Download TopicsInAnalysis

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.