Option Strategies &Exotics

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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

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3

Types ofTypes of Strategies 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)

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Positions in an Option & the Positions in an Option & the UnderlyingUnderlying

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c) (d)

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

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Bull Spread Using CallsBull Spread Using Calls

K1 K2

Profit

ST

Bull Spread Using Calls Bull Spread Using Calls ExampleExample

• 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

Long Call (at K1)

plus

Short Call (at K2 > K1)

equals

Call Bull Spread

+10

+1

Profit

Share Price

K1

5

-3

K1=102

K2=110

SBE=105

00 -1

K2

+1

0

0 Gamble on stock price rise and offset cost with sale of call

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

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Bear Spread Using PutsBear Spread Using Puts

K1 K2

Profit

ST

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Bull Spreads with puts Bull Spreads with puts & Bear Spreads with Calls& Bear Spreads with Calls

• Of course can do bull spreads with puts and bear spreads with calls (put-call parity)

• Figured out how?

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Bull Spread Using PutsBull Spread Using Puts

K1 K2

Profit

ST

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Bear Spread Using CallsBear Spread Using Calls

K1 K2

Profit

ST

04/18/23

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}

Equity CollarEquity Collar

+1+1

+1

-10 0

Long Stock

Long Put

Short Call0 0 -1

00

+1Equity Collar

plus

plus

equals

Equity Collar: Payoff ProfileEquity Collar: Payoff Profile

ST < K1K1 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 Profit K1 – (P – C) ST – (P – C) K2 – (P – C)

Net Profit = Gross Payoff – (P – C)

Equity Collar Payoffs

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Box SpreadBox 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 A Basic Combination: A Synthetic Forward/FuturesForward/Futures

Range Forward ContractsRange 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 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

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Range Forward ContractRange Forward Contract

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Payoff

Asset Price

K1 K2

Payoff

Asset Price

K1 K2

Short Position

Long Position

Volatility CombinationsVolatility Combinations

• Mainly• Straddle• Strangles • These are strategies that show the true ‘character’ of

options

• But also• Strip • Straps• Etc.

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A Straddle CombinationA Straddle Combination

Profit

STK

Long Long ((buybuy)) Straddle StraddleData: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 HFStraddles 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

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A Strangle CombinationA Strangle Combination

K1 K2

Profit

ST

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K ST K ST

Strip Strap

Strip & StrapStrip & Strap

ProfitProfit

Time Decay CombinationsTime 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

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Calendar Spread Using CallsCalendar Spread Using Calls

ST

K

Profit

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Calendar Spread Using PutsCalendar Spread Using Puts

ST

K

Profit

‘‘Quasi-Elementary’ SecuritiesQuasi-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

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Butterfly Spread Using CallsButterfly Spread Using Calls

K1 K3 STK2

Profit

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Butterfly Spread Using PutsButterfly Spread Using Puts

K1 K3

Profit

STK2

Butterflies ReplicationButterflies 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

Short Butterflies ReplicationShort 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

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Short Butterfly Spread Using CallsShort Butterfly Spread Using Calls

K1 K3

Profit

STK2

Variations Using Interest Rate Variations Using Interest Rate OptionsOptions

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Interest Rate OptionsInterest 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)

CapsCaps

• 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 Nk max(Lk - LK, 0) where N is the notional amount, k= 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

Caplet PayoffCaplet Payoff

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t0 = 0 t1 = 30 t2 = 120 days

Expiry \ Valuationof option, (LIBOR1 - LK)

Strike rate LK

fixed inthe contract

δ = 90 days

Planned Borrowing + Caplet (Call on Bond)

4681012141618

5 7 9 11 13 15

LIBOR at expiry

Annu

alis

ed C

ost o

f B

orro

win

g

Loan + Interest Rate Floorlet (Put on Bond)

0

5

10

15

20

4 6 8 10 12 14 16

LIBOR at expiry

Annu

aliz

ed re

turn

on

loan

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Funding cost

iTK

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 Underlying Positions in an Option & the Underlying ((notice variables on vertical axisnotice variables on vertical axis))

CollarCollar

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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 swapsCombining 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

SwaptionsSwaptions

• 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).

Swaptions ExampleSwaptions 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.

ExampleExample• 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]

ExoticsExotics

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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.

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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

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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)

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Forward Start Options

• Option starts at a future time, T1

• Implicit in employee stock option plans• Often structured so that strike price equals asset

price at time T1

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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

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Chooser Option “As You Like It”

• Option starts at time 0, matures at T2

• At T1 (0 < T1 < T2) buyer chooses whether it is a put or call

• This is a package!

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Chooser Option as a Package

54

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TTqr

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strike with

timeat maturingput a plus timeat maturing call a is This

therefore is timeat valueThe

parity call-put From

is valuethe timeAt

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

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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

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Parity Relations

c = cui + cuo

c = cdi + cdo

p = pui + puo

p = pdi + pdo

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Binary Options

• Cash-or-nothing: pays Q if ST > K, otherwise pays nothing. Value according to B&S = e–rT Q N(d2)

• Asset-or-nothing: pays ST if ST > K, otherwise pays nothing. Value according to B&S = S0e-qT N(d1)

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Decomposition of a Call Option

Long Asset-or-Nothing option

Short Cash-or-Nothing option where payoff is K

Value according to B&S = S0e-qT N(d1) – e–rT KN(d2)

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Asian Options

• Payoff related to average stock price• Average Price options pay:

Call: max(Save – K, 0)

Put: max(K – Save , 0)

• Average Strike options pay: Call: max(ST – Save , 0)

Put: max(Save – ST , 0)

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Asian Options

• No exact analytic valuation• Can be approximately valued by assuming that

the average stock price is lognormally distributed

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Lookback Options• Floating lookback call pays ST – Smin at time T (Allows buyer to

buy stock at lowest observed price in some interval of time)

• Floating lookback put pays Smax– ST at time T

(Allows buyer to sell stock at highest observed price in some interval of time)

• Fixed lookback call pays max(Smax−K, 0)

• Fixed lookback put pays max(K −Smin, 0)

• Analytic valuation for all types

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Shout Options• Buyer can ‘shout’ once during option life• Final payoff is either

Usual option payoff, max(ST – K, 0), or Intrinsic value at time of shout, S – K

• Payoff: max(ST – S, 0) + S – K• Similar to lookback option but cheaper

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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(VT – UT, 0)

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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

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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

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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

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2)(ˆ)var(

8

11ˆ)(ˆ

VE

VVEE

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

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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)

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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

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Example

• A 9-month up-and-out call option an a non-dividend paying stock where S0 = 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

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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

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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

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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

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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

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Exercises

• 8.1

• 10.1

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