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1

Lecture 2

From Plain Vanilla to Exotic Derivatives

Giampaolo Gabbi

Financial Engineering

MSc in Finance

2015 - 2016

2

Outline

• Introduction and notations

• Option strategies

• Exotic Derivatives

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

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)

5

Positions in an Option & the

Underlying

Profit

ST K

Profit

ST

K

Profit

ST

K

Profit

ST K

(a) (b)

(c) (d)

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

6

Bull Spread Using Calls

K1 K2

Profit

ST

7

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

8

Long Call (at K1)

plus

Short Call (at K2 > K1)

equals

Call Bull Spread

+1 0

+1

Profit

Share Price

K1

5

-3

K1=102

K2=110

SBE=105

0 0

-1

K2

+1

0

0 Gamble on stock price rise and offset cost

with sale of call

9

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

10

Bear Spread Using Puts

K1 K2

Profit

ST

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?

12

Bull Spread Using Puts

K1 K2

Profit

ST

13

Bear Spread Using Calls

K1 K2

Profit

ST

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 Collar

15

+1 +1

+1

-1 0 0

Long Stock

Long Put

Short Call 0 0

-1

0

0

+1 Equity Collar

plus

plus

equals

Equity Collar: Payoff Profile

16

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

Net Profit = Gross Payoff – (P – C)

Equity Collar Payoffs

17

Short Put

plus

Long Call

equals

Long Futures

+1

+1

0

0 +1

+1

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 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

Asset

Price

K1 K2

Payoff

Asset

Price

K1 K2

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

A Straddle Combination

Profit

ST K

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

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.

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.

26

Payoff Features

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 12 – 24 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

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

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).

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

31

Classical Variants. Twin Win

capped

32

A Strangle Combination

K1 K2

Profit

ST

33

K ST K ST

Strip Strap

Strip & Strap

Profit Profit

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. S0 = K) this

spread will make money from time decay, but will

loose money if the stock price moves substantially

35


• 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.

36


• 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.

37


• 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).

38


• 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

39

Calendar Spread Using Calls

ST

K

Profit

40

Calendar Spread Using Puts

ST

K

Profit

41


• Example:

with the stock at 135.13 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.

42


43


• 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.

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 50.

45


• 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 3.05.

• 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.

46


Example of a Losing Trade

• Using the same charts, we see that establishing a spread just before

earnings would not have worked out.

47


• In August, we saw the price heading up through 220 and implied volatility at 62 percent. The October 220 call was purchased for 22.60 and the September 220 call sold for 15.90, for a net debit of 6.70.

• 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.

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

49

Butterfly Spread Using Calls

K1 K3 ST K2

Profit

50

Butterfly Spread Using Puts

K1 K3

Profit

ST K2

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

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

53

Short Butterfly Spread Using Calls

K1 K3

Profit

ST K2

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?

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)

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?



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)

58


(Iron condor)

59


(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)

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

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.




62

What’s the strategy?

Sell 1 ITM Put

Buy 1 ATM Put

Buy 1 OTM Put

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.




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 LK)

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 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

66

Caplet Payoff

t0 = 0 t1 = 30 t2 = 120 days

Expiry \ Valuation

of option, (LIBOR1 - LK)

Strike rate LK

fixed in

the contract

δ = 90 days

67

Planned Borrowing + Caplet (Call on

Bond)

4

6

8

10

12

14

16

18

5 7 9 11 13 15

LIBOR at expiry

An

nu

alise

d C

ost

of

Bo

rro

win

g

68

Loan + Interest Rate Floorlet (Put

on Bond)

0

5

10

15

20

4 6 8 10 12 14 16

LIBOR at expiry

An

nu

alize

d r

etu

rn

on

loan

69

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 Underlying

(notice variables on vertical axis)

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 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%.

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

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).

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.

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 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]

75

Exotics

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.

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

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)

79

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

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

81

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!

82

Chooser Option as a Package

))((

12

,0max

1))(()(

1

)(1

)(

1

12

)12(1

)12(

1212

1212

),0max(),max(

),max(

TTqr

eSKe

TTqrTTq

TTqTTr

Ke

TT

SKeecpc

T

eSKecp

pcT

TTqTTr

strike with

timeat maturingput a plus timeat maturing call a is This

therefore is timeat valueThe

parity call-put From

is valuethe timeAt

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

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

85

Parity Relations

c = cui + cuo

c = cdi + cdo

p = pui + puo

p = pdi + pdo

86

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)

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 = S0e-qT N(d1) – e–rT

KN(d2)

88

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)

89

Asian Options

• No exact analytic valuation

• Can be approximately valued by assuming

that the average stock price is lognormally

distributed

90

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

91

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, St – K

• Payoff: max(ST – St , 0) + St – K

• Similar to lookback option but cheaper

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(VT – UT, 0)

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

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

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

2)(ˆ

)var(

8

11ˆ)(ˆ

VE

VVEE

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

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)

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

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

il mapping e l’alm - unisi.it€¢ create a bull spread on ibm using the ... • twin-win...

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