8 -0 Finance 457 Properties of Stock Options

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

8 -2 Finance 457 8. 1 Option Value Determinants 1. 2. 3. 4. 5. 6. Call Stock price + Exercise price – Interest rate + Volatility in the stock price + Expiry (American) + Dividends expected prior to expiry – Put – + + + 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. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-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. Mc. Graw-Hill/Irwin • Notation – – – – – S 0 current stock price ST 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 Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

8 -4 Finance 457 8. 3 Upper and Lower Bounds for Option Prices Profit The value of a call option C 0 must fall within max (S 0 – K, 0) < C 0 < S 0. ST C > Max[ST - K, 0] Market Value Time value loss Mc. Graw-Hill/Irwin Out-of-the-money ST -K Intrinsic value E In-the-money ST Copyright © 2002 by The Mc. Graw-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-r. T, 0] Mc. Graw-Hill/Irwin • For Put Options • Upper Bounds – P max[Ke-r. T –S 0, 0] Copyright © 2002 by The Mc. Graw-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-r. T 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-r. T = p 0 + S 0 Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-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-r. T – One European put plus one share of stock • This means that at time zero c 0 + Ke-r. T = p 0 + S 0 • The following slide shows the payoffs Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

8 -8 Finance 457 Put-Call Parity: Payoffs at Expiry Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-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-r. T • Violations of put-call parity represent arbitrage opportunities Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

8 -10 Finance 457 Put-Call Parity: Option Values S 0 – K < C 0 – P 0 < S 0 – Ke-r. T Sell a put with an exercise price of $40 $0 Buy a call option with an exercise price of $40 –K + p 0 Mc. Graw-Hill/Irwin Value of stock at expiry K K – p 0 –K Copyright © 2002 by The Mc. Graw-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-dividendpaying 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. Mc. Graw-Hill/Irwin • Implicit of course is the fact that if a dividend is big enough, it would be optimal to exercise early. Copyright © 2002 by The Mc. Graw-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. American put price A Mc. Graw-Hill/Irwin K Copyright © 2002 by The Mc. Graw-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 • Therefore, a European put exercise an American put option must be sometimes early, but early exercise is worth less than its intrinsic forbidden with European value. puts. European put price K E B Mc. Graw-Hill/Irwin K Copyright © 2002 by The Mc. Graw-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. , exchangetraded 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. Mc. Graw-Hill/Irwin • c 0 > S 0 –D -Ke-r. T • p 0 > D + Ke-r. T –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. Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

8 -15 Finance 457 8. 8 Empirical Research • There 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! Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-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 Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-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 Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.