Copyright © 2002 by John Stansfield All rights reserved. 9-1 F i n a n c e 4 5 7 9 Chapter Nine Trading Strategies Involving Options
Dec 22, 2015
Copyright © 2002 by John Stansfield All rights reserved.
9-1
Finance 457
9
Chapter Nine
Trading Strategies Involving Options
Copyright © 2002 by John Stansfield All rights reserved.
9-2
Finance 457
Chapter Outline
9.1 Strategies Involving a Single Option and a Stock
9.2 Spreads
9.3 Combinations
9.4 Other Payoffs
9.5 Summary
Copyright © 2002 by John Stansfield All rights reserved.
9-3
Finance 457
Notation
• NotationS0 current stock price (at time zero: the beginning of life)
ST stock price at expiry
K is the exercise price
T is the time to expiry
r is the nominal risk-free rate; continuously compounded; maturity T
C0 value of an American call at time zero
c0 value of a European call at time zero
P0 value of an American put at time zero
p0 value of a European put at time zero
Copyright © 2002 by John Stansfield All rights reserved.
9-4
Finance 457
9.1 Strategies Involving a Single Option and a Stock: Writing a Covered Call• Long position in a stock bought at $K• Short position in a call
– K
K0cK
–K + c0
0c
ST
K – c0
This is also known as writing a synthetic put
Copyright © 2002 by John Stansfield All rights reserved.
9-5
Finance 457
9.1 Strategies Involving a Single Option and a Stock: Synthetic Put
Short position in a stock
Long position in a call
K
K0cK
K – c0
0c
K – c0
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-6
Finance 457
9.1 Strategies Involving a Single Option and a Stock: Protective Put (Synthetic Call)
Long position in a stock
Long position in a put– K
K
K – p0
0p
0pK
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-7
Finance 457
9.1 Strategies Involving a Single Option and a Stock: Synthetic Call
• Short position in a stock • Short position in a put
K
K
K – p0
0p
ST
0pK
Copyright © 2002 by John Stansfield All rights reserved.
9-8
Finance 457
9.2 Spreads
• A spread involves taking a position in two or more options of the same type (e.g. two calls or three puts)– Bull Spreads
– Bear Spreads
– Butterfly Spreads
– Calendar Spreads
– Diagonal Spreads
Copyright © 2002 by John Stansfield All rights reserved.
9-9
Finance 457
Bull Spreads: Created with Calls
211 ccK
(K2 – K1) + ( c2 – c1)
ST
1. Buy a call option on a stock with a certain strike price, K1
2. Sell a call option on the same stock with a higher strike price K2 > K1.
• Both options have the same expiry
• Since calls with lower strikes are worth more, cash outflow today: c2 – c1
1c
1K
c2
K2
)( 21 cc
Copyright © 2002 by John Stansfield All rights reserved.
9-10
Finance 457
Bull Spreads: Created with Calls
1c
)( 21 cc K2
c2
1K
211 ccK
(K2 – K1) + ( c2 – c1)
ST
• The maximum profit is c2 less the profit on the call we buy with a strike price of K1 at terminal stock price of K2 :
1122 ][ cKKc
1122 ][ cKKc • If the maximum profit > 0, then
Copyright © 2002 by John Stansfield All rights reserved.
9-11
Finance 457
Figure 9.3 Bull Spreads: Created with Puts
1. Buy a put option with a low strike K1
2. Sell a put option with a higher strike K2
K1 – p1
K1– p1
K2
– (K2 – p2)
p2
p2 p1
K1 – p1
–[(K2 – K1) – (p2– p1)]
p2 – p1
K2 – p2 + p1
ST
Cash inflow today p2 – p1
Copyright © 2002 by John Stansfield All rights reserved.
9-12
Finance 457
Bull Spreads: Created with Puts
To get a better maximum profit:
1. Buy a put option with a lower strike K1
2. Sell a put option with a higher strike K2
K1 – p1
K1
– p1K2
– (K2 – p2)
p2
p2
p1
–[(K2 – p2) – (K1 – p1)]
p2 – p1
K2 – p2 + p1
K1 – p1
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-13
Finance 457
Bear Spreads Using Calls
1. Buy a call with strike K2
2. Sell a call with a lower strike
2K
2cK1
c1
21 cc
–[(K2 – K1) + (c2 – c1)]
211 ccK
c2
c2
K2 – K1
(K2 – (K1 +c1 – c2 )
21 cc ST
Copyright © 2002 by John Stansfield All rights reserved.
9-14
Finance 457
Bear Spreads Using Puts
1. Buy a put for p1 strike K1
2. Sell a put with a lower strike K2
1pK2
11 pK
K1
2p(K1– p1) – (K2 – p2)
2p1p
22 pK
K2 – p2
– (p1– p2)
K1– (p1– p2)
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-15
Finance 457
Butterfly Spreads: With Calls1. Buy a call with a low strike, K1
2. Buy a call with a high strike, K3
3. Sell 2 calls with an average strike, 231
2
KKK
2c2+ (K2 – K1 – c1) – c3
(K2 – K1 – c1)
–c1K1+ c1
K1
2c2
K2+c2
K2–c3 K3
K3+ c3
2c2 – c1 – c3
K1 + c1 + c3– 2c2K3 + 2c2 – c1 – c3
Copyright © 2002 by John Stansfield All rights reserved.
9-16
Finance 457
Butterfly Spreads: With Calls
The above graph shows an arbitrage. It occurs because
231
2
ccc
–c3 K3
K3+ c3
2c2
K2+c2
K2–c1
K1+ c1
K1
c2
c1
c3
c2
What’s the no arbitrage condition? 2c2 < c1 + c3
Copyright © 2002 by John Stansfield All rights reserved.
9-17
Finance 457
Intermezzo
The red line represents the payoff of a portfolio of 2 call options (one call with a strike of K1 , and one call with a strike price of K3). The average strike price of the options in the portfolio is K2
The green line represents 2 call options on the portfolio with a strike price of K2
where
K3K1
231
2
KKK
A portfolio of options is worth more than an option on a portfolio:
K2
K2 – K1 K3 – K2=
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-18
Finance 457
Butterfly Spreads: With Puts1. Buy a put with a low strike, K1
2. Buy a put with a high strike, K3
3. Sell 2 puts with an average strike, 231
2
KKK
2p2+ (K3 – K2 – p3) – p1
Let’s evaluate this
K1– p1
–p1
K1– p1
K1
2p2
K2– p2
K2
–p3
K3
K3–p3
(K3 – K2 – p3)
Copyright © 2002 by John Stansfield All rights reserved.
9-19
Finance 457
Butterfly Spreads: With Puts: Max loss
(K3– p3) + (K1– p1) – 2(K2– p2 )
Consider the payoff at ST = 0:
As a summation of the profit on the two calls bought less the two calls sold:
Recall that
2p2 – p1 – p3
231
2
KKK
K3– p3 + K1– p1 – 2K2+2 p2
Copyright © 2002 by John Stansfield All rights reserved.
9-20
Finance 457
Butterfly Spreads: With Puts
1. Buy a put with a low strike, K1
2. Buy a put with a high strike, K3
3. Sell 2 puts with an average strike, 231
2
KKK
2p2+ (K3 – K2 – p3) – p1
K1– p1
–p1
K1
2p2
K2
–p3
K3
2p2 – p1 – p3
K1 + p1 + p3– 2p2
K3 + 2p2 – p1 – p3
Copyright © 2002 by John Stansfield All rights reserved.
9-21
Finance 457
Put-Call Parity Revisited
• Put-call parity shows that the initial investment required for butterfly spreads is the same for butterfly spreads created with calls as with puts.
= 2c2 – c1 – c32p2 – p1 – p3
c1 – p1 = S0 – K1e-rT
c2 – p2 = S0 – K2e-rT
c3 – p3 = S0 – K3e-rT
231
2
KKK
2K2 = K1 + K3
–2 K2e-rT = – K1e-rT – K3e-rT
2S0–2 K2e-rT = S0– K1e-rT + S0 – K3e-rT
2c2 – 2 p2 = c1 – p1 + c3 – p3
Copyright © 2002 by John Stansfield All rights reserved.
9-22
Finance 457
Calendar Spreads: Using Calls
K1
1. Buy a long-lived option strike K1
2. Sell a short-lived option with same strike
–clong
cshort
cshort – clong S short
• The key here is to recall the shape of an American option with speculative value.
In a neutral calendar spread, strike prices close to the current price are chosen. A bullish calendar spread has higher strike prices and a bearish calendar spread has lower strikes.
Copyright © 2002 by John Stansfield All rights reserved.
9-23
Finance 457
Calendar Spreads: Using Puts
1. Buy a long-lived put option strike K1
2. Sell a short-lived put option with same strike
–plong
K1
pshort
S shortpshort – plong
Copyright © 2002 by John Stansfield All rights reserved.
9-24
Finance 457
Diagonal Spreads
1. Long position in one call and a short position in another.
2. Both the expiry and the strike are different
K1
1. Buy a long-lived option strike K1
2. Short a shorter-lived option with strike K2
–c1
K2
c2
c2 – c1
S short
Copyright © 2002 by John Stansfield All rights reserved.
9-25
Finance 457
9.3 Combinations
• Straddle– Buy a call and a put
– Same strike and expiry
• Strips– Buy a call and 2 puts
– Same strike and expiry
• Straps– Buy 2 calls and 1 put
– Same strike and expiry
• Strangles– Buy a call and a puts
– Same expiry and different strikes
Copyright © 2002 by John Stansfield All rights reserved.
9-26
Finance 457
Straddle
1. Buy a call and a put
2. Same strike and expiry
–c1K1+ c1
K1
K1– p1
–p1 K1– p1
–(p1+ c1)
K1– p1– c1
ST
K1 – (p1+ c1) K1 + (p1+ c1)
Copyright © 2002 by John Stansfield All rights reserved.
9-27
Finance 457
Strips
• Strip is long one call and 2 puts with the same strike and
expiry
–c1K1+ c1
K1
–(2p1+ c1)
–p1
K1– p1
–2p1
K1 + 2p1+ c1
2
2 111
cpK
111 )(2 cpK
ST
2(K1– p1 )
K1– p1
Copyright © 2002 by John Stansfield All rights reserved.
9-28
Finance 457
Straps
A Strap is long 2 calls and one put on same strike
and expiry
–c1K1+ c1
K1
–2c1
–p1
K1– p1
K1– p1
–(p1+ 2c1)
K1 – (p1+ 2c1)
21
11
pcK
K1 – (p1+ 2c1)
ST
Copyright © 2002 by John Stansfield All rights reserved.
9-29
Finance 457
Strangles
Buy a put and a call with the same expiry and different exercise prices
ST–p1
K1– p1
–c1
K2K1 K2 + (p1+ c1)
K1 – (p1+ c1)
– (p1+ c1)
K1 – (p1+ c1)
Copyright © 2002 by John Stansfield All rights reserved.
9-30
Finance 457
9.4 Other Payoffs
• We have only scratched the surface of financial engineering in this chapter.
• If European options expiring at time T were available with every single possible strike price, any payoff function at time T could in theory be obtained.
Copyright © 2002 by John Stansfield All rights reserved.
9-31
Finance 457
9.5 Summary
• A number of common trading strategies involve a single option and the underlying stock.– These include synthetic options
– Protective Puts
– Covered Calls
• Taking a position in multiple options– Spreads
– Straddles
– Strips
– Straps
– Et cetera