1. Option Strategies & Exotics 1

2. Note on Notation Here, T denotes time to expiry as well as time of expiry, i.e. we use T to denote indifferently T and δ = T – t Less accurate but handier this way, I think 2

3. 3 Types of Strategies Take a position in the option and the underlying Take a position in 2 or more options of the same type (A spread) Combination: Take a position in a mixture of calls & puts (A combination)

4. 4 Positions in an Option & the Underlying Profit STST K STST K STST K STST K (a) (b) (c)(d) Basis of Put-Call Parity: P + S = C + Cash (  Ke -rT )

5. 5 Bull Spread Using Calls K1K1 K2K2 Profit STST

6. Bull Spread Using Calls Example Create a bull spread on IBM using the following 3- month call options on IBM: Option 1: Strike:K 1 = 102 Price:C 1 = 5 Option 2: Strike:K 1 = 110 Price:C 2 = 2

7. Long Call (at K 1 ) plus Short Call (at K 2 > K 1 ) equals Call Bull Spread +1 0 Profit Share Price K1K1 5 -3 K 1 =102 K 2 =110 S BE =105 0 0 K2K2 +10 0 Gamble on stock price rise and offset cost with sale of call

8. Payoff: Long call (K 1 ) + short call (K 2 ) = Bull Spread: { 0, +1, +1} + {0, 0, -1} = {0, +1, 0 } = Max(0, S T -K 1 ) – C 1 – Max(0, S T -K 2 ) + C 2 = C 2 - C 1 if S T  K 1  K 2 = S T - K 1 + (C 2 - C 1 )if K 1 < S T  K 2 = (S T - K 1 - C 1 ) + (K 2 - S T + C 2 ) = = K 2 - K 1 + (C 2 - C 1 )if S T > K 1 > K 2 ‘Break-even’: S BE = K 1 + (C 1 – C 2 ) = 102 + 3 = 105

9. 9 Bear Spread Using Puts K1K1 K2K2 Profit STST

10. 10 Bull Spreads with puts & Bear Spreads with Calls Of course can do bull spreads with puts and bear spreads with calls (put-call parity) Figured out how?

11. 11 Bull Spread Using Puts K1K1 K2K2 Profit STST

12. 12 Bear Spread Using Calls K1K1 K2K2 Profit STST

13. You already hold stocks but you want to limit downside (buy a put) but you are also willing to limit the upside if you can earn some cash today (by selling an option, i.e. a call) COLLAR = long stock + long put (K 1 ) + short call (K 2 ) {0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1} Equity Collar

14. +1 00 Long Stock Long Put Short Call 00 0 0 +1 Equity Collar plus equals Equity Collar: Payoff Profile

15. S T K 2 Long SharesS T S T S T Long Put (K 1 )K 1 – S T 0 0 Short Call (K 2 )00 – (S T – K 2 ) Gross PayoffK 1 S T K 2 Net Profit K 1 – (P – C) S T – (P – C) K 2 – (P – C) Net Profit = Gross Payoff – (P – C) Equity Collar Payoffs

16. 16 Box Spread A combination of a bull call spread and a bear put spread If all options are European a box spread is worth the present value of the difference between the strike prices Check it out If they are American this is not necessarily so

17. Short Put plus Long Call equals Long Futures +1 0 0 A Basic Combination: A Synthetic Forward/Futures

18. Range Forward Contracts Have the effect of ensuring that the exchange rate paid or received will lie within a certain range When currency is to be paid it involves selling a put with strike K 1 and buying a call with strike K 2 (with K 2 > K 1 ) When currency is to be received it involves buying a put with strike K 1 and selling a call with strike K 2 Normally the price of the put equals the price of the call 18

19. Range Forward Contract 19 Payoff Asset Price K1K1 K2K2 Payoff Asset Price K1K1 K2K2 Short Position Long Position

20. Volatility Combinations Mainly Straddle Strangles These are strategies that show the true ‘character’ of options But also Strip Straps Etc.

21. 21 A Straddle Combination Profit STST K

22. Long (buy) Straddle Data: K = 102P = 3C = 5C + P = 8 profit long straddle:  = Max (0, S T – K) - C + Max (0, K – S T ) – P = 0 for S T > K => S T - K – (C + P) = K + (C + P) = 102 + 8 = 110 for S T < K => K - S T – (C + P) = K - (C + P) = 102 - 8 = 94

23. Straddles and HF Fung and Hsieh (RFS, 2001) empirically show that many hedge funds follow strategies that resemble straddles: ‘Market timers’ returns are highly correlated with the return to long straddles on diversified equity indices and other basic asset classes

24. 24 A Strangle Combination K1K1 K2K2 Profit STST

25. 25 KSTST KSTST StripStrap Strip & Strap Profit

26. Time Decay Combinations Calendar (or horizontal) spreads Options, same strike price (K) but different maturity dates, e.g. buying a long dated option (360-day) and selling a short dated option (180-day), both are at-the money In a relatively static market (i.e. S 0 = K) this spread will make money from time decay, but will loose money if the stock price moves substantially

27. 27 Calendar Spread Using Calls STST K Profit

28. 28 Calendar Spread Using Puts STST K Profit

29. ‘Quasi-Elementary’ Securities Arrow(-Debrew) introduces so called Arrow- Debrew elementary securities, i.e. contingent claims with $1 payoff in one state and $0 in all other states These can be seen as “bet” options Butterflies look a lot like them

30. 30 Butterfly Spread Using Calls K1K1 K3K3 STST K2K2 Profit

31. 31 Butterfly Spread Using Puts K1K1 K3K3 Profit STST K2K2

32. Butterflies Replication Butterfly requires: sale of 2 ‘inner-strike price’ call options (K2) purchase of 2 'outer-strike price’ call options (K1, K3) Butterfly is a ‘bet’ on a small change in price of the underlying in either direction Potential downside of the ‘bet’ is offset by ‘truncating’ the payoff by buying some options Could also buy (go long) a bull and a bear (call or put) spread, same result

33. Short Butterflies Replication Short butterfly requires: purchase of 2 ‘inner-strike price’ call options (K2) sale of 2 'outer-strike price’ call options (K1, K3) Short butterfly is a ‘bet’ on a large change in price of the underlying in either direction (e.g. result of reference to the competition authorities) Cost of the ‘bet’ is offset by ‘truncating’ the payoff by selling some options Could also sell (go short) a bull and a bear (call or put) spread, same result

34. 34 Short Butterfly Spread Using Calls K1K1 K3K3 Profit STST K2K2

35. Variations Using Interest Rate Options 35

36. Interest Rate Options Interest rate option gives holder the right but not the obligation to receive one interest rate (e.g. floating\LIBOR) and pay another (e.g. the fixed strike rate L K )

37. Caps A cap is a portfolio of “caplets” Each caplet is a call option on a future LIBOR rate with the payoff occurring in arrears Payoff at time t k+1 on each caplet is N  k max(L k - L K, 0) where N is the notional amount,  k  = t k+1 - t k, L K is the cap rate, and L k is the rate at time t k for the period between t k and t k+1 It has the effect of guaranteeing that the interest rate in each of a number of future periods will not rise above a certain level

38. Caplet Payoff 38 t 0 = 0t 1 = 30t 2 = 120 days Expiry \ Valuation of option, (LIBOR 1 - L K ) Strike rate L K fixed in the contract δ = 90 days

39. Planned Borrowing + Caplet (Call on Bond)

40. Loan + Interest Rate Floorlet (Put on Bond)

41. 41 Funding cost iTiT K Return rate iTiT K iTiT K (c) (a) (b) Return rate Long caplet Short caplet Long floorlet iTiT K (d) Funding cost Short floorlet Positions in an Option & the Underlying (notice variables on vertical axis)

42. Collar 42 Comprises a long cap and short floor. It establishes both a floor and a ceiling on a corporate or bank’s (floating rate) borrowing costs. Effective Borrowing Cost with Collar (at T  t k+1 = t k + 90) = = [L k – max[{0, L k – L K } + max {0, L K – L k }]N(90/360) = L k,CAP N(90/360)if L k > L k,CAP = L k,FL N(90/360)if L k < L k,FL = L k (90/360)if L k,FL < L k < L k,CAP Collar involves borrowing cost at each payment date of either L k,CAP = 10% or L k,FL = 8% or L k = LIBOR if the latter is between 8% and 10%.

43. Combining options with swaps Cancelable swaps - can be cancelled by the firm entering into the swap if interest rates move a certain way Swaptions - options to enter into a swap

44. Swaptions OTC option for the buyer to enter into a swap at a future date and a predetermined swap rate  A payer swaption gives the buyer the right to enter into a swap where they pay the fixed leg and receive the floating leg (long IRS).  A receiver swaption gives the buyer the right to enter into a swap where they will receive the fixed leg, and pay the floating leg (short IRS).

45. Swaptions Example A US bank has made a commitment to lend at fixed rate $10m over 3 years beginning in 2 years time and may need to fund this loan at a floating rate. In 2 years time, the bank may wish to swap the floating rate payments for a fixed rate, Perhaps at that time, the bank may think that interest rates may rise over the 3 years and hence the cost of the fixed rate payments in the swap will be higher than at inception.

46. Example Bank might need a $10m swap, to pay fixed and receive floating beginning in 2 years time and an agreement that swap will last for further 3 years The bank can hedge by purchasing a 2-year European payer swaption, with expiry in T = 2, on a 3 year “pay fixed-receive floating” swap, at say s K = 10%. Payoff is the annuity value of Nδmax{s T – s K, 0}. So, value of swaption at T is: f = $10m[s T – s K ] [(1 + L 2,3 ) -1 + (1 + L 2,4 ) -2 + (1 + L 2,5 ) -3 ]

47. Exotics 47

48. Types of Exotics Package Nonstandard American options Forward start options Compound options Chooser options Barrier options Binary options Lookback options Shout options Asian options Options to exchange one asset for another Options involving several assets Volatility and Variance swaps etc., etc., etc. 48

49. Packages Portfolios of standard options Classical spreads and combinations: bull spreads, bear spreads, straddles, etc Often structured to have zero cost One popular package is a range forward contract 49

50. Non-Standard American Options Exercisable only on specific dates (Bermudans) Early exercise allowed during only part of life (initial “lock out” period) Strike price changes over the life (warrants, convertibles) 50

51. Forward Start Options Option starts at a future time, T 1 Implicit in employee stock option plans Often structured so that strike price equals asset price at time T 1 51

52. Compound Option Option to buy or sell an option  Call on call  Put on call  Call on put  Put on put Can be valued analytically Price is quite low compared with a regular option 52

53. Chooser Option “As You Like It” Option starts at time 0, matures at T 2 At T 1 (0 < T 1 < T 2 ) buyer chooses whether it is a put or call This is a package! 53

54. Chooser Option as a Package 54

55. Barrier Options Option comes into existence only if stock price hits barrier before option maturity  ‘In’ options Option dies if stock price hits barrier before option maturity  ‘Out’ options 55

56. Barrier Options (continued) Stock price must hit barrier from below  ‘Up’ options Stock price must hit barrier from above  ‘Down’ options Option may be a put or a call Eight possible combinations 56

57. Parity Relations c = c ui + c uo c = c di + c do p = p ui + p uo p = p di + p do 57

58. Binary Options Cash-or-nothing: pays Q if S T > K, otherwise pays nothing. Value according to B&S = e –rT Q N(d 2 ) Asset-or-nothing: pays S T if S T > K, otherwise pays nothing. Value according to B&S = S 0 e -qT N(d 1 ) 58

59. Decomposition of a Call Option Long Asset-or-Nothing option Short Cash-or-Nothing option where payoff is K Value according to B&S = S 0 e -qT N(d 1 ) – e –rT KN(d 2 ) 59

60. Asian Options Payoff related to average stock price Average Price options pay:  Call: max(S ave – K, 0)  Put: max(K – S ave, 0) Average Strike options pay:  Call: max(S T – S ave, 0)  Put: max(S ave – S T, 0) 60

61. Asian Options No exact analytic valuation Can be approximately valued by assuming that the average stock price is lognormally distributed 61

62. Lookback Options Floating lookback call pays S T – S min at time T (Allows buyer to buy stock at lowest observed price in some interval of time) Floating lookback put pays S max – S T at time T (Allows buyer to sell stock at highest observed price in some interval of time) Fixed lookback call pays max(S max −K, 0) Fixed lookback put pays max(K −S min, 0) Analytic valuation for all types 62

63. Shout Options Buyer can ‘shout’ once during option life Final payoff is either  Usual option payoff, max(S T – K, 0), or  Intrinsic value at time of shout, S  – K Payoff: max(S T – S , 0) + S  – K Similar to lookback option but cheaper 63

64. Exchange Options Option to exchange one asset for another For example, an option to exchange one unit of U for one unit of V Payoff is max(V T – U T, 0) 64

65. Basket Options A basket option is an option to buy or sell a portfolio of assets This can be valued by calculating the first two moments of the value of the basket and then assuming it is lognormal 65

66. Volatility and Variance Swaps Agreement to exchange the realized volatility between time 0 and time T for a pre-specified fixed volatility with both being multiplied by a pre-specified principal Variance swap is agreement to exchange the realized variance rate between time 0 and time T for a pre-specified fixed variance rate with both being multiplied by a prespecified principal Daily expected return is assumed to be zero in calculating the volatility or variance rate 66

67. Variance Swaps The (risk-neutral) expected variance rate between times 0 and T can be calculated from the prices of European call and put options with different strikes and maturity T Variance swaps can therefore be valued analytically if enough options trade For a volatility swap it is necessary to use the approximate relation 67

68. VIX Index The expected value of the variance of the S&P 500 over 30 days is calculated from the CBOE market prices of European put and call options on the S&P 500 This is then multiplied by 365/30 and the VIX index is set equal to the square root of the result 68

69. How Difficult is it to Hedge Exotic Options? In some cases exotic options are easier to hedge than the corresponding vanilla options (e.g., Asian options) In other cases they are more difficult to hedge (e.g., barrier options) 69

70. Static Options Replication (Hard Topic) This involves approximately replicating an exotic option with a portfolio of vanilla options Underlying principle: if we match the value of an exotic option on some boundary, we have matched it at all interior points of the boundary Static options replication can be contrasted with dynamic options replication where we have to trade continuously to match the option 70

71. Example A 9-month up-and-out call option an a non-dividend paying stock where S 0 = 50, K = 50, the barrier is 60, r = 10%, and  = 30% Any boundary can be chosen but the natural one is c (S, 0.75) = MAX(S – 50, 0) when S  60 c (60, t ) = 0 when 0  t  0.75 71

72. Example (continued) We might try to match the following points on the boundary c(S, 0.75) = MAX(S – 50, 0) for S  60 c(60, 0.50) = 0 c(60, 0.25) = 0 c(60, 0.00) = 0 72

73. Example continued We can do this as follows: +1.00 call with maturity 0.75 & strike 50 –2.66 call with maturity 0.75 & strike 60 +0.97 call with maturity 0.50 & strike 60 +0.28 call with maturity 0.25 & strike 60 73

74. Example (continued) This portfolio is worth 0.73 at time zero compared with 0.31 for the up-and out option As we use more options the value of the replicating portfolio converges to the value of the exotic option For example, with 18 points matched on the horizontal boundary the value of the replicating portfolio reduces to 0.38; with 100 points being matched it reduces to 0.32 74

75. Using Static Options Replication To hedge an exotic option we short the portfolio that replicates the boundary conditions The portfolio must be unwound when any part of the boundary is reached 75

76. Exercises 8.1 10.1 76