Chapter Eight

1. Properties of Stock Options 8 Chapter Eight

2. 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 Chapter Outline

3. 8.1 Option Value Determinants Call Put • Stock price + – • Exercise price – + • Interest rate + – • Volatility in the stock price + + • Expiry (American) + + • Dividends expected prior to expiry – + The value of a call option C0 must fall within max (S0 – K, 0) <C0<S0. The precise position will depend on these factors.

4. 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 S0 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 8.2 Assumptions and Notation

5. 8.3 Upper and Lower Bounds for Option Prices C > Max[ST - K, 0] The value of a call option C0 must fall within max (S0 – K, 0) <C0<S0. Profit ST ST - K Market Value Time value Intrinsic value ST E loss Out-of-the-money In-the-money

6. For Call Options Upper Bounds C < S0 c < S0 Lower Bounds on Non-Dividend Paying Stock c > max[S0 – 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–S0, 0] 8.3 Upper and Lower Bounds

7. Philosophically The RIGHT to buy a stock together with the ABILITY to buy it. c0 + Ke-rT Should be worth the same as The RIGHT to sell a stock together with the ABILITY to sell it. p0 + S0 This notion can be formalized as Put-Call Parity c0 + Ke-rT =p0 + S0 8.4 Put-Call Parity

8. 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 c0 + Ke-rT =p0 + S0 • The following slide shows the payoffs

9. Put-Call Parity: Payoffs at Expiry

10. Put-Call Parity for American Options • Put–Call parity only holds for European options. • For American options, we can say: S0 – K<C0– P0< S0 – Ke-rT • Violations of put-call parity represent arbitrage opportunities

11. Put-Call Parity: Option Values • S0 – K<C0– P0< S0 – Ke-rT Sell a put with an exercise price of $40 $0 Value of stock at expiry Buy a call option with an exercise price of $40 K K –p0 –K + p0 –K

12. 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. 8.5 Early Exercise: Calls on a non-dividend paying stock

13. It can be optimal to exercise an American put early. This occurs when the put is “deep enough” in the money. 8.6 Early Exercise: Puts on a non-dividend paying stock American put price A K

14. 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. 8.6 Early Exercise: Puts on a non-dividend paying stock European put price K E B K

15. 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. c0> S0 –D -Ke-rT p0> D + Ke-rT –S0 We can no longer say that early exercise of calls is a bad idea. IF the dividend is “big enough” we should exercise. 8.7 Effects of Dividends

16. 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! 8.8 Empirical Research

17. 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. 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