From Plain Vanilla to Exotic Derivatives

Transcription

1 Lecture 2 From Plain Vanilla to Exotic Derivatives Giampaolo Gabbi Financial Engineering MSc in Finance

2 Outline Introduction and notations Option strategies Exotic Derivatives 2

3 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 3

4 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

5 Positions in an Option & the Underlying Profit Profit K K S T S T Profit (a) Profit (b) K S T K S T (c) (d) Basis of Put-Call Parity: P + S = C + Cash ( Ke -rt ) 5

6 Bull Spread Using Calls Profit K 1 K 2 S T 6

7 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

8 Gamble on stock price rise and offset cost with sale of call 0 0 Profit 5 K 1 = K 1 K Long Call (at K 1 ) plus Short Call (at K 2 > K 1 ) equals Call Bull Spread -3 0 S BE =105 K 2 =110 Share Price 8

9 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 ) = = 105 9

10 Bear Spread Using Puts Profit K 1 K 2 S T 10

11 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

12 Bull Spread Using Puts Profit K 1 K 2 S T 12

13 Bear Spread Using Calls Profit K 1 K 2 S T 13

14 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 (K 1 ) + short call (K 2 ) {0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1}

15 Equity Collar: Payoff Profile Long Stock plus Long Put plus Short Call equals Equity Collar 15

16 Equity Collar Payoffs S T < K 1 K 1 S T K 2 S T > K 2 Long Shares S T S T S T Long Put (K 1 ) K 1 S T 0 0 Short Call (K 2 ) 0 0 (S T K 2 ) Gross Payoff K 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) 16

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

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 Payoff Payoff K 1 K 2 Asset Price K 1 K 2 Asset Price Short Position Long Position 19

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

21 A Straddle Combination Profit K S T 21

22 Long (buy) Straddle Data: K = 102 P = 3 C = 5 C + 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) = = 110 for S T < K => K - S T (C + P) = K - (C + P) = = 94 22

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 23

24 From Straddle Strategies to Twin Win Certificates An example Twin-Win certificates are a special variant of the bonus certificate that generate a positive return on their investment in bullish as well as bearish markets, as long as the underlying asset price doesn't decrease by too much and breaches through a predetermined barrier. Usually, the product has 100% participation to the upside (not capped) and 100% participation to the downside in absolute terms. In other words, should the underlying asset rise by 25%, the return on investment is +25%, and should the underlying asset fall by 25%, the return on investment is also +25%, as long as the barrier (which could for example be set at 40% below spot) wasn t breached during its lifetime. If the barrier is breached (usually anytime during the lifetime of the product, i.e. American style barrier), the Twin-Win transforms itself in a certificate tracking the underlying asset. Any downside absolute participation disappears (the barrier has been knocked-out ). The maturity of the product plays an important role, as the delta of the product amounts to approx. 95% at the product's inception. Hence, the positive performance that should be reflected in the mark-to-market price of the product in a bearish market only grips when around 70% of the time to maturity has expired. 24

25 Payoff Features The structure is constructed by means of a long zero strike call, and long two down & out puts, where the strike is set at-the-money. The barrier level of the puts determines both the maximum downside performance (in absolute terms) the product could reach and the level at which the product "knocks out" and loses the downside absolute participation. 25

26 Payoff Features 26

27 Do s Use this product when you're not sure about where the market is headed, but you think that it won t crash by more than the barrier level. You must still consider the worst-case scenario (a crash through the barrier) and be able to bear the loss in case it happens Carefully consider the downside participation up to the barrier, and pitch it against the potential bonus of a classical bonus certificate: which is more attractive? Ask the structurer for variants. Limit the maturity to Months Use a worst-of feature on underlying assets if their correlation is low and you believe that none of them will breach the barrier. Use 2, max 3 underlying assets, never 4, 5 or even more. Use a barrier level that gives a reasonable payoff in case the underlying effectively drops. If the barrier is too near the spot for your liking, use a cap on the upside to lower the barrier further. If a 20% decrease seems possible, take a barrier that will protect the investment up to a decrease of 35%. Better be safe than sorry. A Twin-win certificate on stock indices is especially well suited if you think that dividends will decrease in the future 27

28 Don ts Don t use this structure when you re either strongly bullish or strongly bearish. Other structures are better suited for such scenarios. Don t invest into Twin-Win Certificates with maturities exceeding 2 or 3 years. Forecasting that an underlying will end up between zero and X% (X% being the level of the barrier) in years from the moment ysou invest is practically impossible. The value of the protection therefore diminishes with the time to maturity 28

29 Impact Factors Dividend Yield (for Equity): the higher, the better. A Twin-Win cannot be achieved on stocks paying no dividends. Of course, the dividends are not paid out to the Twin-Win Certificate's holder, as they are used to buy the two down & out puts. Yield (for Excess Return Indices): the higher the better. Volatility: the higher the better. Fact is that the mathematical models assume that the higher the volatility, the more likely it is that a barrier of the down & out put will be breached. Therefore, all other things remaining equal, the put options' value is considered lower the higher the volatility. In other words, a higher volatility allows to increase the protection puffer of the barrier (lower barrier level). 29

30 Classical Variants Worst-of Twin Win: two or more underlying assets form the underlying assets of the product. If one (the worst-of) hits the barrier, the downside participation is lost and the payoff is linked to the worst-of performin asset. The value extracted from the correlation is used primarily to shorten the maturity or to lower the barrier. Capped Twin Win (represented on the right): the upside participation is capped at a certain level. The extracted value from the cap is used primarily to shorten the maturity or lower the barrier. Lock-in Twin Win: as soon as a certain upside is realized, the whole structure becomes capital guaranteed. Interesting but expensive feature, and can usually only be realized with longer maturities. Could be used in combination with Worst-of 30

31 Classical Variants. Twin Win capped 31

32 A Strangle Combination Profit K 1 K 2 S T 32

33 Strip & Strap Profit Profit K S T K S T Strip Strap 33

34 Calendar (or horizontal) spreads 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 34

35 Calendar (or horizontal) spreads Calendar spreads can be done with calls or puts and, if using the same strikes, put and call calendar spreads are virtually equivalent. Implementing the strategy involves buying one option and selling another option of the same type and strike, but with different expiration. A long calendar spread would entail buying an option (not a "front month" contract) and selling a nearer-expiration option of the same strike and type. Long calendar spreads are traded for a debit, meaning you pay to open the overall position. 35

36 Calendar (or horizontal) spreads This strategy profits in a limited range around the strike used. The trade can be set up with a bullish, bearish or neutral bias. The greatest profit will come when the underlying is at the strike used at expiration. Calendar spreads also profit from a rise in implied volatility, since the long option has a higher Vega than the short option. 36

37 Calendar (or horizontal) spreads Calendar spreads lose if the underlying moves too far in either direction. The maximum loss is the debit paid, up until the option you sold expires. After that, you are long an option and your further risk is the entire value of that option. Options in nearer-month expirations have more time decay than later months (they have a higher theta). 37

38 Calendar (or horizontal) spreads The calendar spread profits from this difference in decay rates This trade is best used when implied volatility is low and when there is implied volatility "skew" between the months used, specifically when the near-month sold has a higher implied volatility than the later-month bought 38

39 Calendar Spread Using Calls Profit K S T 39

40 Calendar Spread Using Puts Profit K S T 40

41 Calendar (or horizontal) spreads Example: with the stock at euros, the September 135 call is purchased for 15.45, and the July 135 call is sold for 10.45, for a net debit of 5, which is the maximum risk. 41

42 Calendar (or horizontal) spreads 42

43 Calendar (or horizontal) spreads This is a neutral trade used when the outlook is for a range-bound underlying. The maximum risk is known from the outset of the trade, and is equal to the debit paid (until the first expiration). If the implied volatility does not change, the position profits from roughly 121 to 154. Rises in implied volatility will increase the profit and the range. Time decay is on your side with this trade. 43

44 Calendar (or horizontal) spreads Example of a Winning Trade RIMM (Research In Motion) moved up to 108 in late February, while implied volatility moved down below

45 Calendar (or horizontal) spreads With the stock at 108, we would buy the April 110 calls for 7.50 and sell the March 110 calls for 4.45, for a net debit of The maximum risk is the 305 we paid (remembering that options contacts come in lots of 100). The risk would be realized if the stock moves "too far" in either direction. In this case, RIMM was at 101 at March expiration, with implied volatility up to 65. So the March 110 call expired worthless, while the April 110 call was worth 4.30, for a 41% return. 45

46 Calendar (or horizontal) spreads Example of a Losing Trade Using the same charts, we see that establishing a spread just before earnings would not have worked out. 46

47 Calendar (or horizontal) spreads In August, we saw the price heading up through 220 and implied volatility at 62 percent. The October 220 call was purchased for and the September 220 call sold for 15.90, for a net debit of After earnings, the price plummeted down into the 80 range and implied volatility dropped below 50. The implied volatility recovered by the September and October expirations, but the price did not, so the maximum loss of 670 was realized. 47

48 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 48

49 Butterfly Spread Using Calls Profit K 1 K 2 K 3 S T 49

50 Butterfly Spread Using Puts Profit K 1 K 2 K 3 S T 50

51 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 51

52 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 52

53 Short Butterfly Spread Using Calls Profit K 1 K 2 K 3 S T 53

54 Butterfly example Suppose XYZ stock is trading at 40 in June. An options trader executes a long call butterfly by purchasing a JUL 30 call for 1100, writing two JUL 40 calls for 400 each and purchasing another JUL 50 call for 100. The net debit taken to enter the position is 400, which is also his maximum possible loss. Questions 1. What is the trader s profit/loss if the XYZ stock at the expiry trades at 40? 2. What is the trader s profit/loss if the XYZ stock at the expiry trades below 30 or above 50? 54

55 Condor spread The condor option strategy is a limited risk, non-directional option trading strategy that is structured to earn a limited profit when the underlying security is perceived to have little volatility. Sell 1 ITM Call Buy 1 ITM Call (Lower Strike) Sell 1 OTM Call Buy 1 OTM Call (Higher Strike) 55

56 Condor spread example Suppose XYZ stock is trading at 45 in June. An options trader enters a condor trade by buying a JUL 35 call for 1100, writing a JUL 40 call for 700, writing another JUL 50 call for 200 and buying another JUL 55 call for 100. The net debit required to enter the trade is 300, which is also his maximum possible loss. What is the trader s profit/loss if the XYZ stock at the expiry trades at 35? What is the trader s profit/loss if the XYZ stock at the expiry trades at 55? What is the trader s profit/loss if the XYZ stock at the expiry trades at 45? 56

57 Condor spread con put (Iron condor) The iron condor is a limited risk, non-directional option trading strategy that is designed to have a large probability of earning a small limited profit when the underlying security is perceived to have low volatility. The iron condor strategy can also be visualized as a combination of a bull put spread and a bear call spread. Sell 1 OTM Put Buy 1 OTM Put (Lower Strike) Sell 1 OTM Call Buy 1 OTM Call (Higher Strike) 57

58 Condor spread con put (Iron condor) 58

59 Condor spread con put (Iron condor) The iron condor is a limited risk, non-directional option trading strategy that is designed to have a large probability of earning a small limited profit when the underlying security is perceived to have low volatility. The iron condor strategy can also be visualized as a combination of a bull put spread and a bear call spread. Sell 1 OTM Put Buy 1 OTM Put (Lower Strike) Sell 1 OTM Call Buy 1 OTM Call (Higher Strike) 59

60 Ladder The long call ladder, or bull call ladder, is a limited profit, unlimited risk strategy in options trading that is employed when the options trader thinks that the underlying security will experience little volatility in the near term. To setup the long call ladder, the options trader purchases an in-the-money call, sells an at-the-money call and sells another higher strike out-of-the-money call of the same underlying security and expiration date. Buy 1 ITM Call Sell 1 ATM Call Sell 1 OTM Call 60

61 Ladder example Suppose XYZ stock is trading at 35 in June. An options trader executes a long call ladder strategy by buying a JUL 30 call for 600, selling a JUL 35 call for 200 and a JUL 40 call for 100. The net debit required for entering this trade is 300. What is the trader s profit/loss if the XYZ stock at the expiry trades at 35? What is the trader s profit/loss if the XYZ stock at the expiry trades at 50? What is the trader s profit/loss if the XYZ stock at the expiry trades at 30? 61

62 What s the strategy? Sell 1 ITM Put Buy 1 ATM Put Buy 1 OTM Put 62

63 What s the strategy? Suppose XYZ stock is trading at 40 in June. An options trader executes a short put ladder strategy by selling a JUL 45 put for 600, buying a JUL 40 put for 200 and a JUL 35 put for 100. The net credit received for entering this trade is 300. What is the trader s profit/loss if the XYZ stock at the expiry trades at 40? What is the trader s profit/loss if the XYZ stock at the expiry trades at 45? What is the trader s profit/loss if the XYZ stock at the expiry trades at 25? 63

64 Interest Rate Options Interest rate options give 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 ) 64

65 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 Nd k max(l k - L K, 0) where N is the notional amount, d 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 65

66 Caplet Payoff Strike rate L K fixed in the contract Expiry \ Valuation of option, (LIBOR 1 - L K ) t 0 = 0 t 1 = 30 t 2 = 120 days δ = 90 days 66

67 Annualised Cost of Borrowing Planned Borrowing + Caplet (Call on Bond) LIBOR at expiry 67

68 Annualized return on loan Loan + Interest Rate Floorlet (Put on Bond) LIBOR at expiry 68

69 Positions in an Option & the Underlying (notice variables on vertical axis) Short caplet Return rate K Return rate K Long floorlet i T i T Long caplet Funding cost (a) Funding cost (b) Short floorlet K K i T i T (c) (d) 69

70 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 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%. 70

71 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 71

72 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). 72

73 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. 73

74 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 ] 74

75 Exotics 75

76 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. 76

77 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 77

78 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) 78

79 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 79

80 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 80

81 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! 81

82 Chooser Option as a Package At time p c e T 1 From put- call r( T max( c, p) with strike the value is max( c, p) 2 T 1 ) q( T 1 ( r q)( T 1 T 2 q( T ) ) max 0, e ) r ( T2 T1 ) 2 ( r q)( T c e max( 0, Ke S Ke parity K S The value at time T e T T is therefore This is a call maturing at time K S e T 1 q ( T2 T1 ) T ) ) 1 plus a put maturing at time T 82 1

83 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 83

84 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 84

85 Parity Relations c = c ui + c uo c = c di + c do p = p ui + p uo p = p di + p do 85

86 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 ) 86

87 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 ) 87

88 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) 88

89 Asian Options No exact analytic valuation Can be approximately valued by assuming that the average stock price is lognormally distributed 89

90 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 90

91 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 t K Payoff: max(s T S t, 0) + S t K Similar to lookback option but cheaper 91

92 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) 92

93 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 93

94 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 94

95 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 Eˆ( ) Eˆ V 1 var( V ) 1 8 Eˆ( V ) 2 95

96 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 96

97 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) 97

98 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 98

99 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 99