Selected Topics in Applied Mathematics I (Mathematical Finance)

Autumn 2003

Professor Paul Feehan

Course Outline

Mathematical tools | Derivatives and the Black-Scholes model | Numerical methods | Interest rate models | Course Home

A. Mathematical tools

The first part of the course will focus primarily on required mathematical tools, though we shall draw our examples from finance:

1. Introduction to probability theory, random variables, and stochastic processes.

2. Stochastic processes, Markov processes, Poisson process, Brownian motion

3. Measure theory, martingales, Itô integral, Itô formula, martingale representation theorem

4. Stochastic differential equations

5. Girsanov’s theorem

6. Stopping times, first passage times, optimal stopping

References:

· Baxter & Rennie

· Karlin & Taylor I

· Karatzas & Shreve

· Øksendal

· Papoulis

· Shreve

B. Derivatives and the Black-Scholes model

The second part of the course comprises an introduction to the basic ideas of mathematical finance, with extensions and applications of the theory we developed in Part A.

1. Black-Scholes model, hedging, arbitrage

2. Black-Scholes formulae for European calls and puts

3. Formulae for the “Greeks”

4. Early exercise and American options

5. Path dependent options: Barrier, Asian, and lookback options

6. Extending the Black-Scholes model

7. Stochastic volatility

8. Jump diffusion processes

9. Portfolio insurance

References:

· Wilmott

· Hull

· Baxter & Rennie

C. Numerical methods

The third part of the course comprises an introduction to numerical methods for solving the stochastic differential equations and models introduced in Part B.

1. Finite difference methods

2. Monte Carlo simulation

References:

· Wilmott

· Hull

· Clewlow & Strickland

· Research papers

D. Interest-rate models

The last part of the course comprises an introduction to interest-rate models, subject to availability of time.

1. Fixed-income instruments

2. Interest rate modeling

3. Hull & White model

4. Heath, Jarrow & Morton model

References:

· Wilmott

· Hull