Properties of Stock Option Prices Chapter 10, 7 th edition Chapter 9, pre 7 th edition

1. Properties ofStock Option PricesChapter 10, 7th editionChapter 9, pre 7th edition

2. Assets Underlying Options • Stocks • Foreign Currency • Stock Indices • Futures

3. c : European call option price p : European put option price S0: Stock price currently K : Strike price T: Life of option : Volatility of stock price C : American Call option price P : American Put option price ST:Stock price at option maturity D : Present value of dividends during option’s life r: Risk-free rate for maturity Twith continuous compounding Notation

4. S0: Current stock price K : Strike price T: Life of option : Volatility of stock price r: Risk-free rate for maturity Twith continuous compounding D : Present value of dividends during option’s life Factors affecting Stock Option Prices

5. Payoff Payoff K K ST ST Payoff Payoff K K ST ST Payoff Patterns from OptionsWhat is the Option Position in Each Case? K = Strike price, ST = Price of asset at maturity

6. Profit Profit Profit K ST ST K Profit K K ST ST Profit Patterns from Options Long Call Short Call Long Put Short Put

7. Call option payoff: amount by which stock price exceeds strike price (K) Call options therefore become more valuable as stock price ___________ Put option payoff: amount by which strike price (K) exceeds stock price (S0 ) Put options therefore become more valuable as stock price ____________ S0 : Current stock price

8. Call options: a lower strike price results in a more valuable call option Put options: a higher strike price results in a more valuable put option K: Strike or Exercise Price

9. Intrinsic value and time value of an option Both put and call options become more valuable as time to expiration __________ Does this apply to both American and European? European options can only be exercised at expiration thus dividend payments could have a negative effect on the holder of the option (specifically the holder of European calls) T: Time to expiration

10. Volatility: uncertainty of stock price movements As volatility increases, the chance that the stock will do well or poorly increases (Bombardier, Abitibi) Does the value of a call option increase with increased volatility? Does the value of a put option decrease with decreased volatility? : Volatility of stock price

11. 30-day T-bill rate as an example CAPM and discounted cash flows As ‘r’ increases, value of future cash flows decreases causing the stock price to drop How does this affect a call option? How does this affect a put option? As ‘r’ increases, and all other variables, including the stock price, are held constant, the value of a call option increases while the value of a put option decreases r: Risk free interest rate

12. Dividends reduce the stock price on the ex-dividend date (what is the ex-dividend date) Is this good news for the holder of call options? Is this bad news for the holder of put options? D: Cash dividends

13. Declaration date– This is the date on which the board of directors announces to shareholders and the market as a whole that the company will pay a dividend. • Ex-date or Ex-dividend date– On (or after) this date the security trades without its dividend. If you buy a dividend paying stock one day before the ex-dividend you will still get the dividend, but if you buy on the ex-dividend date, you won't get the dividend. Conversely, if you want to sell a stock and still receive a dividend that has been declared you need to sell on (or after) the ex-dividend day. The ex-date is the second business day before the date of record. • Date of record– This is the date on which the company looks at its records to see who the shareholders of the company are. An investor must be listed as a holder of record to ensure the right of a dividend payout. • Date of payment (payable date) – This is the date the company mails out the dividend to the holder of record. This date is generally a week or more after the date of record so that the company has sufficient time to ensure that it accurately pays all those who are entitled.

14. Variable c p C P – – + + S0 – – + + K + + ? ? T + + + +  - - + + r* – – + + D Effect of Increasing Each Variable on Option Pricing • As ‘r’ increases, and all other variables, including the stock price, are held constant, the value of a call option increases while the value of a put option decreases • In general, however, an increase in interest rates leads to a drop in stock prices which decreases the value of a call option and increases the value of a put option

15. American vs European Options An American option is worth at least as much as the corresponding European option CcPp

16. Upper Bound for Call Option Prices; No Dividends • Call option (European or American) gives the holder the right to buy one share of a stock for a certain price c S0 C S0 • Otherwise buy stock instead of option

17. Lower Bound for Call Option Prices; No Dividends cS0 –Ke -rT

18. Upper and Lower bounds of Options Prices If an option price is above the upper bound and below the lower bound, there are profitable opportunities for arbitrage.

19. Calls: An Arbitrage Opportunity? • Suppose that c = 3 S0= 20 T= 1 r= 10% K = 18 D= 0 • Is there an arbitrage opportunity?

20. Formal Argument for Lower Bound Call Option Price • Portfolio A: European call on a stock + PV of the strike price in cash (c + Ke-rT) • Portfolio B: one share (SO)

21. Upper Bound for Put Option Prices; No Dividends • Put option (European or American) gives the holder the right to sell one share of a stock for a certain price (K) p K P K

22. Lower Bound for Put Prices; No Dividends pKe-rT–S0

23. Puts: An Arbitrage Opportunity? • Suppose that p = 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 • Is there an arbitrage opportunity?

24. Formal Argument for Lower Bound Put Option Price • Portfolio C: European put on the stock + the stock (p + SO) • Portfolio D: PV of the strike price in cash, Ke-rT

25. 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(ST, K ) at the maturity of the options • European, thus, they must be worth the same today • This means that c + Ke -rT = p + S0

26. Arbitrage Opportunities • Suppose that c = 3 S0= 31 T = 0.25 r= 10% K =30 D= 0 • What are the arbitrage possibilities when p = 2.25 ? p= 1 ?

27. Option Positions • Long call = max (St – K, 0) • Short call = min (K – St, 0) • Long put = max (K – St, 0) • Short put = min (St – K, 0)

28. Early Exercise of American calls • Never optimal to exercise an American call option early on non-dividend stock (does this guideline apply to European options also?) • Example: stock price is $50 and strike price is $40 with 1 month to expiration; investor plans to hold the stock • Reasons not to exercise early a call option: • $40 could be invested for 1 month and paid at expiration (delay paying the strike price) • Stock price could drop before the end of month (Holding the call provides insurance against stock price falling below strike price )

29. Early Exercise of American puts • Always optimal to exercise an American put option early (does this guideline apply to European options also?) • Strike price is $25 and stock price is $0.02 • May be optimal to forgo the insurance and exercise early to realize the strike price immediately (invest the money).

30. Extension of Put-Call Parity • European options; D> 0 c + D + Ke -rT = p + S0 D = present value of all dividends during the life of the option

31. Questions5th and 6th edition: 9.2, 9.3, 9.7, 9.11, 9.12, 9.14, 9.157th edition: 10.2, 10.3, 10.7, 10.11, 10.12, 10.14, 10.15