Math 339, Topics in Applied Mathematics

Fall 2008: Mathematical Finance

Tentative Syllabus

Topic References
Introduction to the course Browse in S&P guide (sections on money, stocks, and options)
The motion of stock prices, return and log return -
Discrete and continuous compound interest; present and future value Hull 4.1, 4.2; also any calculus book
Random variables, expectation, variance, the normal distribution -
Brownian motion as limit of coin tossing (random walk). Geometric Brownian motion. Handouts, also Hull 12.1--12.4
Normality and independence of returns as a consequence of GBM -
Sample mean, variance, standard deviation, confidence intervals -
Tests for normality: KS test, qq-plot -
Analyse stock data using SPSS; estimating the parameters in GBM -
Simulating GBM using Excel Handout
Covarience, correlation -
Tests for independence: scatterplot, sample correlation coefficient -
Market portfolio, indices (eg Dow-Jones, S&P 500) -
Forwards Hull 1.3
Futures Hull 1.4, also S&P guide
Introduction to options: long and short european puts and calls Hull 1.5
Bachelier (payoff) diagrams -
Put-call parity Hull 9.4
One-step binomial trees Hull Ch. 11
Risk-neutral valuation Hull 11.2
Two-step binomial trees Hull 11.3
Multi-step binomial trees and Derivagem 11.8
From trees to the BS formula Handout
Black-Scholes formula Hull 13.8
Stochastic calculus, Ito's lemma Hull 13.5
Black-Scholes PDE Hull 13.6
American options, evaluation with a tree Hull 11.5
Comparing actual option prices with their BS values; implied volatility, volatility smiles Hull 13.11
The Greeks Hull Ch. 15
Exotic options Hull Ch. 22
Monte Carlo methods Hull 17.6