1. an arbitrageur, an arbitrage opportunity an advantage continuous compounding corresponding to delay to derive exception to exercise an ex-dividend date an expiration date extreme flat to hold, holding period immediate intuitive a lower/an upper bound maturity parity present value prior to a property risk-free rate sacrifice strike price term structure time value of money trade-off volatility

2. Chapter 9. Properties of Stock Options The Question of This Chapter: What can we say about the option price?

3. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.3 Notation c : European call option price p :European put option price S 0 :Stock price today K :Strike price T :Life of option  :Volatility of stock price C :American Call option price P :American Put option price S T :Stock price at option maturity D :Present value of dividends during option’s life r :Risk-free rate for maturity T with cont. comp.

4. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.4 Effect of Variables on Option Pricing cpCP Variable S0S0 K T  r D ++ – + ?? ++ ++++ + – + – – –– + – + – +

5. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.5 American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p

6. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.6 Calls: An Arbitrage Opportunity? Suppose that c = 3 S 0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity?

7. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.7 Lower Bound for European Call Option Prices; No Dividends c  S 0 –Ke -rT

8. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.8 Puts: An Arbitrage Opportunity? Suppose that p = 1 S 0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity?

9. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.9 Lower Bound for European Put Prices; No Dividends p  Ke -rT –S 0

10. Question 9.2 What is a lower bound for the price of a four-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?

11. Question 9.3 What is a lower bound for the price of a one-month European put option on a non-dividend-paying stock when the stock price is $12, the strike price is $15, and the risk-free interest-rate is 6% per annum?

12. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.12 Put-Call Parity; No Dividends Consider the following 2 portfolios: –Portfolio A: European call on a stock + PV of the strike price in cash –Portfolio C: European put on the stock + the stock Both are worth max( S T, K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S 0

13. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.13 Arbitrage Opportunities Suppose that c = 3 S 0 = 31 T = 0.25 r = 10% K =30 D = 0 What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

14. Put-Call Parity; No Dividends; American S 0 - K < C - P < S 0 - Ke -rT

15. Question 9.7 The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a three-month European put option with a strike price of $20?

16. Question 9.16 The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in three months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.

17. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.17 Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a non- dividend paying stock This should never be exercised early

18. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.18 For an American call option: S 0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months? An Extreme Situation

19. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.19 Reasons For Not Exercising a Call Early (No Dividends) No income is sacrificed We delay paying the strike price Holding the call provides insurance against stock price falling below strike price

20. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.20 Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0.25; r =10% K = 100; D = 0

21. Question 9.5 “The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put.” Explain this statement.

22. Question 9.13 Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk- free rate increases and volatility decreases.

23. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.23 The Impact of Dividends on Lower Bounds to Option Prices

24. Question 9.11 A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur?

25. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 9.25 Put-Call Parity; Dividends European options; D > 0 c + D + Ke -rT = p + S 0 American options; D > 0 S 0 - D - K < C - P < S 0 - Ke -rT

26. Question 9.14 The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. The term structure is flat, with all risk-free interest rates being 10%. What is the price of a European put option that expires in six month and has a strike price of $30?

27. Early Exercise Sometimes it is optimal to exercise an American call immediately prior to an ex-dividend date It is never optimal to exercise a call at other times A put may be exercised any time

28. Question 9.23 Suppose that c(1), c(2), and c(3) are the prices of European call options with strike prices K(1), K(2), and K(3), respectively, where K(3)>K(2)>K(1) and K(3)- K(2)=K(2)-K(1). All options have the same maturity. Show that c(2) ≤ 0.5*[ c(1)+c(1) ]