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Unformatted text preview: 1 MECHANICS OF OPTIONS MARKETS Options, Futures, and Other Derivatives, 7th Ed., John C. Hull (2008):
Chapter 910 Options 2 A call option is an option to buy a certain asset by a certain date for a certain price (the strike price) A put option is an option to sell a certain asset by a certain date for a certain price (the strike price) An American option can be exercised at any time during its life A European option can be exercised only at maturity Options vs Futures/Forwards 3 A futures/forward contract gives the holder the obligation to buy or sell at a certain price An option gives the holder the right to buy or sell at a certain price Intel Option Prices (Sept 12, 2006; Stock Price=19.56); Source: CBOE 4 Strike
Price Oct
Call Jan
Call Apr
Call Oct
Put Jan
Put Apr
Put 15.00 4.650 4.950 5.150 0.025 0.150 0.275 17.50 2.300 2.775 3.150 0.125 0.475 0.725 20.00 0.575 1.175 1.650 0.875 1.375 1.700 22.50 0.075 0.375 0.725 2.950 3.100 3.300 25.00 0.025 0.125 0.275 5.450 5.450 5.450 Option Types 5 A call is an option to buy A put is an option to sell Option Positions Long call Long put Short call Short put Long Call 6 Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months 30 Profit ($)
20
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0
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stock price ($)
110 120 130 Short Call 7 Profit from writing one European call option: option price = $5, strike price = $100 Profit ($)
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70 80 90 100 Terminal
stock price ($) Long Put 8 Profit from buying a European put option: option price = $7, strike price = $70 30 Profit ($)
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stock price ($)
40 50 60 70 80 90 100 Short Put 9 Profit from writing a European put option: option price = $7, strike price = $70 Profit ($)
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stock price ($) 60
70 80 90 100 Payoffs from Options 10 Payoff from a long position in European call is: Max(ST – K, 0) Conversely, payoff from a short position in European call is: Min (K‐ ST, 0) Payoff from a long position in a European put is: Max (K‐ ST, 0) Payoff from a short position in a European put is: Min (ST – K, 0) Payoffs from Options What is the Option Position in Each Case? 11 K = Strike price, ST = Price of asset at maturity Payoff K long call Payoff K long put Payoff
K
ST ST short call Payoff
K ST ST short put Money‐ness 12 When the payoff is positive, the option is said to be trading in‐the‐money (e.g ST>K for a long call) When the payoff is exactly zero (ST = K), the option is said to be trading at‐the‐money A long call option is said to be out‐of‐the‐money if ST < X. Similarly a short call is out‐of‐the‐money if ST > X. And vice versa for puts. Types of Traders 13 Hedgers Reduce risk faced from future movements in a market variable Speculators Bet on future direction of a market variable Arbitrageurs Take offsetting positions to lock in a profit Hedging Examples 14 A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract An investor owns 1,000 Microsoft shares currently worth $28 per share. A two‐month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts 15 Value of Microsoft Shares with and without Hedging 40,000 Value of Holding ($) 35,000 No Hedging 30,000 Hedging Stock Price ($) 25,000 20,000
20 22 24 26 28 30 32 34 36 38 Long & Short Hedges 16 A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future and want to lock in the price Arguments For and Against Hedging 17 Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables But: Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult Speculation Example 18 An investor with $2,000 to invest feels that a stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2‐month call option with a strike of 22.50 is $1 What are the alternative strategies? Buy 100 shares Buy 2,000 call options Speculation Example 19 Suppose the hunch is correct and price increases to $27. The payoff from the first alternative is: 100 x (27 – 20) = $ 700 The payoff from the second alternative is: 2000 x (27 – 22.5) = $ 9,000 Profit = $ 9000 – 2000 = $ 7000 Options strategy also leads to a larger loss Consider what happens if price falls to $ 15. Arbitrage Example 20 A stock price is quoted as £100 in London and $200 in New York The current exchange rate is 2.0300 What is the arbitrage opportunity? Simultaneously buy 100 shares of stock in NY and sell them in London. Riskless profit of 100 x [(2.03 x 100) – 200] = $300 Notation 21 c : European call option price p : European put option price S0 : Stock price today 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 T with cont comp K : Strike price T : Life of option C : : Volatility of stock price 22 Summary of Effect of Increase on one Variable on the Price of the Option Variable
S0
K
T r
D c p C P +
–
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+
+
– –
+
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+
–
+ +
–
+
+
+
– –
+
+
+
–
+ Upper and Lower Bounds for Options 23 An American option is worth at least as much as the corresponding European option C c
P p A call option gives the holder the right to buy a share. Hence the price of a call can never be more than the stock price: C ≤ S0 c ≤ S0 A put option gives the holder the right to sell a share at K. Hence the value of the put can never be more than this price: p ≤ K P ≤ K Finally, if at maturity, an option can never be worth more than K, today they cannot be more than present value of K: p ≤ Ke‐rT Calls: An Arbitrage Opportunity? 24 Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity? Lower Bound for European Call Option Prices; No Dividends: c max(S0 –Ke –rT, 0), where S0 –Ke –rT= 3.71 For a more formal argument consider two portfolios: Portfolio A: one European call plus an amount of cash equal to Ke‐rT Portfolio B: one share Puts: An Arbitrage Opportunity? 25 Suppose that p = 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity? Lower Bound for European Put Prices; No Dividends: p max(Ke rT–S0, 0) , where Ke rT–S0 = 2.01 For a formal argument consider : Portfolio C: one European put plus one share Portfolio D: an amount of cash equal to Ke‐rT Put‐Call Parity; No Dividends 26 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 They must therefore be worth the same today. This means that c + Ke rT = p + S0 Early Exercise 27 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 An Extreme Situation 28 For an American call option: S0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if you do not feel that the stock is worth holding for the next 3 months? you want to hold the stock for the next 3 months? You will likely not exercise as: No income is sacrificed Payment of the strike price is delayed Holding the call provides insurance against stock price falling below strike price 29 Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S0 = 60; T = 0.25; r=10% K = 100; D = 0 Since K‐ S> Ke‐rT –S for all values of K and S, it may be worth forgoing the insurance a put can provide if the put is sufficiently deep in the money 30 The Impact of Dividends on Lower Bounds to Option Prices c S 0 D Ke rT
p D Ke rT S0 For the PutCall Parity:
European options; D > 0 , where D is the PV of dividends
c + D + Ke rT = p + S0 ...
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 Spring '12
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