WHAT WILL WE DO IN FINANCIAL MATHEMATICS?

(TOPICS IN ANALYSIS, MATH 53 – tentative outline, 11/15/06)

1. Foreign Exchange – No time dimension, no uncertainty

(bring foreign money to class, especially if you’d like to exchange it)

(a) What relationships must prevail in a foreign-exchange table?

Arbitrage

Consequences of the “no-arbitrage” principle

Market friction (bid-asked spread, transactions costs,

measurement error) and accuracy of our results

(b) Connections between models and reality: What’s real?

What’s a theoretical construct?

(c) The “financial principle of relativity” – laws must be valid whatever

the numeraire

FIRST TOPIC: TIME VALUE OF MONEY

2. Government Bonds – time dimension, but no uncertainty

Why would anyone want to own a government bond?

(To receive guaranteed payments of money at specified times)

Terms of payment – face value, coupon rate, coupons, purchase price

Treasury “strips”

How to read the “strips” table in the WSJ

The P(t) function: present value of future payments

Shape of the P(t) function

Cash flows and their present values (based on P(t))

Testing government bonds for consistency

3. Interest rates

Simple interest, compound interest

Continuous compounding vs. annual compounding

“APR” vs. “effective interest rate”

Translating between bankers and mathematicians

Cash flows and their present value (based on a discount rate)

What would P(t) be in a constant-interest-rate environment?

Are we in a constant-interest-rate environment?

4. Ways to talk about the time value of money

Forward (instantaneous) interest rates

Yields

Relationship between P(t), R(t), and Y(t)

How to talk to your broker

Smoothing the R(t) function

(possible deep excursion here: cubic splines

and optimization by calculus of variations)

5. Corporate bonds (briefly) (first dose of uncertainty)

Repayment risk (= credit risk)

Comparing corporate bond prices to government bond prices

Why is the gap so much bigger than the historical probability of default would suggest?

High-yield bonds (“junk” bonds)

6. “Flat dollars” introduced as numeraire

Translating between flat dollars to actual dollars

SECOND TOPIC: UNCERTAIN PAYMENTS

7. Uncertain payments (uncertainty without time)

(Bring money and dice.)

Decision-making among lotteries

“EMV-ers” vs. real people

Risk aversion

Utility functions

The Fundamental Theorem of Decision Theory

(You should maximize expected utility, with respect

to some probability distribution)

Real probabilities vs. “probabilities” in the theorem

“Certainty equivalent” values

Does certainty equivalent depend on mean and variance?

What does Barry Schwartz think of all this?

8. Uncertain future payments (uncertainty and time)

Expected values of cash flows

Higher discount rates as a proxy for uncertainty

(Bringing in the lawyers: “We agree that the truth is

between 3.2% and 27%”)

THIRD TOPIC: MODELING STOCK PRICES

(Time and uncertainty will be with us for the duration)

9. The Stock Market

What is a share of stock?

Why would anyone want to own one?

(dividends, control, cash at liquidation)

Bubbles: The “greater fool” theory

10. Wallowing in data

How to get share-price histories from Yahoo

Daily returns

Arithmetic definition vs. logarithmic definition

The “half-r-squared” term, first appearance

Are daily returns normally distributed?

Are daily returns independent?

What is the mean of daily returns for your company?

Confidence intervals for the mean

What is the standard deviation of daily returns?

Confidence intervals for the standard deviation

How stable is it over time?

Volatility – definition and measurement

Converting between time scales

Relationship between companies

Correlation

Price indexes

Correlation of prices with indexes (“beta”)

How stable is that over time?

11. The Standard Model

Brownian Motion and Geometric Brownian Motion

Einstein, Bachelier, Samuelson

Calculations and predictions based on the standard model

(Here we make heavy use of the normal distribution function)

Story: Apple shares and the estate tax

FOURTH TOPIC: OPTION PRICING AND DERIVATIVE SECURITIES

12. What’s a stock option?

Call and put options; American vs. European; exotic options

Why would anyone buy an option?

Oil wells as options

Home mortgages as options

Observing option prices

Early exercise of call options

13. Decision trees

Chance nodes and decision nodes

Backwards recursion

Valuing options using decision trees

14. Valuing call options assuming risk neutrality

The oil-royalty integral

The Black-Scholes formula (first look)

15. The two-branch model

16. The binary tree model

Valuing options using the binary tree

Standard model as limit of binary trees

17. Black-Scholes without risk-neutrality

Risk-neutral (martingale) measures

The Fundamental Theorem of Derivative Securities

18. Practical decision-tree valuation of options

American options and early exercise

19. Black-Scholes and Partial Differential Equations

(This could get deep. We’ll look at the standard heat equation and

Kolmogorov equation related to Brownian motion.

Main point: If we set up the problem correctly, the Black-Scholes

differential equation is exactly the standard heat equation.)

20. Option pricing and oil-property evaluation

FIFTH TOPIC: PORTFOLIO OPTIMIZATION

21. Measuring reward and risk for a portfolio

Calculating mean and variance of a portfolio from statistics

of the stocks in the portfolio

Optimization by trial and error: Apple and Microsoft

22. Optimizing mean and variance

The feasible set

The efficient frontier

“Markowitz optimization”

Lagrangean multipliers (“e” or “i”?)

Lambdas and tradeoffs

23. Cash as an option

How it changes the efficient frontier

The “market portfolio”

24. The Capital Asset Pricing Model

Diversifiable risk vs. undiversifiable risk

Do we believe this? Hey, it’s a theorem!

25. Practical application: The theory of implied yields

That’s enough, I think.