Homework 5 Revised due Mar 11 Thursday K Foster, Options & Futures, Eco 275, CCNY, Spring 2010

 

Since Matlab doesn't run in the computer lab, we'll come back to these exercises after the Midterm.  So skip Questions 1-4 of Homework 5 for now.  Still do Questions 5-8 of Homework 5.  Also do Questions 1 and 2 of HW 6.

 

You are encouraged to form study groups to work on these problems.  However each student must hand in a separate assignment: the group can work together to discuss the papers and comment on drafts, but each study group member must write it up herself/himself.  When emailing assignments, please include your name and the assignment number as part of the filename.

Please write the names of your study group members at the beginning of your homework to acknowledge their contributions.

1. Create a vector of 100 standard normal random numbers; find its mean and standard error.  Do this 1000 times, recording the means and standard errors each time.  Plot a histogram of the results.  Does it look like a normal distribution?  (Should it?)  Explain.
2. Consider hypothesis testing with the random draws from the previous question.  Suppose you have a new sample with an average of 0.01.  How many of the 1000 random draws have a larger mean (in absolute value)?  This is a bootstrap p-value.  Compare this with a p-value that you would look up in a table; discuss how they relate.
3. Starting from the program you created above, create a vector of 100 random numbers from 5 different distributions (use Matlab's canned random numbers and/or experiment with functions of these); find the means and standard errors.  Do each 1000 times.  Plot histograms of each.  Do they look like normal distributions?  Discuss.  Can you break it and find something clearly non-normal?
4. For each of the five different distributions, get a bootstrap p-value for a new sample with average of 0.01.  Discuss the relationship to the Law of Large Numbers.
5. Please complete Assignment Question 10.19 in Hull.
6. Please complete Assignment Question 10.21 in Hull.
7. Please complete Assignment Question 10.23 in Hull.
8. I am considering an investment in the country of Cunystan, which uses currency called the "zing".  Since my home country offers interest rates of just 1.5%, I am looking to get higher returns.  You are my portfolio manager, whose expertise I depend upon.  The currency markets allow investors to buy and sell zing at a spot rate of $1 to 70 zing, and buy and sell forward (in one year) at a rate of $1 to 75 zing.  Cunystan government bonds pay a riskless interest rate of 4% in one year (the zero rate, assuming discrete semiannual compounding).  What investment strategy would you recommend, in this situation?
9. A stock can be modeled with a one-step discrete-time tree so that after one month the stock, currently trading at $70 per share, will be worth either $75 or $60 (note that this is asymmetric).  Assume that the riskfree interest rate is 3
   
   %.

a.      Find the value of an at-the-money call.

b.      Find the value of an at-the-money put.

c.       Verify that put-call parity holds.

d.      Find the risk-neutral probabilities of up and down movements.

e.      What value of  (shares of the stock) make a riskless portfolio when combined with one short call?  With one short put?

f.        Why are these values of  different?

2.      Consider a one-step tree model, where each step is one month.  Consider a stock currently trading at 1.25 but which has highly asymmetric possible returns: if the stock goes up it will quadruple in price to 5; if it goes down it will drop by one-quarter to 0.3125.  A put option has a strike price of 2.  Assume the risk-free interest rate is 2%.

a.      What does the model imply is the value of this put option?

b.      If the put option is actually priced at 1.25 (an odd coincidence that the stock value and option value are the same!), what values of "up" and "down" would produce this price?