Properties of Stock Options Futures and Other Derivatives

Properties of Stock Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 Chapter 9. 1

Notation l c : European call l l option price p : European put option price S 0 : Stock price today K : Strike price T : Life of option : Volatility of stock price l C : American Call option l l 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 T with cont comp Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 2

Effect of Variables on Option Pricing (Table 9. 1, page 206) Variable S 0 K T r D c + – ? + + – p – +? + – + C + – + + + – Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 P – + + + – + 9. 3

American vs European Options An American option is worth at least as much as the corresponding European option C c P p Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 4

Upper and Lower Bounds on Options Prices Upper Bounds: No matter what, the call option can never be worth more that the stock S 0 c and S 0 C If this is not true, an arbitrageur could easily make a riskless profit by buying the stock and selling the call option Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 5

Upper and Lower Bounds on Options Prices Upper Bounds: No matter what, the put option can never be worth more that the strike price K p and K P Ke-r. T p If this is not true, an arbitrageur could easily make a riskless profit by writing the option and investing the proceeds Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 6

Lower Bounds in Call Options l l c S 0 - Ke-r. T Suppose that c=3 S 0 = 20 T=1 r = 10% K = 18 D=0 Then, S 0 - Ke-r. T = 3. 71 Is there an arbitrage opportunity? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 7

Lower Bounds in Call Options l l If c=3 < S 0 - Ke-r. T, an arbitrageur can short the stock and buy the call to provide a cash inflow of 20 -3=17. If invested for 1 year the $17 grows to $18. 79 At maturity, if the stock price is greater than $18, the arbitrageur exercises the option for $18, closes out the short position, and makes a profit of $18. 79 -$18=$0. 79 If the stock price is lower than $18, the stock is bought in the market and the short position is closed out. If the stock price is $17, the arbitrageur gains: $18. 79 -$17=$1. 79 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 8

Lower Bounds in Call Options Consider the following two portfolios: Portfolio A: one European call option plus one amount of cash equal to Ke-r. T, Portfolio B: one share Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 9

Lower Bounds in Call Options Portfolio A 1. The cash, if invested at r, grows to K. 2. If ST>K the option is exercised and the portfolio worth ST. 3. If ST<K the call expires worthless and the portfolio is worth K. Hence, at time T, portfolio A is worth max(ST, K ) Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 10

Lower Bounds in Call Options Portfolio B 1. Portfolio B is worth ST at time T. Hence, portfolio A is always worth as much as, and can be worth more than portfolio B. This must also be true today. Thus: c + Ke-r. T S 0 or c S 0 - Ke-r. T Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 11

Lower Bounds in Put options p Ke -r. T–S 0 l l Suppose that p =1 T = 0. 5 K = 40 S 0 = 37 r =5% D =0 then, Ke -r. T–S 0 = 2. 01 Is there an arbitrage opportunity? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 12

Lower Bounds in Put Options l l If p=1 < Ke-r. T - S 0, an arbitrageur can borrow $38 for 6 months to buy both the put and the stock. At the end of the 6 months he pays back $38. 96 At maturity, if the stock price is below $40, the arbitrageur exercises the option to sell the stock for $40, repays the loan, and makes a profit of $40 -$38. 96=$1. 04 If the stock price is greater than $40, the arbitrageur discards the option, sells the stock, and repays the loan. If for example the stock price is $42: $42 -$38. 96=$3. 04 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 13

Lower Bounds in Put Options Consider the following two portfolios: Portfolio C: one European put option plus one share, Portfolio D: an amount of cash equal to Ke-r. T Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 14

Lower Bounds in Put Options Portfolio C 1. If ST<K the option is exercised and the portfolio becomes worth K. 2. If ST>K the put expires worthless and the portfolio is worth ST. Hence, at time T, portfolio C is worth max(ST, K ) Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 15

Lower Bounds in Call Options Portfolio D 1. Assuming the cash is invested at r, portfolio D is worth K at time T. Hence, portfolio C is always worth as much as, and can be worth more than portfolio D. This must also be true today. Thus: p + S 0 Ke-r. T or p Ke-r. T - S 0 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 16

Put-Call Parity; No Dividends (Equation 9. 3, page 212) l l l Consider the following 2 portfolios: l Portfolio A: European call on a stock + PV of the strike price in cash l Portfolio C: European put on the stock + the stock Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c+ -r. T Ke = p + S 0 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 17

Arbitrage Opportunities l l Suppose that c =3 S 0 = 31 T = 0. 25 r = 10% K =30 D=0 What are the arbitrage possibilities when p = 2. 25 ? p=1? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 18

Put-Call Parity l From Put-Call Parity: c + Ke -r. T = p + S 0 or p = c + Ke -r. T - S 0 = 1. 26 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 19

Arbitrage when p=2. 25 1. 2. 3. 4. l l Buy call for $3 Short Put to realize $2. 25 Short the stock to realize $31 Invest $30. 25 for 3 months If ST > 30, receive $31. 02 from investment, exercise call to buy stock for $30. Net profit = $1. 02 If ST < 30, receive $31. 02 from investment, put exercised: buy the stock for $30. Net profit = $1. 02 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 20

Arbitrage when p=1 1. 2. 3. 4. l l Borrow $29 for 3 months Short call to realize $3 Buy put for $1 Buy the stock for $31. If ST > 30, call exercised: sell stock for $30. Use $29. 73 to repay the loan. Net profit = $0. 27 If ST < 30, exercise put to sell stock for $30. Use 29. 73 to repay the loan. Net profit = $0. 27 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 21

Early Exercise l l l 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 6 th Edition, Copyright © John C. Hull 2005 9. 22

An Extreme Situation For an American call option: S 0 = 100; T = 0. 25; K = 60; D = 0 Should you exercise immediately? l What should you do if l you want to hold the stock for the next 3 months? you do not feel that the stock is worth holding for the next 3 months? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 23

Reasons For Not Exercising a Call Early (No Dividends) l l l No income is sacrificed Payment of the strike price is delayed Holding the call provides insurance against stock price falling below strike price Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 24

Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0. 25; r=10% K = 100; D = 0 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 25

The Impact of Dividends on Lower Bounds to Option Prices (Equations 9. 5 and 9. 6, pages 218 -219) Consider the following two portfolios: Portfolio A: one European call option plus one amount of cash equal to D+Ke-r. T, Portfolio B: one share Then, using the similar arguments as before: Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 26

The Impact of Dividends on Lower Bounds to Option Prices (Equations 9. 5 and 9. 6, pages 218 -219) Consider the following two portfolios: Portfolio C: one European put option plus one share, Portfolio D: an amount of cash equal to D+ Ke-r. T Then, using the similar arguments as before: Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 27

Put-Call Parity; Dividends (Equation 9. 7, page 219) l l l Consider the following 2 portfolios: l Portfolio A: European call on a stock + PV of the strike price in cash + D l Portfolio C: European put on the stock + the stock Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c+D+ -r. T Ke = p + S 0 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 9. 28