3. Suppose a stock has the following probability distribution of future prices after four months:

Sr 66.23 63.32 60.54 57.88 55.34

PROBABILITY OF OCCURENCE: .0625 .2500 .3750 .2500. 0625

a. Calculate the stocks expected logarithmic return and variance. Assume it is currently priced at $60.

b. Calculate the stocks annualized variance and mean.

c. What are the stocks u and d values for a period of length one month (n=4), one week (n=16), and one day (n=120)?

d. Suppose the stocks mean is equal to zero. What are the stocks u and d values for

the periods of length one month, one week, and one day? Comment on the importance of the mean in calculating u and d when n is large.

4. Suppose ABC stock currently is trading at $60 per share, has an annualized standard deviation of .35, and will not pay any dividends over the next three months; suppose that the continuously compounded annual riskfree rate is 6%.

a. Using the BlackScholes OPM, calculate the equilibrium price for a threemonth ABC 60 European call option

b. Using the BlackScholes OPM, calculate the equilibrium price for a threemonth ABC 60 European put option

c. Show the BlackScholes put price is the same price obtained using the putcall parity model.

d. Describe the arbitrage strategy you would pursue if the ABC 60 call was overpriced and if it was underpriced.

e. What would be the new equilibrium prices of the ABC call and put if ABC stock increased to $60.50?

5. Suppose XYZ stock is trading at $100, its annualized standard deviation is o=.25, and the continuous compounded riskfree rate is 6%.

a. Determine the BS equilibrium call prices for an XYZ European 100 call expiring in 90 days (T=.25) and for an XYZ European 105 call expiring in 90 days.

b. Determine delta, theta, and gamma for the 100 call and the 105 call.

c. How would you construct a neutral delta hedged portfolio? What is the value of the portfolio? What would be your position theta and gamma?

d. Given your neutraldelta hedged portfolio, what would expect the change in the portfolios value to be over a short period of time if there were no changed in the price of XYZ stock? What would you expect the change in value to be if there was a positive change or a negative change in the stock price?

e. What would be the portfolios profit or loss if the price of XYZ stock increased to

$105 and the 100 call and 105 call traded at their BS values of $8.9597 and $6.0162, respectively? What would be the portfolios profit or loss if the price of the XYZ stock decreased to $95 and the 100 call and 105 call traded at their BS values of $3.2890 and $1.8133, respectively? Are your results consistent with your answer to part (d)?