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
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:
Senior TA: David Fournie fournie@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. As a rare exception projects can be individual.
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
(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 Windows, Matlab, R.
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 30. It is due on October 14.
Final exam
will have 2 parts. The take-home part will be handed on November 18, it is due December
16. In-class 1.5 hour final exam will be given on Wednesday, December 16,
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
9/9 Introductory lecture. Overview. Basic assets: cash, stocks, bonds, currencies,
commodities. How they are traded. Forward contracts. Arbitrage.
9/14 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.
9/16 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
9/21 Options and options combinations. Straddles, strangles,
spreads etc.
9/23 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.
9/28 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.
9/30 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.
9/30
Take-home midterm handed. Midterm is due 10/14.
10/5
10/7 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.
10/7 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one
step binomial model. Examples.
10/12 Trading and hedging of options. Greeks (sensitivities with respect
to the inputs of the Black-Scholes): Delta, Gamma,
Theta, Vega,
10/13 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.
10/14 Ito lemma and its use. Martingales.
10/14
Take-home midterm due.
10/19 Derivation of the Black-Scholes equation
using risk-free portfolio. Black-Scholes price as a solution
of that equation using appropriate boundary conditions.
10/26 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.
10/28 Portfolio theory II. CAPM and APM. Examples. Use of portfolio theories in
investment management. Traditional and alternative investments and Hedge Funds.
11/2
University Holiday before election day. No Lecture.
Self-study topic (handout given 10/31): Elements of bond math. Duration and
Convexity.
11/4 Active portfolio management. Forecasting and
information analysis. Portfolio construction. Long/Short investing. Portfolio
optimization.
11/9 Portfolio insurance. Constant proportion portfolio insurance
of Black-Jones-Perold. Time
invariant and other portfolio insurance.
11/11 Additional topics in portfolio management.
11/16 Further topics on Brownian motion.
11/18 Kolmogorov and Fokker-Planck equations
and relation to Black-Scholes equation. Application to exotic options.
11/23 Use of derivatives techniques. Derivatives abuses and disasters.
Take-home final exam handed. In-class practice final handed.
11/24 Additional
topics in portfolio management.
11/25 No lecture. Thanksgiving 11/26.
11/30 Special topics.
12/2 Student
projects presentations 7.40-10pm (Attendance compulsory)
12/7 Student
projects presentations 7.40-10pm (Attendance compulsory)
12/9 Student
projects presentations 7.40-10pm (Attendance compulsory)
12/14 Student
projects presentations 7.40-10pm (Attendance compulsory)
12/16 Wednesday. Final exam. In Class part
Some books
for further reading and reference:
1.
Albert N. Shiryaev, Essentials of Stochastic Finance:
Facts, Models, Theory.
2.
Christina
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,
Recommended
articles:
F.Black, M.Scholes , The pricing of options and corporate liabilities,
Journal of Political Economy , 81 (1973) 637-654