Elementary Introduction to Mathematics of Finance W 4071

Elementary Introduction to Mathematics of Finance W 4071.

Head 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 Wednesday 9pm-10pm and by appointment

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

Teaching Assistant: TBA

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.

Fixed income markets (Professor Costas Hamakiotes) Fixed Income Market Overview, Time Value of Money, Forward Pricing, Risk Measures of a Bond (Duration, Dollar Value of a Basis Point (DV01), Convexity, Analogies to Options (Delta, Gamma, etc.). Duration-Weighed Trades. Aggregating a Portfolio. The Yield Curve. Swaps, Floating Rate Instruments, Mortgages, Bonds With Embedded Optionalities.

Interesting Stories From History (The Salomon Brothers Treasury Scandal of 1991, The Blow Up of Hedge Funds of 1998)

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.

Texts:

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

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.

Recommended Hardware: Hewlett Packard calculator HP 12C (only this model not

newer models) useful for some 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 1,it is due December 20. In-class 1 hour final exam will be given on Monday December 20, 8pm to 9pm. The practice exam for in-class part will be handed on December 1.

Individual and group project. Each student will be given an individual project that is due December 13. The groups of 2-5 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 : 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

9/8 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.

9/13 Fixed Income Markets 1. Overview. Prof. Costas Hamakiotes, Lehman Brothers.

9/15 Basic probability theory 1. Probabilistic models, random variables.Expectation, variance, standard deviation. Normal random variables. Types of derivative securities. Futures, options, swaps, exotic derivatives.

9/20- Fixed Income Markets 2. Prof. Costas Hamakiotes.

9/22 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.

9/27 Fixed Income Markets 3. Prof. Costas Hamakiotes.

9/29 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.

9/29 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.

10/4 Fixed Income Markets 4. Prof. Costas Hamakiotes.

10/6 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.

10/11 Take-home midterm handed.

10/11- Fixed Income Markets 5. Prof. Costas Hamakiotes.

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.

10/18 Fixed Income Markets 6. Prof. Costas Hamakiotes.

10/20 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.

10/20 Take-home midterm due.

10/25- Fixed Income Markets 7. Prof. Costas Hamakiotes.

10/27 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.

11/1-11/3 Election holiday. No lecture 11/1 and 11/3

11/8-11/10 Portfolio Management. Value-At-Risk. Calculation and usage of Value-At-Risk. Modern portfolio theories

11/15-11/17 Modern portfolio theories. CAPM and APM. Examples.

11/22 Guest speaker or lecture moved because of the guest speaker.

11/24 No lecture

11/29-12/1 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.

12/1 Take-home final exam handed. In-class practice final handed.

12/6 Review.

12/8 Special topics in derivatives.

12/13 Special topics in derivatives.

12/13 Individual Projects Due

Further reading:

Probability theory and stochastic processes:

1. B. Oksendal, Stochastic Differential Equations, Springer, 1995

2. D.Cox, H.Miller The theory of stochastic processes, L 1965

Bond Math

1. Christina Ray, Bond Markets, 1997

Complex options

1. E. G. Haug, The complete guide to option pricing formulas, McGraw-Hill , 1997 Book+Excel Disc

Practical aspects of options trading

1. S.Natenberg, Option Volatility and Pricing. Advanced Trading Strategies & Techniques, Probus,1994 or later

CAPM etc.,

1. A.Damordan, Investment Valuation. Wiley 1996

Additional recommended books

C. Luca, Trading in the Global Currency Markets, New York Institute of Finance

F.Fabozzi ed, Handbook of Fixed Income Instruments, McGraw Hill

H. Hothakker and P. Williamson, The Economics of Financial MarketsOxford

Highly recommended:

Campbell, Lo and MacKinlay, The Econometrics of Financial Markets, Princeton University Press