Stochastic Processes and the Mathematics of Finance

Stochastic Processes and the Mathematics of Finance

Jonathan Block April 1, 2008

2

Information for the class

Office: DRL3E2-A Telephone: 215-898-8468 Office Hours: Tuesday 1:30-2:30, Thursday, 1:30-2:30. Email: blockj@math.upenn.edu References:

1. Financial Calculus, an introduction to derivative pricing, by Martin Baxter and Andrew Rennie.

2. The Mathematics of Financial Derivatives-A Student Introduction, by Wilmott, Howison and Dewynne.

3. A Random Walk Down Wall Street, Malkiel.

4. Options, Futures and Other Derivatives, Hull.

5. Black-Scholes and Beyond, Option Pricing Models, Chriss

6. Dynamic Asset Pricing Theory, Duffie

I prefer to use my own lecture notes, which cover exactly the topics that I want. I like very much each of the books above. I list below a little about each book.

1. Does a great job of explaining things, especially in discrete time.

2. Hull--More a book in straight finance, which is what it is intended to be. Not much math. Explains financial aspects very well. Go here for details about financial matters.

3. Duffie-- This is a full fledged introduction into continuous time finance for those with a background in measure theoretic probability theory. Too advanced. But you might want to see how our course compares to a PhD level course in this material.

3

4. Wilmott, Howison and Dewynne--Immediately translates the issues into PDE. It is really a book in PDE. Doesn't really touch much on the probabilistic underpinnings of the subject.

Class grade will be based on homework and a take-home final. I will try to give homework every week and you will have a week to hand it in.

4

Syllabus

1. Probability theory. The following material will not be covered in class. I am assuming familiarity with this material (from Stat 430). I will hand out notes regarding this material for those of you who are rusty, or for those of you who have not taken a probability course but think that you can become comfortable with this material. (a) Probability spaces and random variables. (b) Basic probability distributions. (c) Expectation and variance, moments. (d) Bivariate distributions. (e) Conditional probability.

2. Derivatives. (a) What is a derivative security? (b) Types of derivatives. (c) The basic problem: How much should I pay for an option? Fair price. (d) Expectation pricing. Einstein and Bachelier, what they knew about derivative pricing. And what they didn't know. (e) Arbitrage and no arbitrage. The simple case of futures. Arbitrage arguments. (f) The arbitrage theorem. (g) Arbitrage pricing and hedging.

3. Discrete time stochastic processes and pricing models. (a) Binomial methods without much math. Arbitrage and reassigning probabilities.

5

(b) A first look at martingales. (c) Stochastic processes, discrete in time. (d) Conditional expectations. (e) Random walks. (f) Change of probabilities. (g) Martingales. (h) Martingale representation theorem. (i) Pricing a derivative and hedging portfolios. (j) Martingale approach to dynamic asset allocation.

4. Continuous time processes. Their connection to PDE.

(a) Wiener processes. (b) Stochastic integration.. (c) Stochastic differential equations and Ito's lemma. (d) Black-Scholes model. (e) Derivation of the Black-Scholes Partial Differential Equation. (f) Solving the Black Scholes equation. Comparison with martingale

method. (g) Optimal portfolio selection.

5. Finer structure of financial time series.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download