Elementary Introduction to Mathematics of Finance

Elementary Introduction to Mathematics of Finance.

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 415 Mathematics

Office hours Monday 9pm-10pm

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

Teaching Assistant: Kenji Kamizono kenji@math.columbia.edu

This course focuses on mathematical methods in pricing of derivative securities 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.

Bonds: Duration, Modified duration, Convexity etc. Swaps. Yield curve

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.

Although many questions of "continuous time finance" require more advanced mathematical methods of stochastic calculus, it is still possible to develop the mathematical theory up to a certain point using only elementary methods and basic calculus. The course emphasize this elementary approach.

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 1995

N.Taleb, Dynamic Hedging, Wiley NY,1996

Additional mathematical articles will be distributed and assigned in class.

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.

Hardware: Hewlett Packard calculator HP 12C (only this model no

newer models) required in November for bond calculations. (Fair street price $60-75)

Problem sets: Homework will be assigned on Wednesdays every 2 weeks, it is due on Wednesdays 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 12. 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 December 2,it is due December 21. In-class 1 hour final exam will be given on Monday December 21, 8pm to 9pm. The practice exam for in-class part will be handed on December 2.

Individual project. Each student will be given an individual project that is due December 2. The topic should be discussed with instructor (appointment should be made preferably during office hours) before September 30.

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

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

Guest speakers: there will be several guest speakers. They will be announced during the course.

SYLLABUS

9/9 Introductory lecture

9/14-9/16 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. Expectation, variance, standard deviation. Normal random variables. Types of derivative securities. Futures, options, swaps, exotic derivatives.

9/21-9/23 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.

9/28-9/30 Options and options combinations. Straddles, strangles, spreads etc. The Black-Scholes model. Parameters of the model. Historical volatility, implied volatility, volatility smile. Put-Call parity. More complex option strategies.

9/30 The last day to get an individual project.

10/5-10/7 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. Log-Normal distribution as a resulting price distribution. Black-Scholes formula through expected payoff.

10/12 Take-home midterm handed.

10/12-10/14 Risk-Free portfolio. Risk-Neutral valuation of options. (Key concept). A one step binomial model. Examples. Review of key concepts learned so far.

10/19-10/21 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.

10/21 Take-home midterm due.

10/26-10/28 Derivation of the Black-Scholes equation using risk-free portfolio. 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.

11/2-11/4 Election holiday. No lecture 11/2 and 11/4

11/9-11/11 Bond Math. Coupon bearing bonds, zero coupon bonds, yield, duration, convexity. Bootstrap method. Yield curve. Interest rate swaps, caps, floors and swaptions.

11/16-11/18 Modern portfolio theories. CAPM and APM. Examples.

11/25 No lecture

11/30-12/2 Further topics on Brownian motion. Monte Carlo simulations. Examples. Transition probability function. Examples from physics. Application to complex derivatives. Kolmogorov and Fokker-Planck equations and relation to Black-Scholes equation. Application to barrier options. Greeks near the barrier.

12/2 Take-home final exam handed. In-class practice final handed. Individual projects due.

12/7 Review.

12/9 Special topics in derivatives.

12/14 Special topics in derivatives.

Texts:

Forthcoming text:

Further reading: