Option Strategies & Exotics

1. Option Strategies &Exotics

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. 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) 3

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

5. Bull Spread Using Calls Profit ST K1 • K2 5

6. Bull Spread Using Calls Example • Create a bull spread on IBM using the following 3-month call options on IBM: Option 1: Strike: K1 = 102 Price: C1 = 5 Option 2: Strike: K1 = 110 Price: C2 = 2

7. Gamble on stock price rise and offset cost with sale of call +1 Long Call (at K1) plus Short Call (at K2 > K1) equals Call Bull Spread +1 0 K1 K2 0 0 -1 Profit 0 5 +1 K1=102 Share Price K2=110 -3 SBE=105 0

8. Payoff: Long call (K1) + short call (K2) = Bull Spread: { 0, +1, +1} + {0, 0, -1} = {0, +1, 0 } = Max(0, ST-K1) – C1 – Max(0, ST-K2) + C2 = C2 - C1 if ST  K1 K2 = ST - K1 + (C2 - C1) if K1 < ST K2 = (ST - K1 - C1)+ (K2 - ST + C2) = = K2 - K1 + (C2 - C1) if ST > K1 > K2 ‘Break-even’: SBE = K1 + (C1 – C2) = 102 + 3 = 105

9. Bear Spread Using Puts Profit K1 K2 ST 9

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? 10

11. Bull Spread Using Puts Profit K1 K2 ST 11

12. Bear Spread Using Calls Profit K1 K2 ST 12

13. Equity Collar 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 (K1) + short call (K2) {0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1}

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

15. Equity Collar Payoffs ST < K1 K1  ST  K2 ST > K2 Long Shares ST ST ST Long Put (K1) K1 – ST 0 0 Short Call (K2) 0 0 – (ST – K2) Gross Payoff K1 ST K2 Net ProfitK1 – (P – C) ST – (P – C) K2 – (P – C) Net Profit = Gross Payoff – (P – C)

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 16

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

18. Range Forward Contracts 18 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 K1 and buying a call with strike K2 (with K2 > K1) When currency is to be received it involves buying a put with strike K1 and selling a call with strike K2 Normally the price of the put equals the price of the call

19. Range Forward Contract Payoff Payoff Asset Price K1 K2 K1 K2 Asset Price Long Position Short Position 19

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

21. A Straddle Combination Profit K ST 21

22. Long (buy) Straddle Data: K = 102 P = 3 C = 5 C + P = 8 profit long straddle:  = Max (0, ST – K) - C + Max (0, K – ST) – P = 0 for ST > K => ST - K – (C + P) = K + (C + P) = 102 + 8 = 110 for ST < K => K - ST – (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. A Strangle Combination Profit K1 K2 ST 24

25. Strip & Strap Profit Profit K ST K ST Strip Strap 25

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. S0 = K) this spread will make money from time decay, but will loose money if the stock price moves substantially

27. Calendar Spread Using Calls Profit ST K 27

28. Calendar Spread Using Puts Profit ST K 28

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. Butterfly Spread Using Calls Profit K1 K2 K3 ST 30

31. Butterfly Spread Using Puts Profit K1 K2 K3 ST 31

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. Short Butterfly Spread Using Calls Profit K1 K2 K3 ST 34

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 LK)

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 tk+1 on each caplet is Ndk max(Lk - LK, 0) where N is the notional amount, dk= tk+1 - tk, LK is the cap rate, and Lk is the rate at time tk for the period between tk and tk+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 Strike rate LK fixed in the contract Expiry \ Valuation of option, (LIBOR1 - LK) t0 = 0 t1 = 30 t2 = 120 days δ = 90 days 38

39. Planned Borrowing + Caplet (Call on Bond)

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

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

42. Collar 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  tk+1 = tk + 90) = = [Lk – max[{0, Lk – LK} + max {0, LK – Lk}]N(90/360) = Lk,CAP N(90/360) if Lk > Lk,CAP = Lk,FL N(90/360) if Lk < Lk,FL = Lk (90/360) if Lk,FL < Lk < Lk,CAP Collar involves borrowing cost at each payment date of either Lk,CAP = 10% or Lk,FL = 8% or Lk = LIBOR if the latter is between 8% and 10%. 42

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 payerswaption gives the buyer the right to enter into a swap where they pay the fixed leg and receive the floating leg (long IRS). A receiverswaption 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 sK = 10%. Payoff is the annuity value of Nδmax{sT – sK, 0}. So, value of swaption at T is: f = $10m[sT – sK] [(1 + L2,3)-1 + (1 + L2,4)-2 + (1 + L2,5)-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.

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

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)