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Chapter Nine

Trading Strategies Involving Options. 9. Chapter Nine. 9.1 Strategies Involving a Single Option and a Stock 9.2 Spreads 9.3 Combinations 9.4 Other Payoffs 9.5 Summary. Chapter Outline. Notation S 0 current stock price (at time zero: the beginning of life) S T stock price at expiry

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Chapter Nine

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  1. Trading Strategies Involving Options 9 Chapter Nine

  2. 9.1 Strategies Involving a Single Option and a Stock 9.2 Spreads 9.3 Combinations 9.4 Other Payoffs 9.5 Summary Chapter Outline

  3. 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 Notation

  4. Long position in a stock bought at $K Short position in a call 9.1 Strategies Involving a Single Option and a Stock: Writing a Covered Call ST K –K + c0 K –c0 –K This is also known as writing a synthetic put

  5. Short position in a stock Long position in a call 9.1 Strategies Involving a Single Option and a Stock: Synthetic Put K K –c0 ST K K –c0

  6. Long position in a stock Long position in a put 9.1 Strategies Involving a Single Option and a Stock: Protective Put (Synthetic Call) K –p0 ST K – K

  7. Short position in a stock Short position in a put K –p0 9.1 Strategies Involving a Single Option and a Stock: Synthetic Call K ST K

  8. 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 9.2 Spreads

  9. Buy a call option on a stock with a certain strike price, K1 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 c2 (K2 – K1) + (c2 – c1) K2 Bull Spreads: Created with Calls ST

  10. 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 : • If the maximum profit > 0, then Bull Spreads: Created with Calls c2 (K2 – K1) + (c2 – c1) ST K2

  11. Buy a put option with a low strike K1 Sell a put option with a higher strike K2 K1– p1 p2 p2 p2 – p1 p1 K1 K2 – p1 K1– p1 – (K2 –p2) Figure 9.3 Bull Spreads: Created with Puts ST K2–p2 + p1 –[(K2 –K1) – (p2– p1)] Cash inflow today p2 –p1

  12. To get a better maximum profit: Buy a put option with a lower strike K1 Sell a put option with a higher strike K2 p2 p2 p2 – p1 p1 K1 K2 K1– p1 – (K2 –p2) Bull Spreads: Created with Puts K1– p1 ST – p1 K2–p2 + p1 –[(K2 –p2) – (K1– p1)]

  13. Buy a call with strike K2 Sell a call with a lower strike K2 – K1 c1 c2 c2 (K2 – (K1 +c1– c2 ) K1 Bear Spreads Using Calls ST –[(K2 – K1) + (c2 – c1)]

  14. Buy a put for p1 strike K1 Sell a put with a lower strike K2 K2– p2 K2 K1 Bear Spreads Using Puts (K1– p1) – (K2 – p2) ST – (p1– p2) K1– (p1– p2)

  15. (K2 –K1 –c1) 2c2+ (K2 –K1 –c1) – c3 2c2 –c3 K1 K3 K2 –c1 K2+c2 K3+ c3 K1+ c1 • Buy a call with a low strike, K1 • Buy a call with a high strike, K3 • Sell 2 calls with an average strike, Butterfly Spreads: With Calls 2c2 –c1 – c3 K1 + c1 + c3– 2c2 K3 + 2c2 –c1 – c3

  16. 2c2 c2 c2 c3 –c3 c1 K1 K2 K3 –c1 K2+c2 K1+ c1 K3+ c3 Butterfly Spreads: With Calls • The above graph shows an arbitrage. It occurs because What’s the no arbitrage condition? 2c2 <c1 + c3

  17. K2 – K1 = K3 – K2 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 ST K1 K2 K3

  18. K1– p1 (K3 –K2 –p3) 2p2+ (K3 –K2 –p3) – p1 K1 –p1 2p2 K1– p1 K2 • Buy a put with a low strike, K1 • Buy a put with a high strike, K3 • Sell 2 puts with an average strike, K3 K2– p2 –p3 K3–p3 Butterfly Spreads: With Puts Let’s evaluate this

  19. 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 ) Recall that K3– p3 + K1– p1– 2K2+2p2 2p2 –p1 – p3

  20. 2p2+ (K3 –K2 –p3) – p1 2p2 –p1 – p3 Butterfly Spreads: With Puts K1– p1 2p2 K1 K2 K3 –p1 –p3 K3 + 2p2 –p1 – p3 • Buy a put with a low strike, K1 • Buy a put with a high strike, K3 • Sell 2 puts with an average strike, K1 + p1 + p3– 2p2

  21. 2p2 –p1 – p3 = 2c2 –c1 – c3 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. • 2K2 = K1 + K3 • c1–p1 = S0 –K1e-rT • c2–p2 = S0 –K2e-rT • –2K2e-rT = –K1e-rT –K3e-rT • c3–p3 = S0 –K3e-rT • 2S0–2K2e-rT = S0–K1e-rT + S0–K3e-rT 2c2– 2p2 =c1–p1 + c3–p3

  22. The key here is to recall the shape of an American option with speculative value. K1 Calendar Spreads: Using Calls 1. Buy a long-lived option strike K1 2. Sell a short-lived option with same strike cshort cshort – clong S short –clong 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. pshort K1 –plong Calendar Spreads: Using Puts 1. Buy a long-lived put option strike K1 2. Sell a short-lived put option with same strike pshort – plong S short

  24. Long position in one call and a short position in another. Both the expiry and the strike are different K1 Diagonal Spreads c2 S short c2 – c1 K2 –c1 • Buy a long-lived option strike K1 • Short a shorter-lived option with strike K2

  25. 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 9.3 Combinations

  26. Buy a call and a put Same strike and expiry K1– p1 –p1 K1 K1– p1 –c1 K1+ c1 –(p1+ c1) K1– (p1+ c1) K1+ (p1+ c1) Straddle K1– p1– c1 ST

  27. Strip is long one call and 2 puts with the same strike and expiry 2(K1– p1 ) K1– p1 K1 –c1 –p1 K1+ c1 –2p1 K1– p1 –(2p1+ c1) Strips ST K1 + 2p1+ c1

  28. A Strap is long 2 calls and one put on same strike and expiry K1– p1 K1 –c1 –p1 K1+ c1 K1– p1 –(p1+ 2c1) –2c1 Straps K1 – (p1+ 2c1) ST K1 – (p1+ 2c1)

  29. Buy a put and a call with the same expiry and different exercise prices K2 –c1 Strangles K1– p1 K1 – (p1+ c1) ST –p1 K1 K2 + (p1+ c1) – (p1+ c1) K1 – (p1+ c1)

  30. 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. 9.4 Other Payoffs

  31. 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 9.5 Summary

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