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.