Transcript + K

9-0
Finance 457
Trading Strategies Involving Options
9
Chapter Nine
Copyright © 2002 by John Stansfield All rights reserved.
9-1
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.
9-2
Finance 457
Notation
• Notation
S0 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.
9-3
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
c0
ST
K  c0
–K + c0
–K
K
K – c0
This is also known as writing a synthetic put
Copyright © 2002 by John Stansfield All rights reserved.
9-4
Finance 457
9.1 Strategies Involving a Single Option
and a Stock: Synthetic Put
K
K – c0
ST
 c0
K  c0
K
Short position in a stock K – c0
Long position in a call
Copyright © 2002 by John Stansfield All rights reserved.
9-5
Finance 457
9.1 Strategies Involving a Single Option
and a Stock: Protective Put (Synthetic Call)
K – p0
ST
 p0
K  p0
K
–K
Long position in a stock
Long position in a put
Copyright © 2002 by John Stansfield All rights reserved.
9-6
Finance 457
9.1 Strategies Involving a Single Option
and a Stock: Synthetic Call
K
• Short position in a stock
• Short position in a put
p0
ST
K  p0
K
K – p0
Copyright © 2002 by John Stansfield All rights reserved.
9-7
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.
9-8
Finance 457
Bull Spreads: Created with Calls
c2
(K2 – K1) + ( c2 – c1)
K1
 (c1  c2 )
K2
ST
 c1
K1  c1  c2
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
Copyright © 2002 by John Stansfield All rights reserved.
9-9
Finance 457
Bull Spreads: Created with Calls
c2
(K2 – K1) + ( c2 – c1)
 (c1  c2 )
K1
K2
ST
 c1
K1  c1  c2
• 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 :
c2  [ K 2  K1 ]  c1
• If the maximum profit > 0, then c2  [ K 2  K1 ]  c1
Copyright © 2002 by John Stansfield All rights reserved.
9-10
Finance 457
Figure 9.3 Bull Spreads: Created with Puts
K1 – p1
p2
p2 – p1
– p1
p2
p1
K1
K2
ST
K2 – p2 + p1
–[(K2 – K1) – (p2– p1)]
K1 – p1
– (K2 – p2)
Cash inflow today p2 – p1
1. Buy a put option with a low strike K1
2. Sell a put option with a higher strike K2
Copyright © 2002 by John Stansfield All rights reserved.
9-11
Finance 457
Bull Spreads: Created with Puts
K1 – p1
p2
p2 – p1
– p1
p1
p2
K1
K2
ST
K2 – p2 + p1
–[(K2 – p2) – (K1 – p1)]
K1 – p1
– (K2 – p2)
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
Copyright © 2002 by John Stansfield All rights reserved.
9-12
Finance 457
Bear Spreads Using Calls
1. Buy a call with strike K2
2. Sell a call with a lower strike
K2 – K1
c1
c2
c1  c2
 c2
–[(K2 – K1) + (c2 – c1)]
c2
(K2 – (K1 +c1 – c2 )
c1  c2
ST
K1
K2
K1  c1  c2
Copyright © 2002 by John Stansfield All rights reserved.
9-13
Finance 457
Bear Spreads Using Puts
1. Buy a put for p1 strike K1
2. Sell a put with a lower strike K2
K1  p1
K2 – p2
(K1– p1) – (K2 – p2)
p2
– (p1– p2)
 p1
 K 2  p2
K2
K1
p1
K1– (p1– p2)
Copyright © 2002 by John Stansfield All rights reserved.
ST
p2
9-14
Finance 457
Butterfly Spreads: With Calls
1. Buy a call with a low strike, K1
2. Buy a call with a high strike, K3
K1  K 3
3. Sell 2 calls with an average strike, K 2 
2
(K2 – K1 – c1)
2c2+ (K2 – K1 – c1) – c3
2c2
2c2 – c1 – c3
–c3
–c1
K1
K2
K3
K2+c2 K3+ c3
K1+ c1
K1 + c1 + c3– 2c2
K3 + 2c2 – c1 – c3
Copyright © 2002 by John Stansfield All rights reserved.
9-15
Finance 457
Butterfly Spreads: With Calls
2c2
–c3
–c1
c2
c2
c1
c3
K1
K2
K3
K2+c2
K1+ c1
K3+ c3
The above graph shows an arbitrage. It occurs
because
c1  c3 What’s the no arbitrage condition?
c2 
2
2c2 < c1 + c3
Copyright © 2002 by John Stansfield All rights reserved.
9-16
Finance 457
Intermezzo
A portfolio of options is worth more than an option on a portfolio:
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
K1  K 3
K2 
2
K1
K2
K3
K2 – K1 = K3 – K2
Copyright © 2002 by John Stansfield All rights reserved.
ST
9-17
Finance 457
Butterfly Spreads: With Puts
1. Buy a put with a low strike, K1
2. Buy a put with a high strike, K3
K  K3
3. Sell 2 puts with an average strike, K 2  1
2
K1– p1
(K3 – K2 – p3)
2p2
2p2+ (K3 – K2 – p3) – p1
Let’s evaluate this–p
1
–p3
K1
K1– p1
K2
K2– p2
K3–p3
Copyright © 2002 by John Stansfield All rights reserved.
K3
9-18
Finance 457
Butterfly Spreads: With Puts: Max loss
Consider the payoff at ST = 0:
As a summation of the profit on the two calls bought less
the two calls sold:
(K3– p3) + (K1– p1) – 2(K2– p2 )
K1  K 3
Recall that K 2 
2
K3– p3 + K1– p1 – 2K2+2 p2
2p2 – p1 – p3
Copyright © 2002 by John Stansfield All rights reserved.
9-19
Finance 457
Butterfly Spreads: With Puts
K1– p1
2p2
2p2+ (K3 – K2 – p3) – p1
2p2 – p1 – p3
–p1
–p3
K1
K2
K3
K3 + 2p2 – p1 – p3
1. Buy a put with a low strike, K1
K1 + p1 + p3– 2p2
2. Buy a put with a high strike, K3
K1  K 3
3. Sell 2 puts with an average strike, K 2 
2
Copyright © 2002 by John Stansfield All rights reserved.
9-20
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.
K1  K 3
2K2 = K1 + K3
K2 
2
c1 – p1 = S0 – K1e-rT
c2 – p2 = S0 – K2e-rT
–2 K2e-rT = – K1e-rT – K3e-rT
2S0–2 K2e-rT = S0– K1e-rT + S0 – K3e-rT
c3 – p3 = S0 – K3e-rT
2c2 – 2 p2 = c1 – p1 + c3 – p3
2p2 – p1 – p3 =
2c2 – c1 – c3
Copyright © 2002 by John Stansfield All rights reserved.
9-21
Finance 457
Calendar Spreads: Using Calls
1. Buy a long-lived option strike K1
2. Sell a short-lived option with same strike
cshort
cshort – clong
–clong
K1
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.
9-22
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
pshort
pshort – plong
–plong
S short
K1
Copyright © 2002 by John Stansfield All rights reserved.
9-23
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
c2
c2 – c1
–c1
K1
K2
S short
1. Buy a long-lived option strike K1
2. Short a shorter-lived option with strike K2
Copyright © 2002 by John Stansfield All rights reserved.
9-24
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.
9-25
Finance 457
Straddle
1. Buy a call and a put
2. Same strike and expiry
K1– p1
K1– p1– c1
–p1
–c1
–(p1+ c1)
ST
K1– p1 K1
K1+ c1
K1 – (p1+ c1)
K1 + (p1+ c1)
Copyright © 2002 by John Stansfield All rights reserved.
9-26
2(K1– p1 )
Finance 457
Strips
2( K1  p1 )  c1
K1– p1
–p1
–2p1 –c1
–(2p1+ c1)
• Strip is long one call and 2
puts with the same strike and
expiry
ST
K1
K1– p1
K1+ c1
K1 + 2p1+ c1
2 p1  c1
K1 
Copyright © 2002 by John Stansfield All rights reserved.
2
9-27
Finance 457
Straps
K1– p1
A Strap is long 2 calls and
one put on same strike
and expiry
K1 – (p1+ 2c1)
ST
–p1
K1
–c1
K1– p1 K1+ c1
–2c1
–(p1+ 2c1)
K1 – (p1+ 2c1)
p1
K1  c1 
2
Copyright © 2002 by John Stansfield All rights reserved.
9-28
Finance 457
Strangles
Buy a put and a call with the same expiry and
different exercise prices
K1– p1
K1 – (p1+ c1)
–p1
K1
–c1
– (p1+ c1)
K2
ST
K2 + (p1+ c1)
K1 – (p1+ c1)
Copyright © 2002 by John Stansfield All rights reserved.
9-29
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.
9-30
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
Copyright © 2002 by John Stansfield All rights reserved.