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.