absence of arbitrage
call option, put option, cap (other answers are possible)
binomial model
simulation
simulating investment strategies, pricing options (other answers are possible)
Riskless bond (interest rate is 50%):
100 --- 150
Stock:
|- 125 50 -| |- 50
Derivative security (call option with a strike of 80):
|- 45 ? -| |- 0
Suppose the replicating portfolio has S dollars worth of stock and B dollars worth of bonds, replicating in both states implies that: (125/50) S + (150/100) B = 45 (up state) and (50/50) S + (150/100) B = 0. Solving for S and B, we have that 1.5 S = 45 or S = 30 (which is purchase of 3/5 of a share) and 30 + 1.5 B = 0 or B = -20 (which shorting 1/5 of a bond).
The price of the replicating portfolio is 30 - 20 = 10.
Let p be the risk-neutral probability of the up state. Then 1-p is the risk-neutral probability of the down state. To price the stock correctly, we must have that 50 = (1/1.5) x (125 p + 50 (1-p)). Solving for p we have that p = 1/3 and 1-p = 2/3.
If we short a portfolio of options and bonds that replicates the stock, we can undo the risk exposure without violating at least the letter of the terms of our inheritence.
The questions in this section are based on the asset-allocation simulation in Homework 4. You should not need to use the complete program listings, but they are included at the end of the exam in case you would like to have a look at them [note: not included in this answer sheet]. All of the references in this section are to classes defined in the file AssetAlloc.java.
TextField r,mu,sigma,inrisky;in the class constructor ValuePlotFrame():
inputs.add(inrisky = new TextField("70",10));in the "start simulation" method startSimu():
ir = text2double(inrisky)/100.0;in the method reset()
inrisky.setText("70");What is the TextField inrisky? Briefly, how is it used?
The TextField inrisky is the input cell where the user puts the proportion to be put in the risky asset. This is put in the inputs Panel of the ValuePlotFrame and the text is set as a default to 70 both initially and by reset(). When needed for computations, it is converted to the double ir using text2double() and dividing by 100 to convert from a percentage to a decimal. (Note: this answer is more comprehensive than is necessary.)
double stockTotRet() { return mean1per + std1persqrt12 * (Math.random()-0.5);}where the constandt std1persqrt12 would be set previously by the method newPars in the line
std1persqrt12 = sigma * Math.sqrt(12.0 * tinc);of code. What is the point of the 0.5 and the square root of 12.0 in these lines?
The 0.5 adjusts the mean of the Math.random() to 0.0 and the Math.sqrt(12.0) adjusts the std deviation to 1.0 to make it easy to convert to the mean and std deviation we want.
wealth = stock + cash; stock = wealth * inrisky; cash = wealth - stock;What do these lines do?
The first line computes the wealth, the second line then gives the amount to hold (after trading) in the stock, and the third line computes the cash as a residual.
These questions are intended to be more challenging conceptual questions. Answering these questions correctly can give you up to 20 points to substitute for points missed in Parts A-C.
The stock is an investment (a store of value) and we need to get interest on the value we put in, while we do not put any money into a futures contract.
For the asset allocation simulation we care about returns in the actual probabilities investors will face, while for option pricing we want to use the risk-neutral probabilities that price derivative securities correctly.