Discrete Time Models in Finance W 3050 SPRING 2002

Discrete Time Models in Finance W 3050 SPRING 2002 Discrete Time Models in Finance W 3050 SPRING 2002

Instructor: Professor Mikhail Smirnov

Time: Tuesday, 6.10-7.25 PM

Thursday 7.40-8.55 PM with Practitioners Seminar in Math 203 (recommended only)

email smirnov@math.columbia.edu

web site www.math.columbia.edu/~smirnov

phone (212) 854-4303, fax (212) 665-0839

Office 415 Mathematics

Office hours Tuesday 9pm-10pm, Thursday 7.40-9pm and by appointment

Prerequisites: elements of calculus,

Teaching Assistant: Jesus Ruiz-Matajesusr@stat.columbia.edu

This course covers market conventions and instruments, Black-Scholes option pricing model, practical aspects of trading and hedging of derivatives. Other topics include modern portfolio theory, portfolio management, portfolio risks, Value At Risk and its applications.

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, arbitrage, risk neutralvaluation, binomial models, modifications of binomial models. Exotic options, Asians, Barrier options, Binary options. Fixed Income Market Overview, Time Value of Money, Risk Measures of a Bond. Elements of Bond Math.Portfolio theory and optimization of simple portfolios. Portfolio risks. Value At Risk and its applications. Other alternative measures of risk.

All the necessary definitions and concepts from the probability theory: random variables, normal and log-normal distributions etc,will be explained in the course.

Texts:

J.Hull, Options Futures and other derivatives Prentice Hall NJ 1999

(required)

N.Taleb, Dynamic Hedging, Wiley NY,1996

Forthcoming text:

M.Smirnov, C.Malureanu, L.Atkinson. Introduction to derivatives pricing and hedging.

Software: Excel 5.0 or higher (better for PC). Mathematica 2.2 or higher optional. Recommended Hardware: Hewlett Packard calculator HP 12C (only this model not the newer models) useful for some bond calculations. (Fair street price $60-75)

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: Take-home midterm will be handed on February 19. It is due on March 12.

Final exam will have 2 parts. The take-home part will be handed on April 16,it is due May 14. In-class 1 hour 30 min final exam will be given on Tuesday, May 14, 8pm to 10pm.

Individual and group project. Each student will be given an individual project that is due May 9. The groups of 2-4 students should be formed according to student's preferences. Topic should be discussed with instructor (appointment should be made preferably during office hours).

Grading : Homework grades (25%), Individual Project (20%), Midterm exam (15%), Final exam (25%), Class participation (15%).

Reviews : Part of the class before each of the two exams will be devoted to review.

Guest speakers: 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 203, 7.40-8.55pm on Tuesdays and Thursdays

SYLLABUS

1/22 Introductory lecture. Overview. Basic assets: cash, stocks, bonds, currencies, commodities. How they are traded. Arbitrage. Idealized assumptions of mathematical finance vs. market reality. Basic probability theory 1. Probabilistic models, random variables. Market conventions 1. Trading of different financial instruments.

1/29 Basic probability theory 2. Probabilistic models, random variables.Expectation, variance, standard deviation. Normal random variables. Types of derivative securities. Futures, options, bonds, swaps, exotic derivatives.

2/5 More probability. 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. A model of the behavior of stock prices. Options and options combinations. Straddles, strangles, spreads etc.

2/12 The Black-Scholes model. Parameters of the model. Historical volatility, implied volatility, volatility smile. Put-Call parity. More complex option strategies.

2/12 The last day to form a group for a project. After the group is formed its representatives should discuss project with professor Smirnov before the end of February.

2/19 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. Black-Scholes formula through expected payoff.

2/20 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one step binomial model. Examples. Review of key concepts learned so far.

3/5 Trading and hedging of options. Greeks (sensitivities with respect to the inputs of the Black-Scholes): Delta, Gamma, Theta, Vega, Rho.

Trading Gamma. Examples. Hedging of other greeks.

3/12 Derivation of the Black-Scholes equation using risk-free portfolio.

3/26 Black-Scholes price as a solution of that equation using appropriate boundary conditions. American options. Early exercise. Options on dividend paying stocks, currencies and futures.

4/2 Introduction to portfolio theory. Risk and return. Portfolio Management. Leverage, Sharpe ratio.

4/9 Construction of optimal portfolios. Modern portfolio theories. CAPM and APM. Examples

4/16 Value-At-Risk. Calculation and usage of Value-At-Risk. Conditional Value-At-Risk. Alternative risk measures. Examples.

4/23 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function.

4/30 Review.