Assignments - Winter 2011

problem set

Solutions

[Both modified slightly on Apr 11]

- Problems 13.2 and 13.3 [Note that overnight profit/loss is the change from one day to the next, if you don't rebalance the hedging portfolio]
- In problem 13.2, calculate Delta two ways, once using the formula from class, and once by working out the BSM price at two very close values of the stock price.
- Calculate the implied volatility, accurate to the nearest percentage point, in the setting of problem 12.3 (T=1 year), if the call sells for $8.00

The following problems all refer to the setup of the 11.16 in the book, which you studied earlier on Assignment 4. In particular, you have an 8-period binomial tree, with expiration T=1, initial stock price S=100, interest rate r=0.08, volatility sigma=0.3; In assignment 4 you obtained u=1.112 and d=0.899 using the CRR calibration. You will be looking at European and American calls, with strike K=95. You will look at two dividend rates, div=0 and div=0.05; You may wish to start with the tree you built for assignment 4. Or, if you wish, you may take as a starting point the tree I posted in the solutions to assignment 4, but you do that, you will need to modify it to incorporate dividends.

- On assignment 4 and 5 you computed the European and BSM prices when del=0. Now find them both when del=0.05
- Take div=0. Verify that there is no early exercise and find the American option price in this case.
- Take div =0.05 and find the American option price. Find the early exercise premium. Indicate at which nodes it is optimal to exercise early.
- Take div=0.05; Which of the two American option prices 15.0 or 15.5 gives an arbitrage you can exploit by buying the option? Describe the cash flows and give the optimal exercise time, if the actual sequence of stock prices changes observed is uuuuudud.
- Take div=0.05; Which of the two American option prices 15.0 or 15.5 gives an arbitrage you can exploit by selling the option? Describe the cash flows, if the actual sequence of stock prices changes observed is uuuuudud, and if the buyer exercises after 4 periods. Do the same if the buyer holds the option to expiration.

- Copy the analysis we went through in class, and obtain the Black-Scholes-Merton pricicng formula for a European put (no dividends). Then check that put-call parity still holds.
- Copy the analysis we went through in class, and obtain a price for a binary option in the Black-Scholes-Merton model (no dividends). The option pays A if S(T)>K and 0 otherwise.
- This problem refers to Assignment 4. For problems 10.4 (European call), the binary option, and the European put portion of problem 11.16, compute the Black-Scholes-Merton price. Determine what percentage the binomial price is of the Black-Scholes-Merton price.
- Problem 12.3
- Problem 12.13

All 5 questions were graded, for a total of 50 marks.

- Chapter 9: Problems 1, 9
- Chapter 10: Problems 3, 4.

Now repeat problem 4, but for a binary option whose payoff is $10 if S(T)>$102, and $0 otherwise.

[I assumed people would do all of this with the given u and d, in which case sigma is irrelevant. But if you opted to use sigma to calibrate a revised u and d, that's fine too]. - Chapter 11: Problem 16 (except take the dividend rate to be 0). You may use software for this. In addition, find the risk neutral probability p of an up-move, as well as the risk-neutral probability of finishing in the money (ie of S(T)>K).

[My suggestion is to use Excel to build the tree, rather than trying to program it in C++ or Matlab. For an 8 period tree, this is quite feasible. I meant for you to only do the European call and put. But if you've done the American version too, this isn't wasted effort; save it because I will be asking you that on a subsequent assignment]

All 6 questions were graded, for a total of 60 marks.

- Chapter 5: Problems 12, 16, 19
- Chapter 7: Problems 16, 23

All 5 questions were graded, for a total of 50 marks.

A question came up about problem 5.16 and whether the factor in the denominator is really 917.57 (the futures price) or 875 (the current level of the index). In fact the solution I gave is correct, and is consistent with the formula given in the notes. But I realize I never actually proved this formula for you. I will try to write up a derivation over reading week.

- Chapter 5: Problems 5, 8
- Chapter 6: Problem 9
- Chapter 7: Problems 10, 12
- Chapter 8: Problem 14

5 questions were graded (5.8, 6.9, 7.10, 7.12, 8.14) for a total of 50 marks.

Note that several of the questions require you to read portions of the text, to understand what is being asked, eg bid/ask prices, asymmetric butterfly spreads, and the examples of chapter 4.

- Chapter 1: Problems 3, 6
- Chapter 2: Problem 2
- Chapter 3: Problems 3, 6, 14, 17
- Chapter 4: Problems 1, 3

7 questions were graded (1.6, 2.2, 3.3, 3.6. 3.14, 3.17, 4.3) for a total of 70 marks.