1. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-0 Finance 457 8 Chapter Eight Properties of Stock Options

2. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-1 Finance 457 Chapter Outline 8.1 Factors Affecting Option Prices 8.2 Assumptions and Notation 8.3 Upper and Lower Bounds for Option Prices 8.4 Put-Call Parity 8.5 Early Exercise: Calls on a non-dividend paying stock 8.6 Early Exercise: Puts on a non-dividend paying stock 8.7 Effects of Dividends 8.8 Empirical Research

3. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-2 Finance 457 8.1 Option Value Determinants Call Put 1.Stock price+ – 2.Exercise price– + 3.Interest rate + – 4.Volatility in the stock price+ + 5.Expiry (American)+ + 6.Dividends expected prior to expiry– + The value of a call option C 0 must fall within max (S 0 – K, 0) < C 0 < S 0. The precise position will depend on these factors.

4. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-3 Finance 457 8.2 Assumptions and Notation Assumptions –There are no transactions costs. –All trading profits (net of trading losses) are subject to the same tax rate. –Borrowing and Lending are possible at the risk-free rate of interest. Notation –S 0 current stock price –S T stock price at expiry –K is the exercise price –T is the time to expiry –r is the nominal risk-free rate; continuously compounded; maturity T –C value of an American call –c value of a European call –P value of an American put –p value of a European put

5. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-4 Finance 457 8.3 Upper and Lower Bounds for Option Prices C > Max[S T - K, 0] Profit loss E STST Market Value Intrinsic value S T - K Time value Out-of-the-moneyIn-the-money STST The value of a call option C 0 must fall within max (S 0 – K, 0) < C 0 < S 0.

6. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-5 Finance 457 8.3 Upper and Lower Bounds For Call Options Upper Bounds –C < S 0 –c < S 0 Lower Bounds on Non- Dividend Paying Stock c > max[S 0 – Ke -rT, 0] For Put Options Upper Bounds –P < K –p < K Lower Bounds for European Puts on Non- Dividend-Paying Stock p > max[Ke -rT –S 0, 0]

7. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-6 Finance 457 8.4 Put-Call Parity Philosophically –The RIGHT to buy a stock together with the ABILITY to buy it. c 0 + Ke -rT Should be worth the same as –The RIGHT to sell a stock together with the ABILITY to sell it. p 0 + S 0 This notion can be formalized as Put-Call Parity c 0 + Ke -rT = p 0 + S 0

8. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-7 Finance 457 Put Call Parity The following two portfolios have the same payoffs at expiry: –One European call plus an amount of cash equal to Ke -rT –One European put plus one share of stock This means that at time zero c 0 + Ke -rT = p 0 + S 0 The following slide shows the payoffs

9. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-8 Finance 457 Put-Call Parity: Payoffs at Expiry

10. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-9 Finance 457 Put-Call Parity for American Options Put–Call parity only holds for European options. For American options, we can say: S 0 – K < C 0 – P 0 < S 0 – Ke -rT Violations of put-call parity represent arbitrage opportunities

11. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-10 Finance 457 Put-Call Parity: Option Values Sell a put with an exercise price of $40 Buy a call option with an exercise price of $40 $0 – K– K K – p 0 K –K + p 0 Value of stock at expiry S 0 – K < C 0 – P 0 < S 0 – Ke -rT

12. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-11 Finance 457 8.5 Early Exercise: Calls on a non-dividend paying stock It is never optimal to exercise an American call early on a non-dividend- paying stock. Basically, you would prefer to sell the option instead of exercising so that you capture the speculative value as well as the intrinsic value. Another argument is that holding a call instead of the stock provides insurance. Implicit of course is the fact that if a dividend is big enough, it would be optimal to exercise early.

13. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-12 Finance 457 8.6 Early Exercise: Puts on a non-dividend paying stock It can be optimal to exercise an American put early. This occurs when the put is “deep enough” in the money. AK American put price

14. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-13 Finance 457 8.6 Early Exercise: Puts on a non-dividend paying stock Since it can be optimal to exercise an American put early, but early exercise is forbidden with European puts. Therefore, a European put option must be sometimes worth less than its intrinsic value. BK European put price K E

15. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-14 Finance 457 8.7 Effects of Dividends So far, our results were derived for options written on non-dividend-paying stocks. In the U.S., exchange- traded options generally are short enough in maturity that dividends can be predicted with accuracy. Let’s use D to denote the present value of the expected dividends. c 0 > S 0 –D -Ke -rT p 0 > D + Ke -rT –S 0 We can no longer say that early exercise of calls is a bad idea. IF the dividend is “big enough” we should exercise.

16. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-15 Finance 457 8.8 Empirical Research There are a number of complications: –Asynchronous price quotes –Transactions costs –Put-call parity holds only for European options –Dividends paid over the life of the option must be estimated The results support the notion that we can’t make money sitting here 15 miles north of Ashland, MO. However, market makers may get the occasional arbitrage, but that’s what makes the market efficient!

17. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-16 Finance 457 Summary and Conclusions The most familiar options are puts and calls. –Put options give the holder the right to sell stock at a set price for a given amount of time. –Call options give the holder the right to buy stock at a set price for a given amount of time. Put-Call parity

18. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 8-17 Finance 457 Summary and Conclusions The value of a stock option depends on six factors: 1. Current price of underlying stock. 2. Dividend yield of the underlying stock. 3. Strike price specified in the option contract. 4. Risk-free interest rate over the life of the contract. 5. Time remaining until the option contract expires. 6. Price volatility of the underlying stock. Exactly how is taken up in later chapters