L3 - Option Payoffs

7/29/2019 L3 - Option Payoffs

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Option Strategies &

Exotics

1



Note on Notation

• Here, T denotes time to expiry as well as time

of expiry, i.e. we use T to denote indifferentlyT 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)



4

Positions in an Option & the

Underlying

Profit

S T K

Profit

S T

K

Profit

S T

K

Profit

S T K

(a) (b)

(c) (d)

Basis of Put-Call Parity: P + S = C + Cash ( Ke-rT )



5

Bull Spread Using Calls

K 1 K 2

Profit

S T



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: C1 = 5

Option 2:

Strike: K 1 = 110

Price: C2 = 2



Long Call (at K 1)

plus

Short Call (at K 2 > K 1)

equals

Call Bull Spread

+10

+1

Profit

Share Price

K 1

5

-3

K 1=102

K 2=110

S BE=105

00

-1

K 2

+1

0

0

Gamble on stock price rise and offset cost

with sale of call



Payoff:

Long call (K 1) + short call (K

2) = Bull Spread:

{ 0, +1, +1} + {0, 0, -1} = {0, +1, 0 }

= Max(0, ST-K 1) – C1 – Max(0, ST-K 2) + C2

= C2 - C1 if ST K 1 K 2

= ST - K 1 + (C2 - C1) if K 1 < ST K 2

= (ST - K 1 - C1) + (K 2 - ST + C2) =

= K 2 - K 1 + (C2 - C1) if ST > K 1 > K 2

„Break -even‟:

SBE = K 1 + (C1 – C2) = 102 + 3 = 105



Bear Spread Using Puts

K 1 K 2

Profit

S T



Bull Spreads with puts

& Bear Spreads with Cal ls

• Of course can do bull spreads with puts and bear

spreads with calls (put-call parity)

• Figured out how?



Bull Spread Using Puts

K 1 K 2

Profit

S T



Bear Spread Using Calls

K 1 K 2

Profit

S T



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}

Equi ty Collar



+1+1

+1

-10 0

Long Stock

Long Put

Short Call

0 0-1

0

0

+1Equity Collar

plus

plus

equals

Equi ty Collar : Payoff Profi le



ST < K 1 K 1 ST K 2 ST > K 2

Long Shares ST ST ST

Long Put (K 1) K 1 – ST 0 0

Short Call (K 2) 0 0 – (ST – K 2)

Gross Payoff K 1 ST K 2

Net Profit K 1 – (P – C) ST – (P – C) K 2 – (P – C)

Net Profit = Gross Payoff – (P – C)

Equity Collar Payoffs



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



Short Put

plus

Long Call

equals

Long Futures

+1

+1

0

0+1

+1

A Basic Combination: A Synthetic

Forward/Futures



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



Range Forward Contract

19

Payoff

Asset

Price K 1 K 2

Payoff

Asset

Price

K 1 K 2

Short

Position

Long

Position



Volati l i ty Combinations • Mainly

• Straddle

• Strangles

• These are strategies that show the true „character‟ of

options

• But also

• Strip

• Straps

• Etc.



21

A Straddle Combination

Profit

S T K



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



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

K 1 K 2

Profit

S T



25

K S T K S T

Strip Strap

Strip & Strap

ProfitProfit



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 ashort 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

S T

K

Profit



28

Calendar Spread Using Puts

S T

K

Profit



„Quasi - Elementary‟ Securities • Arrow(-Debrew) introduces so called Arrow-

Debrew elementary securities,

i.e. contingent claims with $1 payoff in one state and $0in all other states

• These can be seen as “bet” options

• Butterflies look a lot like them



30

Butterf ly Spread Using Calls

K 1 K 3 S T K 2

Profit



31

Butterf ly Spread Using Puts

K 1 K 3

Profit

S T K 2



Butter f l ies 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 theunderlying 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



Short Butterf l ies 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 theunderlying in either direction (e.g. result of reference tothe 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 Butterf ly Spread Using Calls

K 1 K 3

Profit

S T K 2



Var iations Using I nterest Rate

Options

35



I nterest Rate Options

• Interest rate option

gives holder the right but not the obligation to receiveone interest rate (e.g. floating\LIBOR) and pay

another (e.g. the fixed strike rate LK )



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 dk max( Lk - L K , 0) where

N is the notional amount, dk = t k +1 - t k , L K is the cap rate, and

Lk 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



Caplet Payoff

38

t0 = 0 t1 = 30 t2 = 120 days

Expiry \ Valuation

of option, (LIBOR 1 - LK )

Strike rate LK

fixed in

the contract

δ = 90 days



Planned Borrowing + Caplet (Call

on Bond )

4

6

8

10

12

14

16

18

5 7 9 11 13 15

LIBOR at expiry

A n n u a l i s e d C

o s t o f

B o r r o w i n g



Loan + I nterest Rate F loor let (Put

on Bond )

0

5

10

15

20

4 6 8 10 12 14 16

LIBOR at expiry

A n n u a l i z e d r e

t u r n o n

l o a n



41

Funding cost

iT K

Return rate

iT

K

iT

K

(c)

(a) (b)

Return rate

Long

caplet

Short

caplet

Long

floorlet

iT

K

(d)

Funding cost

Short

floorlet

Positions in an Option & the Under lying

(notice variables on vertical axis )



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 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%.



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 intoa swap



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 toenter into a swap where they pay the fixed leg andreceive the floating leg (long IRS).

A receiver swaption gives the buyer the right toenter into a swap where they will receive the fixedleg, and pay the floating leg (short IRS).



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 fundthis 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 riseover the 3 years and hence the cost of the fixed rate payments in theswap will be higher than at inception.



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 N δmax{ sT – s K , 0}. So, value of swaption atT is:

• f = $10m[ sT – s K ] [(1 + L2,3)-1 + (1 + L2,4)

-2 + (1 + L2,5)-3]



Exotics

47



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 oneasset for another

• Options involving several

assets

• Volatility and Varianceswaps

• etc., etc., etc.

48



Packages

• Portfolios of standard options

• Classical spreads and combinations: bullspreads, bear spreads, straddles, etc

• Often structured to have zero cost

• One popular package is a range forwardcontract

49



Non-Standard American Options • Exercisable only on specific dates

(Bermudans)

• Early exercise allowed during only partof life (initial “lock out” period)

• Strike price changes over the life

(warrants, convertibles)

50



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



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



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



Chooser Option as a Package

54

))((

12

,0max

1))(()(

1

)(1

)(

1

12

)12(1

)12(

1212

1212

),0max(),max(

),max(

T T qr

eS K e

T T qr T T q

T T qT T r

Ke

T T

S Keec pc

T

eS K ec p

pcT

T T qT T r

strikewith

timeatmaturingputaplustimeatmaturingcallaisThis

thereforeistimeatvalueThe

paritycall-putFrom

isvaluethetime At



Barr ier 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



Barr ier 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



Parity Relations

c = cui + cuo

c = cdi + cdo

p = pui + puo

p = pdi + pdo

57



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 0e-qT N (d 1)

58



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 0e-qT N (d 1) – e – rT KN (d 2)

59



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



Asian Options • No exact analytic valuation

• Can be approximately valued by assuming that

the average stock price is lognormally distributed

61



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



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

63



Exchange Options

• Option to exchange one asset for another

• For example, an option to exchange oneunit of U for one unit of V

• Payoff is max(V T – U T , 0)

64



Basket Options • A basket option is an option to buy or sell a

portfolio of assets

• This can be valued by calculating the first twomoments of the value of the basket and then

assuming it is lognormal

65



Volati l i ty and Var iance 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



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

2)(ˆ

)var(

8

11ˆ)(ˆ

V E

V V E E



VIX I ndex • 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 onthe 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



How Diff icul t is it to

Hedge Exotic Options?

• In some cases exotic options are easier tohedge than the corresponding vanilla options

(e.g., Asian options)

• In other cases they are more difficult to hedge

(e.g., barrier options)

69



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 exoticoption on some boundary , we have matched it at allinterior points of the boundary

• Static options replication can be contrasted withdynamic options replication where we have to trade

continuously to match the option

70



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



Example (continued)

We might try to match the following points on

the boundary

c(S , 0.75) = MAX(S – 50, 0) for S < 60c(60, 0.50) = 0

c(60, 0.25) = 0

c(60, 0.00) = 0

72



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



73



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



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



Exercises

• 8.1

• 10.1

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L3 - Option Payoffs

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