Discrete Time Models in Finance W 3050 SPRING 2004

Discrete Time Models in Finance W 3050 SPRING 2004

Instructor: Professor Mikhail Smirnov

Time: Tuesday, Thursday 6.10-7.25 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 Tuesday, Thursday 9pm-10pm, by appointment.

Prerequisites: Calculus III,

Teaching Assistant: Nikos Egglezos negglez@math.columbia.edu

This course covers market conventions and instruments, Black-Scholes option pricing model, practical aspects of trading and hedging of derivatives and Bond Math.

Basic financial instruments. The distribution of the rate of return of stocks. Random walk model of stock prices, ideas of L. Bachelier, 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. Fixed Income Market Overview, Time Value of Money, Risk Measures of a Bond. Elements of Bond Math.

Texts: J.Hull, Options Futures and other derivatives Prentice Hall NJ Any Edition

Software: Excel, Mathematica 3 or higher optional.

Problem sets: Homework will be assigned on Tuesdays every 2 weeks, it is due on Tuesdays 2 weeks later. Problem sets will be distributed in class. Summary of some lectures will be distributed in class.

Midterm exam: February 12, March 25

Final exam will have 2 parts. The take-home part will be handed on April 15,it is due May 11. In-class 2 hour final exam will be given on Tuesday, May 11, 8pm to 10pm. The practice exam for in-class part will be handed on April 15.

Grading : Homework grades (30%), Midterm exam (20%), Final exam (40%), Class participation (10%).

There might be several guest speakers. They will be announced during the course. Students are welcome to attend talks by guest speakers at Finance Practitioners seminar at Math 207, 7.40-8.55pm on Tuesdays

SYLLABUS

1/20 Introduction.

1/22 Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Arbitrage. Idealized assumptions of mathematical finance vs. market reality.

1/27 Basic probability theory 1. Trading of different financial instruments.

1/29 Types of derivative securities. Futures, options, bonds, swaps, exotic derivatives.

2/3 More probability. Review of probability distributions and their properties.

2/5 Market conventions. Trading of different financial instruments. Mechanics of trading. Options and options combinations. Straddles, strangles, spreads etc.

2/10 The Black-Scholes model. Parameters of the model. Historical volatility, implied volatility, volatility smile. Put-Call parity. More complex option strategies.

2/12 Midterm I

2/17 Basic probability theory 2. Probabilistic models, random variables. Expectation, variance, standard deviation. Normal random variables. Log-normal distribution and its properties.

2/19 Continuation of basic probability theory 2. Examples

2/24 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.

2/26 Further properties of Brownian motion. Geometric Brownian Motion and its properties. Log-Normal distribution as a resulting price distribution.

3/2 Black-Scholes formula through expected payoff.

3/4 Review of key concepts learned so far.

3/9 Derivation of the Black-Scholes equation using risk-free portfolio.

3/11 Black-Scholes price as a solution of that equation using appropriate boundary conditions.

3/23 Trading and hedging of options. Greeks (sensitivities with respect to the inputs of the Black-Scholes): Delta, Gamma, Theta, Vega, Rho. American options. Early exercise. Options on dividend paying stocks, currencies and futures.

3/25 Midterm II

3/30-4/1 Bond Mathematics.

4/6 Speaker

4/8 Bond Mathematics. Continuation. (Duration, Convexity etc.)

4/13-4/15 Bond Markets.

4/15 Take-home final exam handed. In-class practice final handed.

4/20-4/22 Mortgages. Derivatives

4/27-4/29 Review.

5/11 Tuesday, 8pm-10pm Final Exam