Options. Investment Management. Fall 2005

Transcription

1 Investment Management Fall 2005

2 A call option gives its holder the right to buy a security at a pre-specified price, called the strike price, before a pre-specified date, called the expiry date. A put option gives its holder the right to sell a security at a pre-specified price, called the strike price, before a pre-specified date, called the expiry date. 2

3 For each option, there is a holder and a writer. The writer issues the option. Whereas the holder of an option has the right to buy or sell the security at a pre-specified price, the writer of the option has the obligation to do the reverse trade. 3

4 A European option can only be exercised at the expiry date. An American option can be exercised at any time before the expiry date. 4

5 Let S 0 current underlying asset price, S T underlying asset price at the option s expiration date, X exercise price, T expiration date of the option, r f risk-free rate of interest, C price of a call option, P price of a put option, 5

6 A call option is out of the money if S 0 < X, At the money if S 0 = X, in the money if S 0 > X. 6

7 A put option is out of the money if S 0 > X, At the money if S 0 = X, in the money if S 0 < X. 7

8 The profit to the holder of a call option, π h, is π h = max { S T X, 0 } C. The profit to the writer of the option, π w, is π w = C max { S T X, 0 }. 8

9 π h (a) π w (b) 0 C X S T C 0 X S T 9

10 The profit to the holder of a put option is π h = max { X S T, 0 } P, and the profit to the writer of a put is π w = P max { X S T, 0 }. 10

11 π h X P (a) π w (b) 0 P X S T P 0 P X X S T 11

12 Put-Call-Spot Parity What is the payoff of a European call option at time T? S T X if S T X 0 if S T < X. 12

13 Put-Call-Spot Parity What is the payoff of a portfolio consisting of a European call option and a pure-discount bond paying X at time T? S T if S T X X if S T < X. 13

14 Put-Call-Spot Parity Can we get a similar payoff structure using a put option? The payoff of a put option (with the same strike price X) at time T is 0 if S T X X S T if S T < X. 14

15 Put-Call-Spot Parity What can be combined with a put option to get the payoff S T if S T X X if S T < X. at time T? 15

16 Put-Call-Spot Parity The payoff at time T of a put option combined with one unit of the underlying asset is S T if S T X X if S T < X. 16

17 Put-Call-Spot Parity Let then Portfolio A: European call option with strike price X and maturity date T combined with a pure-discount bond paying X at time T. Portfolio B: European put option with strike price X and maturity date T combined with one unit of the underlying asset (protective put). 17

18 Put-Call-Spot Parity Since portfolios A and B yield the same payoff at time T, they should have the same value today. The value of portfolio A today is C + The value of portfolio B today is P + S 0. Put-call parity: X (1+r f ) T. C + X (1 + r f ) T = P + S 0. 18

19 Put-Call-Spot Parity Using continuous discounting, the put-call parity can be rewritten as C + Xe r f T = P + S 0. 19

20 Put-Call-Spot Parity with Dividends Suppose the company considered pays a dividend D T one second before the option expires. In this case, portfolio B pays S T + D T X + D T if S T X if S T < X in period T. To match this payoff structure with a call and a bond, we also need to save enough today to withdraw D T at time T. 20

21 Put-Call-Spot Parity with Dividends That is, portfolio A then consists of buying a call option while saving X+D T (1+r f ) T at the risk-free rate. The put call parity in this case is C + X + D T (1 + r f ) T = P + S 0. 21

22 Options Strategies: Straddle A straddle consists of the simultaneous purchase of one put and one call with the same exercise price. The buyer of a straddle expects a significant change in the stock price, either negative or positive. This could be due, for example, to an expected court ruling that can be very good or very bad for the company. 22

23 Options Strategies: Straddle The profit to the holder of a straddle is given by S T X (C + P) if S T X, π h = X S T (C + P) if S T < X, The profit to the writer of a straddle is C + P (S T X) if S T X, π w = C + P (X S T ) if S T < X, 23

24 Options Strategies: Straddle π h X (C+P) 0 (C+P) Buyer X S T π w P+C 0 P+C X Seller X S T 24

25 Covered Call A trader is said to write a covered call when she already owns the stock on which she writes a call option. The profit from this strategy is π w = C + S T S 0 if S T X, C + X S 0 if S T > X. 25

26 Covered Call π w C+X S 0 0 C S 0 S 0 X Stock Only S T Covered Call 26

27 Protective Put We say that a trader buys a protective put if he buys a put option on a stock he already owns. The profit to such a strategy is as follows (here we refer to the option holder, so we use π h ): S T S 0 P if S T X, π h = X S 0 P if S T < X, 27

28 Protective Put Buying a protective is like buying insurance for a house or a car, it limits losses. 28

29 16.5 Options Protective Put π h 0 X S T X S 0 P 29

30 Options with Short Sales Suppose a trader sells short a stock while buying a call option on it. This strategy insures the trader against high increases in the stock price. The profit from this strategy is S 0 S T C if S T < X, π h = S 0 X C if S T X, 30

31 Options with Short Sales A trader could also sell short the stock and write a put option on it. This stragegy reduces the loss if the stock price increases above S 0, but it reduces the trader s profit on the short sale if the price decreases below S 0. Profits in this case are S 0 X + P if S T < X, π w = S 0 S T + P if S T X, 31

32 Options with Short Sales π h S 0 C Sell Short, Buy Call π w Sell Short, Write Put 0 S 0 X C X S T S 0 X+P 0 X S T 32

33 Strips and Straps A strip is a combination of 2 puts and one call. The profit to the holder of such a combination is 2(X S T ) (2P +C) if S T X, π h = S T X (2P +C) if S T > X, 33

34 Strips and Straps A strap consists of one put and two calls. Its holder s profit is X S T (P + 2C) if S T X, π h = 2(S T X) (P + 2C) if S T > X, 34

35 Spreads A money spread is the simultaneous purchase and sale of options with different strike prices. A time spread is the simultaneous purchase and sale of options with different expiration dates. 35

36 Bullish Spreads Suppose a trader buys a call with premium C 1 and strike price X 1 and writes a call with premium C 2 and strike price X 2. Let X 2 > X 1, which implies that C 1 > C 2 (the lower the strike price, the more valuable the option). The profit from such a strategy is (C 1 C 2 ) if S T X 1, π = S T X 1 (C 1 C 2 ) if X 1 < S T X 2, X 2 X 1 (C 1 C 2 ) if S T > X 2. 36

37 Bullish Spreads The trader using such a strategy believes the stock price is likely to increase, i.e. he is bullish about the stock, and this strategy is called a bullish spread. Were the trader only buying a call option with strike price X 1, his loss if S T X 1 would be C 1, which is greater than C 1 C 2. Hence the goal of this spread is to reduce the loss incurred were the option not exercised. The cost of doing so is the reduction in profit when S T > X 2. 37

38 Bullish Spreads π h X 2 X 1 +C 2 C 1 0 C 2 C 1 X 1 X 2 S T 38

39 Bullish Spreads with Puts Note that a similar payoff profile can be achieved using put options. That is, suppose a trader simultaneously buys and writes put options. The strike price of the option purchased, X 1, is smaller than the strike price of the option sold, X 2. If P 1 denotes the price of the option bought and P 2 the price of the option sold, then P 2 > P 1. This spread is bullish because the trader expects to profit when the stock price rises. 39

40 Bullish Spreads with Puts The profit from this stragegy is P 2 P 1 + X 1 X 2 if S T X 1, π = P 2 P 1 + S T X 2 if X 1 < S T X 2, P 2 P 1 if S T > X 2. 40

41 Bullish Spreads π h P 2 P 1 0 P 2 P 1 +X 1 X 2 X 1 X 2 S T 41

42 Bearish Spreads A trader may enter a bearish spread if he believes a stock price is likely to decline. Using call options, this is done by selling a call option with a lower strike price than the call option purchased. Let C 1 and X 1 denote the premium and strike price of the option purchased and let C 2 and X 2 denote the premium and strike price of the option sold. 42

43 Bearish Spreads The profit from this strategy is C 2 C 1 if S T X 2, π = C 2 C 1 + X 2 S T if X 2 < S T X 1, C 2 C 1 + X 2 X 1 if S T > X 1. 43

44 Basic Characteristics of Option Values An American call option cannot be worth less than a European call option on the same stock, with the same strike price and the same expiration date. Consider two American call options written on the same stock with the same strike price. Then the option with a shorter life cannot be worth more than the other. Consider two call options with the same expiration date written on the same stock. Then the option with a higher strike price cannot be worth more than the option with a lower strike price. 44