Copyright © 2002 by John Stansfield All rights reserved. 9-0 Finance 457 9 Chapter Nine Trading Strategies Involving Options

Copyright © 2002 by John Stansfield All rights reserved.

9-1

Finance 457

9

Chapter Nine

Trading Strategies Involving Options


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


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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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)


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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


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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


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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


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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.


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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


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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


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Finance 457

9.3 Combinations

• Straddle– Buy a call and a put

– Same strike and expiry

• Strips– Buy a call and 2 puts


• Straps– Buy 2 calls and 1 put


• Strangles– Buy a call and a puts

– Same expiry and different strikes


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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)


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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


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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


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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)


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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.


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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

Documents

Copyright © 2002 by John Stansfield All rights reserved. 9-0 Finance 457 9 Chapter Nine Trading Strategies Involving Options