INTRODUCTION TO MATHEMATICS OF
FINANCE W 4071.
Instructor: Professor Mikhail Smirnov
Time: Monday, Wednesday
email smirnov@math.columbia.edu
web site www.math.columbia.edu/~smirnov
phone (212)
854-4303, fax (212) 665-0839
Office 425 Mathematics
Office hours: Monday, Wednesday 9pm-10.00pm and by appointment
Prerequisites: working knowledge
of calculus, knowledge of elementary probability theory.
Teaching Assistants:
David Swinarski swinarsk@math.columbia.edu
Irina Goia irina@math.columbia.edu
Yun Zhou zhou@math.columbia.edu
Grading: Homework grades (20%), Midterm exam (20%), Final
exam both parts (25%), Individual or Group project
25%, Class participation (10%).
Each student will be given a project. The groups of 2-4
students should be formed according to student’s preferences. Topic
should be discussed with professor Smirnov (appointment should be made
preferably during office hours).
SYLLABUS AND ADDITIONAL INFORMATION
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,
Elementary derivation of Black-Scholes
formula, arbitrage, risk neutralvaluation, binomial
models, modifications of binomial models. Exotic
options.
Risk Measures. Value-At-Risk, Conditional
Value-At-Risk. Other measures. Factor based
models of risk.
Active portfolio management. Forecasting and information analysis. Portfolio
construction. Long/Short investing. Portfolio optimization.
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.
Required Main Texts:
1. J.Hull,
Options Futures and other derivatives Prentice Hall NJ, 6th Edition (previous
editions as well as international Editions are acceptable)
2.
R.Grinold, R.Kahn, Active Portfolio Management, McGraw-Hill, 1999
3.
Paul Wilmott, Paul Wilmott on Quantitative Finance, 3 Volume Set, John Wiley
& Sons; ISBN:0470018704
Recommended but nor required additional text: N.Taleb, Dynamic Hedging,
Software: Excel for PC, Matlab
6.0 or higher optional and Mathematica 4.0 or higher
optional.
Hardware: Hewlett Packard calculator HP 12C is useful for
some bond calculations and widely used on Wall Street (
Problem sets: Homework will be assigned on Mondays every 2
weeks, 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 3.
It is due on October 17.
Final exam will have 2 parts. The take-home part will be
handed on November 19,it is due December 19. In-class
1.5 hour final exam will be given on Wednesday, December 19, 7.40-9.10pm, The final exam is compulsory and can not be rescheduled
earlier or later. If there are conflicts with other exams please reschedule
other exams.
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 Futures, options, other derivatives. Mechanics of the futures markets. Margins, margin calls. Contango and
backwardation. Futures
trading. CTA’s, their strategies, risk
management of CTA’s, Margin to Equity,
leverage, drawdown. Sharpe and other
ratios. Basics of portfolio optimization
9/17
Options and options combinations. Straddles, strangles, spreads etc.
9/19 The Black-Scholes
model. Parameters of the model. Historical volatility, implied volatility, volatility smile.
Put-Call parity. More complex option
strategies. Investments. Traditional
long investments. Long/Short and Market Neutral
investments. Arbitrage strategies. Use of derivatives for investment management.
9/24 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.
9/26 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.
10/1 Portfolio theory I.
10/3 Portfolio optimization with volatility and with
drawdown constraints.
10/3 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 the middle of October.
10/3 Take-home midterm handed. Midterm is due 10/17.
10/8 Risk-Free portfolio. Risk-Neutral valuation of options. (Key
concept). A one step binomial model. Examples.
10/9 No class scheduled for that day but it is the LAST DAY
TO DROP A CLASS for Barnard, Columbia College, General Studies, SIPA, GSAS, and
Continuing Education.
10/10 Trading and hedging of options. Greeks
(sensitivities with respect to the inputs of the Black-Scholes):
Delta, Gamma, Theta, Vega,
10/15 Ito lemma and its use. Martingales.
10/17 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.
10/22 Risk measurment 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.
10/24 Portfolio theory II. CAPM and APM. Examples. Use of portfolio theories in investment management. Traditional and alternative investments and Hedge Funds.
10/29 Active portfolio management. Forecasting and information analysis. Portfolio
construction. Long/Short investing. Portfolio optimization.
10/31Portfolio insurance. Constant proportion portfolio insurance of Black-Jones-Perold. Time invariant and other
portfolio insurance.
11/5 University Holiday before election
day. No Lecture. Self-study topic (handout given 10/31): Elements of
bond math. Duration and Convexity.
11/7 Guest speaker or lecture moved from another day because
of the guest speaker on another day.
11/12 Additional topics in portfolio management.
11/14 Further topics on Brownian motion.
11/19 Kolmogorov and
Fokker-Planck equations and relation to Black-Scholes
equation. Application to exotic options.
Take-home final exam handed. In-class practice final handed.
11/21 No lecture.
Thanksgiving 11/22.
11/26 Use of derivatives techniques. Derivatives abuses and disasters.
11/28 Special topics.
12/3 Special topics.
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)
12/19 Wednesday. Final
exam. In Class part
Some books for further reading and reference:
1. Albert N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory . World Scientific Pub Co; ISBN: 9810236050
2. Christina
3. B. Oksendal, Stochastic
Differential Equations, Springer, 1995
4. D.Cox, H.Miller
The theory of stochastic processes, L 1965
5. E. G. Haug, The complete guide to
option pricing formulas, McGraw-Hill , 2006 Book+Excel Disc
6. S.Natenberg, Option Volatility
and Pricing. Advanced Trading Strategies & Techniques, Probus,1994 or later
7. Fabozzi, Frank, Investment
Management, Pearson, 1998
8. Sharpe, William F., Gordon J. Alexander, Investments,
Prentice Hall, 1999
9. Taggart, Robert A., Quantitative Analysis for Investment
Management, Prentice Hall, 1996
10. A.Damordan, Investment
Valuation. Wiley 1996
11. C. Luca, Trading in the Global Currency Markets, New
York Institute of Finance
12. F.Fabozzi ed, Handbook
of Fixed Income Instruments, McGraw Hill
13. H. Hothakker and P.
Williamson, The Economics of Financial Markets,
Recommended articles:
F.Black, M.Scholes , The pricing of
options and corporate liabilities, Journal of Political Economy , 81 (1973)
637-654