| Week | Date | Finance | Mathematics | ||
|---|---|---|---|---|---|
| 1 | 7 Sept 2002 |
Introduction: The relation between mathematics and finance, a few
successful mathematical models in finance, a list of useful mathematics in
finance. Option pricing: Definition of an Option, Black-Scholes formula to price a European call option, The defintion of cumulative normal distribution function and its properties. | Elementary Algebra: Number systems, Multiplication and factorization, Fraction. | ||
| 2 | 14 Sept 2002 | Black-Scholes formula: European call, Dimension analysis, How to remember the formula, Put-call parity, Formula for a European put. | Elementary Algebra: Root, Ratio, Inequality, Determinant, System of linear equations, Quadratic equation with one unknown, cubic and quartic equations with one unknown, Progression, Exponent, Logarithm. | ||
| 3 | 21 Sept 2002 | The Chinese Mid-Autumn Festival. No class. Assignment 1: Read a Journal of Finance paper talking about Fischer Black written by Nobel prize laureates Merton and Scholes. Write a reading report (at least 3 A4 pages). The report is due in class next week. | None | ||
| 4 | 28 Sept 2002 | Review of a few properties of cumulative normal distribution function, Black-Scholes formula for a European call, Asymptotic analysis, Geometric picture of a call price, Put-call parity, Formula for a European put, Geometric picture of a put price, Greeks. | Calculus: a few basic functions, Limit, The limit for e, Differentiation. | ||
| 5 | 5 Oct 2002 | Deriving Delta, Theta from Black-Scholes formula | Calculus: Differentiation of multi-variable function, Mean value theorem, Taylor's formula, Integration, Rules, Indefinite integration, Definite integration. | ||
| 6 | 12 Oct 2002 | Modeling the random behavior of stock price, Assignment 2 is due in class in two weeks | Stochastic Calculus: Brownian motion and its properties, Understanding Ito's Lemma, (dB_t)^2=dt +O((dt)^2), Expectation of the power of Brownian motion, Expectation of exponential of Brownian motion. | ||
| 7 | 19 Oct 2002 | Binomial tree method, risk-neutral probability. | Stochastic Calculus in Finance: Probability triple, \sigma algebra, filtration, adapted process, random variable, martingale. | ||
| 8 | 26 Oct 2002 | Risk-neutral valuation formula, derive Black-Scholes formula for European call and put. | Martingale, Probability density function of normal and lognormal random numbers. | ||
| 9 | 2 Nov 2002 | Deriving Black-Scholes equation, Forward price transform, Price-moneyness transformation, Green's function for standard heat equation. | Partial Differential Equation (PDE), heat equation with variable coefficient, Delta function, Green's function. | ||
| 10 | 9 Nov 2002 | Solving standard heat equation with a particular initial condition. Solution to Black-Scholes PDE. Lecture notes on option pricing | Solving PDE | ||
| 11 | 16 Nov 2002 | Exotic options, Assignment 3 is due in class in two weeks. | More on PDE | ||
| 12 | 23 Nov 2002 | Interest rate modeling | Solving ODE | ||
| 13 | 30 Nov 2002 | Vasicek model | Ornstein Uhlenbeck process | ||
| 14 | 7 Dec 2002 | Review | Summary |