Mechanics of Options Markets Chapter 8 Options, Futures, and Other Derivatives, 7th Edition, Copyright John C. Hull 2008 1
Review of Option Types A call is an option to buy A put is an option to sell A European option can be exercised only at the end of its life An American option can be exercised at any time 2008 2
Option Positions Long call Long put Short call Short put 2008 3
Long Call (Figure 8.1, Page 180) Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months 30 Profit ($) 20 10 0-5 70 80 90 100 Terminal stock price ($) 110 120 130 2008 4
Short Call (Figure 8.3, page 182) Profit from writing one European call option: option price = $5, strike price = $100 Profit ($) 5 110 120 130 0 70 80 90 100 Terminal -10 stock price ($) -20-30 2008 5
Long Put (Figure 8.2, page 181) Profit from buying a European put option: option price = $7, strike price = $70 30 Profit ($) 20 10 Terminal stock price ($) 0 40 50 60 70 80 90 100-7 2008 6
Short Put (Figure 8.4, page 182) Profit from writing a European put option: option price = $7, strike price = $70 7 0-10 Profit ($) 40 50 60 70 80 90 100 Terminal stock price ($) -20-30 2008 7
Payoffs from Options What is the Option Position in Each Case? K = Strike price, S T = Price of asset at maturity Payoff Payoff K S T K S T Kaufen, call Payoff Payoff K S T K S T Verkaufen, put Man darf, also long man muss, also short 2008 8
Assets Underlying Exchange-Traded Options Page 183-184 Stocks Foreign Currency Stock Indices Futures 2008 9
Specification of Exchange-Traded d Options Expiration date Strike price European or American Call or Put (option class) 2008 10
Terminology Moneyness : At-the-money option In-the-money option Out-of-the-money option 2008 11
Terminology (continued) Option class Option series Intrinsic value Berechnetet Wert Time value Diff. Preis - Int.Wert. 2008 12
Dividends & Stock Splits (Page 186-188) Suppose you own N options with a strike price of K : No adjustments ts are made to the option o terms for cash dividends When there is an n-for-m stock split, the strike price is reduced to mk/n the no. of options is increased to nn/m Stock dividends are handled in a manner similar to stock splits 2008 13
Dividends & Stock Splits (continued) Consider a call option to buy 100 shares for $20/share How should terms be adjusted: for a 2-for-1 stock split? for a 5% stock dividend? 2008 14
Market Makers Most exchanges use market makers to facilitate options trading A market maker quotes both bid and ask prices when requested The market maker does not know whether the individual id requesting the quotes wants to buy or sell 2008 15
Margins (Page (Page 190-19 191) Margins are required when options are sold When a naked option is written the margin is the greater of: 1 A total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount (if any) by which the option is out of the money 2 A total of 100% of the proceeds of the sale plus 10% of the underlying share price For other trading strategies there are special rules 2008 16
Warrants Warrants are options that are issued by a corporation or a financial institution The number of warrants outstanding is determined by the size of the original issue and changes only when they are exercised or when they expire 2008 17
Warrants (continued) The issuer settles up with the holder when a warrant is exercised When call warrants are issued by a corporation on its own stock, exercise will usually lead to new treasury stock being issued 2008 18
Executive Stock Options Executive stock options are a form of remuneration issued by a company to its executives They are usually at the money when issued When options are exercised the company issues more stock and sells it to the option holder for the strike price 2008 19
Executive Stock Options continued They become vested after a period of time (usually 1 to 4 years) Exercisable They cannot be sold They often last for as long as 10 or 15 years Accounting standards now require the expensing of executive stock options 2008 20
Convertible Bonds Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio Very often a convertible is callable The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose 2008 21
Properties of Stock Options Chapter 9 Options, Futures, and Other Derivatives, 7th Edition, Copyright John C. Hull 2008 22
Notation c : European call option price p : European put option price S 0 : Stock price today K : Strike price T : Life of option σ: Volatility of stock price C : American Call option price P : American Put option price S T :Stock price at option maturity D : Present value of dividends during option s life r : Risk-free rate for maturity T with cont comp 2008 23
Effect of Variables on Option Pricing (Table 9.1, page 202) Variable S 0 K T σ r D c p C P + + + +?? + + + + + + + + 2008 24
American vs European Options An American option is worth at least as much as the corresponding European option C c P p 2008 25
Calls: An Arbitrage Opportunity? Suppose that c = 3 S 0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity? Leerverkauf jetzt, und long call kaufen, restliche Geld anlegen: In 1 Jahr hat man (20-3)*exp(0.1*1) =18,79 Dann nach 1 Jahr kann man die Underlying für max. 18 zurückkaufen 2008 26
Lower Bound for European Call Option Prices; No Dividends id d ( Dividends (Equation 9.1, page 207) c max(s 0 Ke rt, 0) 2008 27
Puts: An Arbitrage Opportunity? Suppose that t p = 1 S 0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity? Jetzt 37+1=38 Geld ausleihen, dann einmal Underlying und long put kaufen: In 1 Jahr muss man (38)*exp(0,05*0,5) =38,96 zurückzahlen, aber man kann die Underlying für mindesten 40 verkaufen 2008 28
Lower Bound for European Put Prices; No Dividends (Equation 9.2, page 208) p max(ke -rt S 0, 0) 2008 29
Put-Call Parity; No Dividends (Equation 9.3, page 208) 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(s T, K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rt = p + S 0 2008 30
Arbitrage Opportunities 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? c + Ke -rt = p + S 0 3+30e(-0,1*0,25) - 31 = 126 1,26 2008 31
Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a nondividend paying stock This should never be exercised early 2008 32
An Extreme Situation For an American call option: S 0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if 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? Möglichkeit 1: Jetzt call kaufen, gleich ausüben und den Underlying verkaufen Wert in 3 Monate ist (100-60)*exp(0 exp(0,25*r) Möglichkeit 2: Jetzt Leerverkauf und die Option erst am Ende ausüben, Wert am Ende ist (100)*exp(0,25*R)-60 2008 33
Reasons For Not Exercising a Call Early (No Dividends) No income is sacrificed Payment of the strike price is delayed Holding the call provides insurance against stock price falling below strike price 2008 34
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 Möglichkeit 1: Jetzt put und Underlying kaufen, gleich ausüben Wert in 3 Monate ist (100-60)*exp(0 exp(0,25*0,1) 1)= 41.01 Möglichkeit 2: Option am Ende ausüben, Wert am Ende ist (-60)*exp(0,25*0,1)+100 = 38,48 2008 35
The Impact of Dividends on Lower Bounds to Option Prices (Equations 9.5 and 9.6, pages 214-215) c S 0 D Ke rt p D + Ke rt S 0 2008 36
Extensions of Put-Call Parity American options; D = 0 S 0 - K < C - P < S 0 - Ke -rt (Equation 9.4, p. 211) European options; D > 0 c + D + Ke -rt = p + S 0 (Equation 9.7, p. 215) American options; D > 0 S 0 - D - K < C - P < S 0 - Ke -rt (Equation 9.8, p. 215) 2008 37