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, fax (212) 665-0839

Office 425 Mathematics

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

Prerequisites: working knowledge of calculus, knowledge of elementary probability theory.

Teaching Assistants:

Nikos Egglezos negglez@math.columbia.edu

Irina Goia irina@cpw.math.columbia.edu

and David Swinarski swinarsk@cpw.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 2-5 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, 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.

Risk Measures. Value-At-Risk. Portfolio construction. 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. Paul Wilmott, Paul Wilmott on Quantitative Finance, 2 Volume Set, John Wiley & Sons; ISBN:0471874388

Reading of chapters from books 1 and 2 will be assigned periodically.

Recommended Additional Text: N.Taleb, Dynamic Hedging, Wiley NY, 1996. Additional mathematical articles will be distributed and assigned in class.

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 (Fair street price $60-75). This calculator is recommended but not required.

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 5. It is due on October 19.

Final exam will have 2 parts. The take-home part will be handed on November 21,it is due December 21. In-class 1.5 hour final exam will be given on Wednesday, December 21, 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 the other exam.

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

9/7 Introductory lecture. Overview. Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Forward contracts. Arbitrage.

9/12 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.

9/14 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

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

9/21 Portfolio theory I.

9/26 Portfolio optimization with volatility and with drawdown constraints. Guest Speaker.

9/28The 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 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 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.

10/5 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.

10/5 Take-home midterm handed. Midterm is due 10/19.

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

10/11 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.

10/12 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.

10/17 Ito lemma and its use.

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.

10/19 Take-home midterm due. Martingales. Martingale methods.

10/24 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.

10/26 Portfolio theory II. CAPM and APM. Examples. Use of portfolio theories in investment management.

Applications to long/short stock portfolios. Traditional and alternative investments and Hedge Funds.

10/31 Portfolio insurance. Constant proportion portfolio insurance of Black-Jones-Perold. Time invariant and other portfolio insurance.

11/2 Elements of bond math. Duration and Convexity. Bond options.

11/7 University Holiday before election day. No Lecture.

11/9 Guest speaker or lecture moved from another day because of the guest speaker.

11/14 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function. Examples from physics. Application to complex derivatives.

11/16 Kolmogorov and Fokker-Planck equations and relation to Black-Scholes equation.

11/21 Application to exotic options. Take-home final exam handed. In-class practice final handed.

11/23 No lecture. Thanksgiving 11/24.

11/28 Use of derivatives techniques. Derivatives abuses and disasters.

11/30 Special topics.

12/5 Special topics.

12/7 Student projects presentations 7.40-10pm (Attendance compulsory)

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

12/14 Student projects presentations 7.40-10pm (Attendance compulsory)

12/21 Wednesday. Final exam. In Class part 7.40-9.10pm. Take-Home final due.