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 7pm-7.40pm, 9pm-10pm and by appointment

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

Teaching Assistants: Nikos Egglezos negglez@math.columbia.edu, Sampsa Samila ssamila@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.

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 (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 7. It is due on October 21. 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 25,it is due December 18. In-class 1.5 hour final exam will be given on Wednesday, December 18, 7.40-9.10pm

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

9/4 Introductory lecture. Overview. Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Forward contracts. Arbitrage.

9/9 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.

9/11 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.

9/18 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.

9/25 Investments. Traditional long investments. Long/Short and Market Neutral investments. Arbitrage strategies. Use of derivatives for investment management.

9/30 The last day to form a group for an individual project. After the group is formed its representatives should discuss project with professor Smirnov before the middle of October.

9/30 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.

10/2 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.

10/7 Take-home midterm handed.

10/7 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one step binomial model. Examples. Review of key concepts learned so far.

10/9 Trading and hedging of options. Greeks (sensitivities with respect to the inputs of the Black-Scholes): Delta, Gamma, Theta, Vega, Rho.

10/14 Trading Gamma. Hedging of other greeks. Dynamic option replication.

10/16 Ito lemma and its use.

10/21 Take-home midterm due.

10/21 Derivation of the Black-Scholes equation using risk-free portfolio. Black-Scholes price as a solution of that equation using appropriate boundary conditions.

10/23 Risk measurment and risk management. Value-At-Risk. Calculation and usage of Value-At-Risk. Methods of calculation Value-At-Risk (covariance matrix, historical, simulation). Examples. Alternative risk measures.

10/28 Modern portfolio theories. CAPM and APM. Examples.

10/30 Use of portfolio theories in investment management.

11/4-11/6 Election day holiday. No Lecture 11/4. No Lecture 11/6

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

11/13 Elements of bond math. Duration and Convexity. Bond options.

11/18 Guest speaker or lecture moved because of the guest speaker.

11/20 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function. Examples from physics. Application to complex derivatives.

11/25 Kolmogorov and Fokker-Planck equations and relation to Black-Scholes equation. Application to barrier options.

11/25 Take-home final exam handed. In-class practice final handed.

11/28 No lecture. Thanksgiving 11/28

12/2 Universe of investments. Modern investment management. Use of derivatives techniques.

12/4 Special topics in derivatives. Review.

12/18 Wednesday. Final exam. In Class part 7.40-9.10pm. Take-Home final due. Individual projects due.