, Investments (Prentice Hall, 1999)
11. Taggart, Robert A., Quantitative Analysis for Investment Management (Prentice Hall, 1996)