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7. 1 Properties of Stock Option Prices Options, Futures, and Other Derivatives, 4 th 7. 1 Properties of Stock Option Prices Options, Futures, and Other Derivatives, 4 th edition © 1999 by John C. Hull

FACTORS AFFECTING OPTION PRICES • The current stock price • The strike price • FACTORS AFFECTING OPTION PRICES • The current stock price • The strike price • The time to expiration • The volatility of the stock price • The risk free rate • The dividend expected during the life of the option 7. 2

7. 3 The owner of a EUROPEAN option can only exercise at the maturity 7. 3 The owner of a EUROPEAN option can only exercise at the maturity of the option. The owner of an AMERICAN option can exercise any time before the maturity of the option

WHICH ONE IS MORE EXPENSIVE ? Dec 2015 CALL or JULY 2016 CALL IF WHICH ONE IS MORE EXPENSIVE ? Dec 2015 CALL or JULY 2016 CALL IF IT IS AMERICAN 7. 4

7. 5 TIME TO EXPIRATION American Put and call options become MORE valuable as 7. 5 TIME TO EXPIRATION American Put and call options become MORE valuable as the time to expiration increase because the holder of the option has all the exercise opportunity and time. European call and put options do not necessarily increase in value as the time to expiration increases because the owner of the option can only exercise at expiration.

7. 6 VOLATILITY It is a measure of how uncertain we are about future 7. 6 VOLATILITY It is a measure of how uncertain we are about future stock price movements. Volatility ( ) = Standard deviation of the return on a stock in a length of time t

DIVIDENDS 7. 7 WHAT DOES DIVIDENDS 7. 7 WHAT DOES "EX DIVIDEND " MEAN ? A stock goes ex-dividend the day the company pays the dividend WHAT HAPPENS TO THE VALUE OF THE STOCK ? Stock price is reduced by the amount of the dividend at the opening HOW DOES IT AFFECT CALL & PUT ? • The value of a put option is positively related to the size of any anticipated dividend • The value of a call option is inversely related to the size of any anticipated dividend.

Notation • c : European call • • • option price p : European Notation • c : European call • • • option price p : European put option price S 0 : Stock price today X : Strike price T : Life of option : Volatility of stock price 7. 8 • C : American Call option • • price P : American Put option price ST : Stock price at time T D : Present value of dividends during option’s life r : Risk-free rate for maturity T with cont comp

7. 9 UPPER AND LOWER BOUNDS FOR OPTION PRICES Can the price of a 7. 9 UPPER AND LOWER BOUNDS FOR OPTION PRICES Can the price of a call option be worth more than the stock ? NO C S 0 The stock price is an upper bound to the option price. What if that did not hold ? ARBITRAGE

7. 10 No matter how low the stock price becomes, the option (p) can 7. 10 No matter how low the stock price becomes, the option (p) can never be worth more than the price of the stock (X) p X For EUROPEAN put options, we know, that at expiration (T), its value will not be worth more than X. Its value today cannot be more than the present value of X : p Xe-r. T

7. 11 LOWER BOUNDS FOR EUROPEAN CALLS or THEORETICAL MINIMUM A lower bound for 7. 11 LOWER BOUNDS FOR EUROPEAN CALLS or THEORETICAL MINIMUM A lower bound for the price of a EUROPEAN call option is : S 0 - -r. T Xe - D

Calls: An Arbitrage Opportunity? • Suppose that c =3 T =1 X = 18 Calls: An Arbitrage Opportunity? • Suppose that c =3 T =1 X = 18 S 0 = 20 r = 10% D=0 • Is there an arbitrage opportunity? 7. 12

EXAMPLE 7. 13 Suppose a stock is trading at $20 (S 0). The strike EXAMPLE 7. 13 Suppose a stock is trading at $20 (S 0). The strike price (X) of the call option is $18, the risk free rate r is 10% and T = 1 year. S - Xe-r. T = $3. 71 The lower bound is : 0 The call option is trading at $3. An arbitrageur can buy the call and short the stock In cash flow analysis 20 - 3 = $17 invested at 10% per annum 17 e 0. 1 = $18. 79 Options, Futures, and Other Derivatives, 4 th edition © 1999 by John C. Hull

$19 At expiration $17 7. 14 $19 At expiration $17 7. 14

7. 15 STOCK IS AT $19 The arbitrageur exercises the option : $18. 79 7. 15 STOCK IS AT $19 The arbitrageur exercises the option : $18. 79 - $18. 00 = $0. 79 RIGHT TO BUY AT 18

7. 16 STOCK = $17 The arbitrageur buys the stock back in the market 7. 16 STOCK = $17 The arbitrageur buys the stock back in the market and the short position is closed out. $18. 79 - $17. 00 = $1. 79

Effect of Variables on Option Pricing Variable S 0 X T r D c Effect of Variables on Option Pricing Variable S 0 X T r D c + – ? + + – p – + ? + – + C + – + + + – P – + + + – + 7. 17

7. 18 Lower Bound for European Call Option c S 0 -Xe -r. T 7. 18 Lower Bound for European Call Option c S 0 -Xe -r. T

LOWER BOUNDS FOR EUROPEAN PUT OPTIONS or THEORETICAL MINIMUM 7. 19 A lower bound LOWER BOUNDS FOR EUROPEAN PUT OPTIONS or THEORETICAL MINIMUM 7. 19 A lower bound for the price of a EUROPEAN put option is : -r. T Xe - S 0 + D

Puts: An Arbitrage Opportunity? • Suppose that p =1 T = 0. 5 X Puts: An Arbitrage Opportunity? • Suppose that p =1 T = 0. 5 X = 40 S 0 = 37 r =5% D =0 • Is there an arbitrage opportunity? 7. 20

EXAMPLE 7. 21 Suppose a stock is trading at $37(S 0). The strike price EXAMPLE 7. 21 Suppose a stock is trading at $37(S 0). The strike price (X) of the put option is $40, the risk free rate r is 5% and T = 0. 5 year. Xe -r. T The lower bound is : Th put option is trading at $1. 00 - S 0 = $2. 01 An arbitrageur can buy the put and the stock by borrowing $38 In cash flow analysis $38 borrowed at 5% per annum 38 e 0. 5 X 0. 05 = $38. 96 will have to be paid back in 6 months.

$45 At expiration $39 7. 22 $45 At expiration $39 7. 22

7. 23 STOCK IS AT $45 The arbitrageur discards the put, sells the stock 7. 23 STOCK IS AT $45 The arbitrageur discards the put, sells the stock in the market and repays the loan : $45 - $38. 96 = $6. 04

7. 24 STOCK IS AT $39 The arbitrageur exercises the option to sell the 7. 24 STOCK IS AT $39 The arbitrageur exercises the option to sell the stock for $40, repays the loan and makes a profit of : $40 - $38. 96 = $1. 04

7. 25 Lower Bound for European Put options p Xe -r. T - S 7. 25 Lower Bound for European Put options p Xe -r. T - S 0

7. 26 PUT - CALL PARITY Put-call parity is a fundamental relationship that must 7. 26 PUT - CALL PARITY Put-call parity is a fundamental relationship that must exist between the prices of a put option and call option if both have the same underlier, strike price and expiration date. The relationship is derived using arbitrage arguments.

7. 27 Put-Call Parity; No Dividends • Consider the following 2 portfolios: – Portfolio 7. 27 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 B: European put on the stock + the stock • Both are worth MAX(ST , X ) at the maturity of the options • They must therefore be worth the same today – This means that c + Xe -r. T = p + S 0

7. 28 Options, Futures, and Other Derivatives, 4 th edition © 1999 by John 7. 28 Options, Futures, and Other Derivatives, 4 th edition © 1999 by John C. Hull

7. 29 BOTH PORTFOLIOS HAVE IDENTICAL PAYOFF PATTERNS THEY MUST HAVE THE SAME VALUE 7. 29 BOTH PORTFOLIOS HAVE IDENTICAL PAYOFF PATTERNS THEY MUST HAVE THE SAME VALUE TODAY ARBITRAGE OPPORTUNITY

7. 30 PUT - CALL PARITY C+ -rt Xe = P + S 0 7. 30 PUT - CALL PARITY C+ -rt Xe = P + S 0 Put-call parity is often used as a simple test of option pricing models. Any option pricing model which produces put and call prices that do not satisfy put-call parity must be rejected as unsound. Such a model will suggest trading opportunities where none exist.

Arbitrage Opportunities • Suppose that c =3 S 0 = 31 T = 0. Arbitrage Opportunities • Suppose that c =3 S 0 = 31 T = 0. 25 r = 10% X =30 D =0 • What are the arbitrage possibilities when p = 2. 25 ? p =1? 7. 31

C+ A -rt Xe = P + S 0 B VALUE OF PORTFOLIO C+ A -rt Xe = P + S 0 B VALUE OF PORTFOLIO "A" ? $32. 26 VALUE OF PORTFOLIO "B" ? $33. 25 7. 32

7. 33 BUY 7. 33 BUY "A" AND SELL "B"

For p= 2. 25 7. 34 Portfolio B is overpriced relative to portfolio A For p= 2. 25 7. 34 Portfolio B is overpriced relative to portfolio A Portf A : c + Xe-r. T = $32. 26 Portf B : p + S 0 = $33. 25 Arbitrage : buy securities in A and sell securities in B Buy the call and short the put and the stock. . . CASH FLOW ANALYSIS…

7. 35 Cash flow : -3 + 2. 25 +31 = 30. 25 (invested 7. 35 Cash flow : -3 + 2. 25 +31 = 30. 25 (invested at 10% for 3 months = 31. 02)

Suppose the stock at expiration is greater than $30. Call is exercised 7. 36 Suppose the stock at expiration is greater than $30. Call is exercised 7. 36 Suppose the stock at expiration is less than $30 You are exercised Put is exercised LONG THE SHARE AT $30 Cash Flow: $31. 02 - $30 = $1. 02 (close out the short position

For p= 1 7. 37 Portfolio A is overpriced relative to portfolio B Portf For p= 1 7. 37 Portfolio A is overpriced relative to portfolio B Portf A : c + Xe-r. T = $32. 26 Portf B: p + S 0 = $32. 00 Arbitrage : buy securities in B and sell securities in A Buy the put and the stock and short the call Investment of : - $31 - $1 + $3 = -$29 (29 e 0. 10 x 0. 25 = $29. 73)

Suppose the stock at expiration is greater than $30. you get ur Call exercised Suppose the stock at expiration is greater than $30. you get ur Call exercised 7. 38 Suppose the stock at expiration is less than $30 Put is exercised SHORT THE SHARE AT $30 Cash Flow: $30. 00 - $29. 73 = $0. 27

7. 39 DOES PUT/CALL PARITY EXIST FOR AMERICAN TYPE OPTIONS ? NO WHY ? 7. 39 DOES PUT/CALL PARITY EXIST FOR AMERICAN TYPE OPTIONS ? NO WHY ? Because American options can be exerciced before Expiration and would not the same value today

7. 40 Early Exercise • Usually there is some chance that an American option 7. 40 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 Options, Futures, and Other Derivatives, 4 th edition © 1999 by John C. Hull

7. 41 An Extreme Situation • For an American call option: S 0 = 7. 41 An Extreme Situation • For an American call option: S 0 = 100; T = 0. 25; X = 60; D = 0 Should you exercise immediately? • What should you do if 1 You want to hold the stock for the next 3 months? 2 You do not feel that the stock is worth holding for the next 3 months?

Reasons For Not Exercising a Call Early (No Dividends ) • No income is 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 7. 42

Should Puts Be Exercised Early ? Are there any advantages to exercising an American Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0. 25; r=10% X = 100; D = 0 7. 43

7. 44 The Impact of Dividends on Lower Bounds to Option Prices 7. 44 The Impact of Dividends on Lower Bounds to Option Prices

7. 45 RELATIONSHIP BETWEEN AMERICAN PUT AND CALL PRICES Put-Call parity apply for European 7. 45 RELATIONSHIP BETWEEN AMERICAN PUT AND CALL PRICES Put-Call parity apply for European options but can be derived to American option : S 0 - X C - P S 0 - Xe-r. T (page 220)

EXAMPLE 7. 46 An American Call option with strike price (X) = $20. 00 EXAMPLE 7. 46 An American Call option with strike price (X) = $20. 00 and maturity in 5 months (T) is worth $1. 50. Current stock price (S 0) = $19. 00 and r = 10% S 0 - X C - P S 0 - Xe-r. T 19 - 20 1. 50 -P 19 - 20 e-0. 10 x 5/12 $1. 68 P $2. 50 Upper and lower bounds for price of an American put with the same strike price and expiration date as the American call are $2. 50 and $1. 68