1. © Paul Koch 1-1 Chapter 11. Trading Strategies with Options I. Basic Combinations. A. Calls & Puts can be combined with other building blocks (Stocks & Bonds) to give any payoff pattern desired. 1. Assume European options with same exp. (T), K, & underlying. 2.Already know payoff patterns for buying & selling calls & puts: a. Calls._______│_______ S _______│________ S __________ K K b. Puts._______│_______ S _______│________ S K ___________ K 3.Consider payoffs for long & short positions on: a. Stocks._______│_______ S _______│________ S K K b. Bonds._______│_______ S _______│________ S K _ _ _ _ _ _ K _ _ _ _ _ _ _ +c -c +p -p +S-S +B -B

2. © Paul Koch 1-2 I.B. Protective Put (S+P) B. Buy Stock (+S) and Buy Put (+P) Value S +S +P S+P

3. © Paul Koch 1-3 I.C. Principal - Protected Note * (B+C) C. Buy Bond (+B) and Buy Call (+C) * If you buy a zero-coupon, deep discount bond, the initial outlay (B) is small (esp. if r is high); If volatility of S is low, call (C) is cheap; Then the initial cost (B+C) may be set ≈ K (PPN). Then your principal is protected (worst outcome; S low, call OTM, get to keep Bond payoff (K). Value S +B +C B+C

4. © Paul Koch 1-4 I.D. Put-Call Parity (S+P = B+C) D. B & C give same payoff pattern (S+P = B+C) Value S +B +C B+C +S +P S+P

5. © Paul Koch 1-5 I.E. Writing a Covered Call (+S - C) E. Buy Stock (+S) and Sell Call (-C) Value S +S -C S - CS - C S+P = B+C ↓ S - C = B - P

6. © Paul Koch 1-6 I.F. Buying a Straddle (+C+P) F. Buy Call (+C) and Buy Put (+P) Value S +C +P C+P

7. © Paul Koch 1-7 I.G. Selling a Straddle (-C-P) G. Sell a Call (-C) and Sell a Put (-P) Value S -C -P -C - P

8. © Paul Koch 1-8 I.H. Buying a Strangle (+C+P) – with Different K’s H. Buy Call with K 2 ; Buy Put with K 1 (K 1 < K 2 ) Value S +C 2 +P 1 C 2 +P 1 K2K2 K1K1

9. © Paul Koch 1-9 II. How to Plot Payoff Pattern for Any Combination Problem: Given any Combination of shares, bonds, & options, graph the Payoff Pattern for the Intrinsic Value; show slopes of line segments; & show break-even points. Three Steps: 1. Compute the initial cost / revenue of the Combination, and get values of S where all options are worth zero (ATM or OTM). For these values of S, Combination is worth the initial cost / revenue. 2. Get values of S where one option is ITM. For these values of S, Combination Value = initial cost / revenue + intrinsic value of this option. 3. Get values of S where next option is ITM. For these values of S, Combination Value = old value + intrinsic value of this option. Continue until you examine all values of S, for all options in combination.

10. © Paul Koch 1-10 II. How to Plot Payoff Pattern for Any Combination Example 1: Strip; Buy 1 Call & 2 Puts with same K = $50; C = $5; P = $6. 1. Initial Cost = (-1) x ($5) + (-2) x ($6) = -$17. At S = K = $50, both options ATM, Combination Value = -$17. 2. If S > $50, Call ITM, Combination Value = -$17 + 1(S - K). (coeff. of S = +1) 3. If S < $50, Puts ITM, Combination Value = -$17 + 2(K - S). (coeff. of S = -2) K = $50 ____________________________________________________________ S $41.50 │ $67 │ slope = -2 │ slope = +1 │ -17 │ │

11. © Paul Koch 1-11 II. How to Plot Payoff Pattern for Any Combination Example 2: Buy 1 Call with K 1 = $40 (C 1 = $8); Sell 2 Calls with K 2 = $45 (C 2 = $5). 1. Initial Cost = (-1) x ($8) + (+2) x ($5) = +$2. If S < K 1 = $40, both options OTM, Combination Value = +$2. (coeff of S = 0) 2. If 40 < S < $45, C 1 is ITM, Value = +$2 + 1(S - K 1 ). (coeff = +1) 3. If S > $45, C 1 & C 2 are ITM, Value = +$2 + 1(S - K 1 ) - 2(S - K 2 ). (coeff = -1) K = $40 K = $45 │ 7 │ │ │ slope = +1 │ │ slope = -1 2 │ slope = 0 │ _____________________________________________________ S │ $45 $52 │

12. © Paul Koch 1-12 II.A. Bull Spread with Calls (C 1 - C 2 ) A. Buy Call with K 1 (pay C 1 ); Sell Call with K 2 (receive C 2 ) (K 1 C 2 ); So (-C 1 +C 2 ) < 0; initial outflow (left) Value S K2K2 K1K1 (-C 1 +C 2 ) +C 2 -C 1

13. © Paul Koch 1-13 II.B. Bull Spread with Puts (P 1 - P 2 ) B. Buy Put with K 1 (pay P 1 ); Sell Put with K 2 (receive P 2 ) (K 1 0; initial inflow (right) Value S K2K2 K1K1 -P 1 +P 2 (-P 1 +P 2 )

14. © Paul Koch 1-14 II.C. Bear Spread with Calls (C 2 - C 1 ) C. Sell Call with K 1 (receive C 1 ); Buy Call with K 2 (pay C 2 ) (K 1 C 2 ); So (+C 1 -C 2 ) > 0; initial inflow (left) Value S K2K2 K1K1 C2C2 C1C1 (+C 1 -C 2 )

15. © Paul Koch 1-15 II.D. Bear Spread with Puts (P 2 - P 1 ) D. Sell Put with K 1 (receive P 1 ); Buy Put with K 2 (pay P 2 ) (K 1 < K 2 ); Thus (P 1 < P 2 ); So (+P 1 -P 2 ) < 0; initial outflow (right) Value S K2K2 K1K1 P1P1 P2P2 (+P 1 -P 2 )

16. © Paul Koch 1-16 II.E. Butterfly Spread with Calls (C 1 - 2C 2 + C 3 ) E. Buy 1 Call with K 1 ; Sell 2 Calls with K 2 ; Buy 1 Call with K 3 (K 1 C 2 > C 3 ); initial outflow (left).

17. © Paul Koch 1-17 II.F. Butterfly Spread with Puts (P 1 - 2P 2 + P 3 ) F. Buy 1 Put with K 1 ; Sell 2 Puts with K 2 ; Buy 1 Put with K 3 (K 1 < K 2 < K 3 ); Thus, (P 1 < P 2 < P 3 ); initial outflow (right).

18. © Paul Koch 1-18 III.A. Graphing Total, Intrinsic, and Extrinsic Value Total Value Intrinsic Value Extrinsic Value S S S K K K 0 < dc/dS < 1 If σ ↑dc/dS = 1 dc/dS = 0

19. © Paul Koch 1-19 III.B. Buy Calendar Spread using Calls (+C 2 - C 1 ) B. Buy Call with maturity, T 2 ; Sell Call with maturity, T 1 ; (T 2 > T 1 ); Thus, (C 2 > C 1 ); initial outflow.

20. © Paul Koch 1-20 III.C. Buy Calendar Spread using Puts (+P 2 - P 1 ) C. Buy Put with maturity, T 2 ; Sell Put with maturity, T 1 ; (T 2 > T 1 ); Thus, (P 2 > P 1 ); initial outflow.

21. © Paul Koch 1-21 IV. Interest Rate Option Combinations (Hull Chap 21) A. Using Options on Eurodollar Futures. 1. ED Futures Contract Characteristics : (Review) a. Underlying Asset - ED deposit with 3-month maturity. b. ED rates are quoted on an interest-bearing basis, assuming a 360-day year. c. Each ED futures contract represents $1MM of face value ED deposits maturing 3 months after contract expiration. d. 40 different contracts trade at any point in time; contracts mature in Mar, Je, Sept, and Dec, 10 years out. e. Settlement is in cash; price is established by a survey of current ED rates. f. ED futures trade according to an index; Q = 100 - R = 100 - (futures rate); e.g., If futures rate = 8.50%, Q = 91.50, and interest outlay promised would be (.0850) x ($1,000,000) x (90 / 360) = $21,250. g. Each basis point in the futures rate means a $25 change in value of contract: [ (.0001) x ($1,000,000) x (90 / 360) ] = $25 ] h. The ED futures is truly a futures on an interest rate. (The T.Bill futures is a futures on a 90-day T.Bill.)

22. © Paul Koch 1-22 IV.A. Using Options on ED Futures 2. Example: Long Hedge with ED futures for a Bank. (more Review) Jan. 6: Bank expects $1 MM payment on May 11 (4 months). Anticipates investing funds in 3-month ED deposits. Cash Market risk exposure: Bank would like to invest @ today’s ED rate, but won’t have funds for 4 mo. If ED rate , bank will realize opportunity loss (will have to invest the $1 MM at lower ED rates). Long Hedge: Buy ED futures today (promise to deposit later @ R). Then if cash rates , futures rates (R) will  & futures prices (Q) will . so long futures position will  to offset opportunity losses in cash mkt. The best ED futures to buy is June contract; expires soonest after May 11. Jan. 6 May 11 June 14 |__________________________________________|_____________| $1 MM receivable due May 11. Cash: Plan to invest $1MM on May 11Invest the $1 MM in ED deposits. Futures: Buy 1 ED futures. Sell futures contract.

23. © Paul Koch 1-23 IV.A. Using Options on ED Futures 3. Data for example – (more Review) Jan. 6: Cash market ED rate (LIBOR) = R S = 3.38% (S 1 = 96.62) June ED futures rate (LIBOR) = R F = 3.85% (F 1 = 96.15) ; Basis = (S 1 - F 1 ) =.47% May 11: Cash market ED rate = 3.03% (S 2 = 96.97) June ED futures rate = 3.60% (F 2 = 96.40) ; Basis = (S 2 - F 2 ) =.57% _______________________________________________________________________________ DateCash Market Futures MarketBasis 1 / 6bank plans to invest $1MM bank buys 1 Je ED futures at cash rate = S 0 = 3.38% at futures rate = R 0 = 3.85%.47% 5 / 11bank invests $1MM in 3-mo ED bank sells 1 June ED futures at cash rate = S 1 = 3.03% at futures rate = R 1 = 3.60%.57% Netopport. loss = 3.38 - 3.03 =.35% futures gain = 3.85 - 3.60 =.25% change Effect(35) x ($25) = $875 (25) x ($25) = $625.10%. Cumulative Investment Income: Interest @ 3.03% = $1,000,000 (.0303) (90/360) = $7,575 Profit from futures trades: = $625 Total: $8,200 Effective Return = [ $8,200 / $1,000,000 ] x (360 / 90) = 3.28% (10 bp worse than spot market = change in basis). This is basis risk.

24. © Paul Koch 1-24 IV.A. Using Options on ED Futures 4. Using Options on ED futures to build Floors, Caps, & Collars. a. ED futures contract: Buy ; Promise to buy ED ( lend @ forward ED rate); Sell ; Promise to sell ED (borrow @ forward ED rate). [ Lock in R. ] b. Call option on ED futures: Right to buy ED futures (lend @ forward ED rate). c. Put option on ED futures: Right to sell ED futures (borrow @ fwd ED rate). d. Lender? Want to buy ED in future. To hedge risk of loss with falling rates: i. Buy ED futures. If rates , lock in (min.) lending rate. But if rates , opportunity loss (could have loaned at higher rates). ii. Buy Call option on ED futures. If rates , lock in min. lending rate. NOW if rates , lend at higher rates! Call is OTM - interest rate Floor. e. Borrower? Want to sell ED in future. To hedge risk of loss with rising rates: i. Sell ED futures. If rates , lock in (max.) borrowing rate. But if rates , opportunity loss (could have borrowed at lower rates). ii. Buy Put option on ED futures. If rates , lock in max. borrowing rate. NOW if rates , borrow at lower rates! Put is OTM - interest rate Cap. f. Combining Call & Put on ED futures gives Collar.

25. © Paul Koch 1-25 IV.A. Using Options on ED Futures 5. Example: Building Interest Rate Collar for a bank. Cap: Buy a Put. Floor: Sell a Call. Both: Collar. Strike Option Strike Option Range of Net Price Premium Price Premium Borrowing Cost Premium. 96.00.13 96.75.02 3¼% - 4%.11 = $275 96.50.40 96.75.02 3¼% - 3½%.38 = $950 96.25.23 96.50.05 3½% - 3¾%.18 = $450. Cap at 4%; Floor at 3¼ %; Collar: Net Cost = 11 basis points. | | 96.00 96.75 |  > Futures Price (Q) | Loss a. CAP borrowing rates @ 4% by buying a Put with K = 96.00 (= 100 - 4). Must pay 13 bp for this Put (13 x $25 = $325). i. If ED rates  above 4%, Q  below 96.00, & Put is ITM – Cap at 4%. ii. If ED rates  below 4%, Q  above 96.00, & Put is OTM – Borrow at < 4%. b. If you don’t think ED rates will  below, say, 3.25%, can recover some of cost by selling a Call with K = 96.75 (= 100 - 3.25). Receive 2 bp ($50). i. If ED rates  below 3.25%, Q  above 96.75%, & Call is ITM – Floor at 3.25%. 0.02 0.11 0.13 Buy put Sell call