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

2. 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

3. Copyright © 2002 by John Stansfield All rights reserved. 9-2 Finance 457 Notation S 0 current stock price (at time zero: the beginning of life) S T 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 C 0 value of an American call at time zero c 0 value of a European call at time zero P 0 value of an American put at time zero p 0 value of a European put at time zero

4. 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 – K– K K –K + c 0 STST K – c 0 This is also known as writing a synthetic put

5. Copyright © 2002 by John Stansfield All rights reserved. 9-4 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 K K – c 0 STST

6. 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) Long position in a stock Long position in a put – K K K – p 0 STST

7. Copyright © 2002 by John Stansfield All rights reserved. 9-6 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 – p 0 STST

8. 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

9. Copyright © 2002 by John Stansfield All rights reserved. 9-8 Finance 457 Bull Spreads: Created with Calls (K 2 – K 1 ) + ( c 2 – c 1 ) STST 1.Buy a call option on a stock with a certain strike price, K 1 2.Sell a call option on the same stock with a higher strike price K 2 > K 1. Both options have the same expiry Since calls with lower strikes are worth more, cash outflow today: c 2 – c 1 c2c2 K2K2

10. Copyright © 2002 by John Stansfield All rights reserved. 9-9 Finance 457 Bull Spreads: Created with Calls K2K2 c2c2 (K 2 – K 1 ) + ( c 2 – c 1 ) STST The maximum profit is c 2 less the profit on the call we buy with a strike price of K 1 at terminal stock price of K 2 : If the maximum profit > 0, then

11. Copyright © 2002 by John Stansfield All rights reserved. 9-10 Finance 457 Figure 9.3 Bull Spreads: Created with Puts 1.Buy a put option with a low strike K 1 2.Sell a put option with a higher strike K 2 K 1 – p 1 K1K1 – p 1 K 2 – (K 2 – p 2 ) p2p2 p2p2 p1p1 K 1 – p 1 –[(K 2 – K 1 ) – (p 2 – p 1 )] p 2 – p 1 K 2 – p 2 + p 1 STST Cash inflow today p 2 – p 1

12. Copyright © 2002 by John Stansfield All rights reserved. 9-11 Finance 457 Bull Spreads: Created with Puts To get a better maximum profit: 1.Buy a put option with a lower strike K 1 2.Sell a put option with a higher strike K 2 K 1 – p 1 K1K1 – p 1 K2K2 – (K 2 – p 2 ) p2p2 p2p2 p1p1 –[(K 2 – p 2 ) – (K 1 – p 1 )] p 2 – p 1 K 2 – p 2 + p 1 K 1 – p 1 STST

13. Copyright © 2002 by John Stansfield All rights reserved. 9-12 Finance 457 Bear Spreads Using Calls 1.Buy a call with strike K 2 2.Sell a call with a lower strike K1K1 c1c1 –[(K 2 – K 1 ) + (c 2 – c 1 )] c2c2 c2c2 K 2 – K 1 (K 2 – (K 1 +c 1 – c 2 ) STST

14. Copyright © 2002 by John Stansfield All rights reserved. 9-13 Finance 457 Bear Spreads Using Puts 1.Buy a put for p 1 strike K 1 2.Sell a put with a lower strike K 2 K2K2 K1K1 (K 1 – p 1 ) – (K 2 – p 2 ) K 2 – p 2 – (p 1 – p 2 ) K 1 – (p 1 – p 2 ) STST

15. Copyright © 2002 by John Stansfield All rights reserved. 9-14 Finance 457 Butterfly Spreads: With Calls 1.Buy a call with a low strike, K 1 2.Buy a call with a high strike, K 3 3.Sell 2 calls with an average strike, 2c 2 + (K 2 – K 1 – c 1 ) – c 3 (K 2 – K 1 – c 1 ) –c1–c1 K 1 + c 1 K1K1 2c22c2 K 2 +c 2 K2K2 –c3–c3 K3K3 K 3 + c 3 2c 2 – c 1 – c 3 K 1 + c 1 + c 3 – 2c 2 K 3 + 2c 2 – c 1 – c 3

16. Copyright © 2002 by John Stansfield All rights reserved. 9-15 Finance 457 Butterfly Spreads: With Calls The above graph shows an arbitrage. It occurs because –c3–c3 K3K3 K 3 + c 3 2c22c2 K 2 +c 2 K2K2 –c1–c1 K 1 + c 1 K1K1 c2c2 c1c1 c3c3 c2c2 What’s the no arbitrage condition? 2c 2 < c 1 + c 3

17. Copyright © 2002 by John Stansfield All rights reserved. 9-16 Finance 457 Intermezzo The red line represents the payoff of a portfolio of 2 call options (one call with a strike of K 1, and one call with a strike price of K 3 ). The average strike price of the options in the portfolio is K 2 The green line represents 2 call options on the portfolio with a strike price of K 2 where K3K3 K1K1 A portfolio of options is worth more than an option on a portfolio: K2K2 K 2 – K 1 K 3 – K 2 = STST

18. Copyright © 2002 by John Stansfield All rights reserved. 9-17 Finance 457 Butterfly Spreads: With Puts 1.Buy a put with a low strike, K 1 2.Buy a put with a high strike, K 3 3.Sell 2 puts with an average strike, 2p 2 + (K 3 – K 2 – p 3 ) – p 1 Let’s evaluate this K 1 – p 1 –p1–p1 K1K1 2p22p2 K 2 – p 2 K2K2 –p3–p3 K3K3 K3–p3K3–p3 (K 3 – K 2 – p 3 )

19. Copyright © 2002 by John Stansfield All rights reserved. 9-18 Finance 457 Butterfly Spreads: With Puts: Max loss (K 3 – p 3 ) + (K 1 – p 1 ) – 2(K 2 – p 2 ) Consider the payoff at S T = 0: As a summation of the profit on the two calls bought less the two calls sold: Recall that 2p 2 – p 1 – p 3 K 3 – p 3 + K 1 – p 1 – 2K 2 +2 p 2

20. Copyright © 2002 by John Stansfield All rights reserved. 9-19 Finance 457 Butterfly Spreads: With Puts 1.Buy a put with a low strike, K 1 2.Buy a put with a high strike, K 3 3.Sell 2 puts with an average strike, 2p 2 + (K 3 – K 2 – p 3 ) – p 1 K 1 – p 1 –p1–p1 K1K1 2p22p2 K2K2 –p3–p3 K3K3 2p 2 – p 1 – p 3 K 1 + p 1 + p 3 – 2p 2 K 3 + 2p 2 – p 1 – p 3

21. 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. =2c 2 – c 1 – c 3 2p 2 – p 1 – p 3 c 1 – p 1 = S 0 – K 1 e -rT c 2 – p 2 = S 0 – K 2 e -rT c 3 – p 3 = S 0 – K 3 e -rT 2K 2 = K 1 + K 3 –2 K 2 e -rT = – K 1 e -rT – K 3 e -rT 2S 0 –2 K 2 e -rT = S 0 – K 1 e -rT + S 0 – K 3 e -rT 2c 2 – 2 p 2 = c 1 – p 1 + c 3 – p 3

22. Copyright © 2002 by John Stansfield All rights reserved. 9-21 Finance 457 Calendar Spreads: Using Calls K1K1 1. Buy a long-lived option strike K 1 2. Sell a short-lived option with same strike –c long c short c short – c long 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.

23. Copyright © 2002 by John Stansfield All rights reserved. 9-22 Finance 457 Calendar Spreads: Using Puts 1. Buy a long-lived put option strike K 1 2. Sell a short-lived put option with same strike –p long K1K1 p short S short p short – p long

24. 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 K1K1 1.Buy a long-lived option strike K 1 2.Short a shorter-lived option with strike K 2 –c1–c1 K2K2 c2c2 c 2 – c 1 S short

25. 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

26. 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 –c1–c1 K 1 + c 1 K1K1 K 1 – p 1 –p1–p1 –(p 1 + c 1 ) K 1 – p 1 – c 1 STST K 1 – (p 1 + c 1 )K 1 + (p 1 + c 1 )

27. Copyright © 2002 by John Stansfield All rights reserved. 9-26 Finance 457 Strips Strip is long one call and 2 puts with the same strike and expiry –c1–c1 K 1 + c 1 K1K1 –(2p 1 + c 1 ) –p1–p1 K 1 – p 1 –2p1–2p1 K 1 + 2p 1 + c 1 STST 2(K 1 – p 1 ) K 1 – p 1

28. Copyright © 2002 by John Stansfield All rights reserved. 9-27 Finance 457 Straps A Strap is long 2 calls and one put on same strike and expiry –c1–c1 K 1 + c 1 K1K1 –2c1–2c1 –p1–p1 K 1 – p 1 –(p 1 + 2c 1 ) K 1 – (p 1 + 2c 1 ) STST

29. 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 STST –p1–p1 K 1 – p 1 –c1–c1 K2K2 K1K1 K 2 + (p 1 + c 1 ) K 1 – (p 1 + c 1 ) – (p 1 + c 1 ) K 1 – (p 1 + c 1 )

30. 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.

31. 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