topic 13 exotic options

8/9/2019 Topic 13 Exotic Options

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Topic 13:Topic 13:

Exotic OptionsExotic Options


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IntroductionIntroduction

The calls and puts that we have examined so far areThe calls and puts that we have examined so far aregenerically known asgenerically known as Plain VanillaPlain Vanilla instruments, meaninginstruments, meaningthat they are standardized, exchange traded optionsthat they are standardized, exchange traded optionsthat basically everybody in the market completelythat basically everybody in the market completely

understands.understands. They are not interesting, but may be useful. Much in the sameThey are not interesting, but may be useful. Much in the same

way that plain vanilla ice cream may be good, but it is notway that plain vanilla ice cream may be good, but it is notterribly unusual or exciting.terribly unusual or exciting.

We have already discussed why exchangeWe have already discussed why exchange--tradedtraded

options would tend to be highly standardized. They mayoptions would tend to be highly standardized. They maynot, however, meet the needs of all market participants.not, however, meet the needs of all market participants.Some may need highly specialized options. TheseSome may need highly specialized options. Theseoptions, known asoptions, known as exotic optionsexotic optionstrade in the OTCtrade in the OTCmarkets.markets.


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So has the market evolved exotic (i.e. nonstandard)So has the market evolved exotic (i.e. nonstandard)options?options?

Hull does a nice job of elucidating some reasons:Hull does a nice job of elucidating some reasons:

Because they may meet specific, and unusual, hedging needs ofBecause they may meet specific, and unusual, hedging needs of

certain clients.certain clients.

They may have tax, accounting or regulatory advantages overThey may have tax, accounting or regulatory advantages overstandard options.standard options.

They may allow a firm to more fully express its sentiments onThey may allow a firm to more fully express its sentiments oncertain aspects of the market.certain aspects of the market.

Because there is more profit in them for investment banks.Because there is more profit in them for investment banks. Because they may allow an investment bank to sell an overpricedBecause they may allow an investment bank to sell an overpriced

product to its less technically sophisticated clients.product to its less technically sophisticated clients.


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There are many types of exotic options, and more areThere are many types of exotic options, and more arecreated all of the time.created all of the time.

We will examine the following types of exotics:We will examine the following types of exotics:

Bermudian optionsBermudian options

Forward start optionsForward start options

Compound optionsCompound options

Chooser optionsChooser options

Barrier options (knockBarrier options (knock--in and knockin and knock--out)out)

Binary optionsBinary options

Lookback optionsLookback options

Shout optionsShout options

Asian optionsAsian options


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When modeling exotic options, one has to make aWhen modeling exotic options, one has to make afundamental decision very early in the process: shouldfundamental decision very early in the process: shouldyou model the option in a continuousyou model the option in a continuous--time, Blacktime, Black--Scholes type of model, or in a binomial model.Scholes type of model, or in a binomial model.

Generally many exotic options are initially priced via a binomialGenerally many exotic options are initially priced via a binomialmodel, and then at some point traders figure out a closedmodel, and then at some point traders figure out a closed--formformpricing model.pricing model.

Sometimes, it turns out that no closed form solution is everSometimes, it turns out that no closed form solution is everfound.found.

Indeed, for certain highly pathIndeed, for certain highly path--dependent options, one cannotdependent options, one cannoteven work backwards in a lattice, instead one must use a Monteeven work backwards in a lattice, instead one must use a Monte--Carlo method to value the option.Carlo method to value the option.

This is especially true for many interest rate derivatives.This is especially true for many interest rate derivatives.


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We will use some of both.We will use some of both.

Hull tends to want to provide lots of closedHull tends to want to provide lots of closed--form continuous timeform continuous timeoptions.options.

McDonald generally provides a richer discussion of the optionsMcDonald generally provides a richer discussion of the options

themselves, but relegates his discussion of the pricing formulasthemselves, but relegates his discussion of the pricing formulasto an appendix at the end of Chapter 14.to an appendix at the end of Chapter 14.


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Bermuda OptionsBermuda Options

A Bermuda option is a generic name for an option that isA Bermuda option is a generic name for an option that isin between an American and a European option.in between an American and a European option. Usually these are options that allow early exercise only onUsually these are options that allow early exercise only on

specific dates.specific dates.

Sometimes they are options that do not allow early exercise untilSometimes they are options that do not allow early exercise untila specific date but then allow early exercise from that point on.a specific date but then allow early exercise from that point on.These are said to contain a lock out period.These are said to contain a lock out period.

Sometimes they are standard options but have a strike price thatSometimes they are standard options but have a strike price thatchanges, typically as a function of time.changes, typically as a function of time.

Typically these are priced through a binomial model.Typically these are priced through a binomial model. Example: A nonExample: A non--dividend paying stock is currently priced at $20,dividend paying stock is currently priced at $20,

and you hold a put that allows early exercise in 2 months and in 4and you hold a put that allows early exercise in 2 months and in 4months. The option expires in 6 months. Volatility is 30%, andmonths. The option expires in 6 months. Volatility is 30%, andr=5%. What is the value of this option?r=5%. What is the value of this option?


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We will use the CRR binomial model. First, we need toWe will use the CRR binomial model. First, we need todetermine our lattice parameters:determine our lattice parameters:

We can then build the lattice itself. Since this is a ratherWe can then build the lattice itself. Since this is a ratherlong option, we will build the lattice in Excel.long option, we will build the lattice in Excel.

1.3

12

1.3

12

( )

1.090463

0.917042

and

1.00417 0.9170420.502439

1.090436 0.917042

dt

dt

r dt

u e e

d e e

e dp

u d

W

W

! ! !

! ! !

! ! !


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The Bermuda option has a value of 1.47.The Bermuda option has a value of 1.47.

Compare this to a similar option with a lockCompare this to a similar option with a lock--out provisionout provisionfor the first two months, which has a value of 1.49.for the first two months, which has a value of 1.49.


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Forward Start OptionsForward Start Options

A forward start option is one in which the option isA forward start option is one in which the option isissued to you but begins at some point in the future.issued to you but begins at some point in the future.

These are widely used in employee/executive compensationThese are widely used in employee/executive compensationschemes.schemes.

Generally you are granted the option today, but it does notGenerally you are granted the option today, but it does notbecome active until some date in the future, Tbecome active until some date in the future, T11. Usually you. Usually youwould have to still be employed by the firm at time Twould have to still be employed by the firm at time T11, or your, or youroption would not activate.option would not activate.

Usually these options are written such that the strike isUsually these options are written such that the strike is

determined to be at the money at time Tdetermined to be at the money at time T11, that is, K=S, that is, K=S11.. The option then expires at time TThe option then expires at time T22..

So how do we value such an instrument?So how do we value such an instrument?


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The key to developing a pricing model is to realize thatThe key to developing a pricing model is to realize thatwe know for certain that the option will be at the moneywe know for certain that the option will be at the moneyat time Tat time T11..

Recall that we are assuming that other parameters, such as rRecall that we are assuming that other parameters, such as r

andand are constant.are constant.

The value of an atThe value of an at--thethe--money call option is proportional to themoney call option is proportional to thestock price.stock price.

To see this, notice the following example. Price a year, at theTo see this, notice the following example. Price a year, at themoney call option on a stock with volatility of .30, and with riskmoney call option on a stock with volatility of .30, and with risk

free rate of 5%. Price this option for different values of Sfree rate of 5%. Price this option for different values of S00 (and(andhence of K).hence of K).


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Here is a list of BlackHere is a list of Black--Scholes prices for SScholes prices for S00 from 1 to 20.from 1 to 20.

Notice that the option price for SNotice that the option price for S00>1 is just the S>1 is just the S00=1=1price multiplied by the ratio of the stock price to 1!price multiplied by the ratio of the stock price to 1!

Stock & Strike

Call

Value

Ratio

to

Value

when

S=1

1 0.096349 1

2 0.192697 2

3 0.289046 3

4 0.385395 4

5 0.481744 5

6 0.578092 6

7 0.674441 7

8 0.77079 8

9 0.867138 9

10 0.963487 10


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What this means is that for any given set of values for r,What this means is that for any given set of values for r,, and T, if an option is at the money, its value can be, and T, if an option is at the money, its value can berelated to any other atrelated to any other at--thethe--money option with the samemoney option with the samer,r, , and T values., and T values.

Denote cDenote c00 as the Blackas the Black--Scholes value of an atScholes value of an at--thethe--money optionmoney optionat time 0, with maturity T = Tat time 0, with maturity T = T22--TT11, on the stock, that is, K=S, on the stock, that is, K=S00..

Since r,Since r, , at T do not change at time T, at T do not change at time T11, only S, only S11 and K, I knowand K, I knowthat the value of an atthat the value of an at--thethe--money option at time Tmoney option at time T11 will be:will be:

cc11=c=c00*(S*(S11/S/S00))

Going back to the previous example, if SGoing back to the previous example, if S00 were $5, thenwere $5, thencc00=0.481744. Thus, if at time T=0.481744. Thus, if at time T11 SS11=7, then we would expect=7, then we would expectcc11=0.481744*(7/5)=0.481744*1.4 = 0.67444.=0.481744*(7/5)=0.481744*1.4 = 0.67444.


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This is nice because it means that all we have to do is toThis is nice because it means that all we have to do is totake determine the risktake determine the risk--neutral expected value for thisneutral expected value for thisoption and discount it at the riskoption and discount it at the risk--free rate, i.e.:free rate, i.e.:

Of course, SOf course, S00 and cand c00 are known with certainty at time 0, and theare known with certainty at time 0, and theexpected stock price at time Texpected stock price at time T11 under riskunder risk--neutrality is simplyneutrality is simply

Plugging in and simplifying yields:Plugging in and simplifying yields:

1 1_ 00

rT forward start Sc e E c

S

! -

1( )1 0 r q

S S e !

1

_ 0

qT

foward startc c e!


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Compound OptionsCompound Options

A compound option is simply an option to buy an option.A compound option is simply an option to buy an option.There are four types of these options:There are four types of these options: Call on a call.Call on a call.

Call on a put.Call on a put.

Put on a call.Put on a call. Put on a put.Put on a put.

There will be two sets of strikes and maturities. The firstThere will be two sets of strikes and maturities. The firstset, Kset, K11 and Tand T11, are the strike and time at which you can, are the strike and time at which you canexercise the option on the option, and Kexercise the option on the option, and K22 and Tand T22 are theare the

strike and time at which you can exercise the option onstrike and time at which you can exercise the option onthe underlying.the underlying.

Of course, you can also be long or short any of theseOf course, you can also be long or short any of theseoptions, so there are really a total of 8 potentialoptions, so there are really a total of 8 potentialpositions.positions.


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Now lets think just for a minute about what theseNow lets think just for a minute about what theseoptions really allow you to do.options really allow you to do. Call on a call: you have the right at time TCall on a call: you have the right at time T11 to pay Kto pay K11 for anfor an

option that will allow you to buy the underlying stock at time Toption that will allow you to buy the underlying stock at time T22at price Kat price K

22..

Call on a put: you have the right at time TCall on a put: you have the right at time T11 to pay Kto pay K11 for anfor anoption that will allow you tooption that will allow you to sellsell the underlying stock at time Tthe underlying stock at time T22for price Kfor price K22..

Put on a call: you have the right at time TPut on a call: you have the right at time T11 to sell for price Kto sell for price K11 ananoption that will allow you to take a short position in a call on theoption that will allow you to take a short position in a call on the

stock at strike Kstock at strike K22 and with maturity Tand with maturity T22.. Put on a call: you have the right at time TPut on a call: you have the right at time T11 to sell for price Kto sell for price K11 anan

option that will allow you to take a short position in a put on theoption that will allow you to take a short position in a put on thestock at strike Kstock at strike K22 and with maturity Tand with maturity T22..


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There are closed form solutions available for theseThere are closed form solutions available for theseoptions. For example, the value of a European Call on aoptions. For example, the value of a European Call on aCall is:Call is:

M is the cumulative bivariate normal distribution. SM is the cumulative bivariate normal distribution. S** is the assetis the assetprice at time Tprice at time T11 for which the option price at time Tfor which the option price at time T11 equals Kequals K11..

2 2 10 1 1 1 2 2 2 2 1 2 1 2

2

01*

1 2 1 1

1

2

0

12

1 2 1 2

2

, ; / , ; / ( )

ln2

, and a

ln2

, and b

qT rT rT

callc S e M a b T T K e M a b T T e K N a

whereS

r q TS

a a TT

Sr q T

Kb b T

T

W

WW

W

WW

!

! !

! !


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Even though a closedEven though a closed--form solution exists, I find it quiteform solution exists, I find it quiteinstructive to examine a call on a call with the binomialinstructive to examine a call on a call with the binomialmodel, just so that we can fully illustrate how thismodel, just so that we can fully illustrate how thisinstrument behaves.instrument behaves.

Additionally, the binomial has the advantage of being relativelyAdditionally, the binomial has the advantage of being relativelyeasy to implement and does not require the bivariate binomialeasy to implement and does not require the bivariate binomialdistribution.distribution.

Lets go back to the base case we have been working. We haveLets go back to the base case we have been working. We havea stock with Sa stock with S00=20, r=.05,=20, r=.05, =.30. Lets assume that the option=.30. Lets assume that the option

on which the option will be written is a call with strike of $20on which the option will be written is a call with strike of $20which expires at time Twhich expires at time T22=6 months. Lets further assume that we=6 months. Lets further assume that wehave a call on that call that expires in three months, and that itshave a call on that call that expires in three months, and that itsstrike is $2. Both options are European.strike is $2. Both options are European.

Thus, SThus, S00=20, K=20, K11=2, T=2, T11=.25, K=.25, K22=20, and T=20, and T22=.50=.50


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From the spreadsheet we can see that the option on theFrom the spreadsheet we can see that the option on theunderlying would be worth $1.86 at time 0, but that theunderlying would be worth $1.86 at time 0, but that theoption on the option would be worth $0.69.option on the option would be worth $0.69.

If we were to value a put on that call, it would be worthIf we were to value a put on that call, it would be worth$0.81.$0.81.

One could just as easily determine the value of a call onOne could just as easily determine the value of a call ona put or a put on a put in the same manner.a put or a put on a put in the same manner.


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Chooser OptionsChooser Options

A chooser option is frequently referred to as anA chooser option is frequently referred to as an as youas youlike itlike it option.option.

This option gives you the right to declare at time TThis option gives you the right to declare at time T11whether the option is a put or a call with maturity atwhether the option is a put or a call with maturity attime Ttime T22..

At time TAt time T11 the option value is the max(c,p).the option value is the max(c,p).

We can use putWe can use put--call parity to determine a valuation formula if thecall parity to determine a valuation formula if thechooser option is based on European options and they have thechooser option is based on European options and they have thesame maturity and strike price.same maturity and strike price.

Lets redefine the TLets redefine the T11 value as:value as:

2 1 2 1

2 1 2 1

2

( ) ( )

1

( ) ( )( )

1

2

- (

max( , ) max ,

max 0,

thus it is equivalent to a package (i.e. portfolio) consisting of:

1. Call with strike price K and maturity T , and

2.

r T T q T T

q T T r q T T

q T

c p c c Ke S e

c e Ke S

e

!

!

1 2 1) -( - )( )

1 put options with strike price and maturity T .T r q T T

Ke


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Chooser OptionsChooser Options

We can verify this easily enough. Lets assume that weWe can verify this easily enough. Lets assume that wehave a three month chooser option on a stock currentlyhave a three month chooser option on a stock currentlypriced at 20, withpriced at 20, with =.30, T=.30, T22--TT11=.25, and with K=20, and=.25, and with K=20, andr=.05, and q=0.r=.05, and q=0.

From BlackFrom Black--Scholes, the value of a call with strike K and maturityScholes, the value of a call with strike K and maturityTT22=.50, (and the above parameters) is: 1.93.=.50, (and the above parameters) is: 1.93.

The number of put options to buy is eThe number of put options to buy is e0(.25)0(.25)=1.=1.

The strike for the put is 20eThe strike for the put is 20e--(.05)(.25)(.05)(.25)=19.75.=19.75.

The BlackThe Black--Scholes value for the put, therefore is: $0.95.Scholes value for the put, therefore is: $0.95.

Thus, the total value of the chooser is 1.93+(1)0.95 = 2.88.Thus, the total value of the chooser is 1.93+(1)0.95 = 2.88. When we build this through the binomial.When we build this through the binomial.

Even with only a 1 month time step, we still get a value for theEven with only a 1 month time step, we still get a value for thechooser of 2.91.chooser of 2.91.


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Barrier OptionsBarrier Options

Barrier options are options that have a payout that isBarrier options are options that have a payout that isdependent not only on the terminal stock price, but alsodependent not only on the terminal stock price, but alsodepend upon whether the stock attains some barrierdepend upon whether the stock attains some barrierduring the life of the option. Two general kinds:during the life of the option. Two general kinds:

KnockKnock--in options: The option comes into being only if the stockin options: The option comes into being only if the stockreaches a given barrier during its life.reaches a given barrier during its life.

KnockKnock--out options: The option ceases to exist if the stockout options: The option ceases to exist if the stockreaches a given barrier during the options life.reaches a given barrier during the options life.

Some examples:Some examples:

DownDown--andand--outout callcall. This is a call with strike K that ceases to exist if. This is a call with strike K that ceases to exist ifthe asset price reaches the barrier level H, where Sthe asset price reaches the barrier level H, where S00>H.>H.

DownDown--andand--in callin call. This is call with strike K that comes into existence. This is call with strike K that comes into existenceonly if the stock price reaches the barrier level.only if the stock price reaches the barrier level.


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Assume for a moment that you held two portfolios:Assume for a moment that you held two portfolios: Portfolio A: One call with strike K.Portfolio A: One call with strike K.

Portfolio B: One down and in call with strike K and barrier H.Portfolio B: One down and in call with strike K and barrier H.One down and out call with strike K and barrier H.One down and out call with strike K and barrier H.

At maturity there are two potential states of the world,At maturity there are two potential states of the world,lets compare the portfolio values in each:lets compare the portfolio values in each: The stock remained above the barrier at all times (state 1).The stock remained above the barrier at all times (state 1).

A: Max(0,SA: Max(0,STT--K).K).

B: Down and In: 0 (never activated).B: Down and In: 0 (never activated).Down and out: max(0,SDown and out: max(0,STT--K)K)

The stock at some point touched the barrier (state 2).The stock at some point touched the barrier (state 2). A: Max(0,SA: Max(0,STT--K)K)

B: Down and In: max(0,SB: Down and In: max(0,STT--K)K)Down and Out: 0 (died when stock touched barrier).Down and Out: 0 (died when stock touched barrier).


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We can see, therefore, that the two portfolios have theWe can see, therefore, that the two portfolios have thesame terminal values in all states of the world, so theysame terminal values in all states of the world, so theymust be equal. Denote c as the value of the normal callmust be equal. Denote c as the value of the normal calloption at time 0, coption at time 0, cdidi as the value of the down and inas the value of the down and in

option at time 0 and coption at time 0 and cdodo as the value of the down andas the value of the down andout option at time 0. Then,out option at time 0. Then,

c = cc = cdidi + c+ cdodo So we really only need to determine the value of one ofSo we really only need to determine the value of one of

these barriers to determine the value of the other atthese barriers to determine the value of the other attime 0.time 0.

Which of the two that we will choose to solve depends upon theWhich of the two that we will choose to solve depends upon therelationship between K and H.relationship between K and H.


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If H=K, then value the down and out call:

2 2 2

0

0 0

22

0

2

( )

ln

2 and

qT rT

di

H Hc S e N y Ke N y T

S S

where

Hr q S K

y TT

P P

W

W

P PWW W

!

- ! !

2 2 2

0 1 1 0 1 1

0 0

20

0

1 12

( ) ( )

lnln2 , = and

q r q r

doc S x e e x S e y e yS S

here

Sr q S

x y

P P

W W

W

P PW PW W W W

!

- - ! !


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AnAn upup--andand--outout call is a regular call that ceases to exist ifcall is a regular call that ceases to exist ifthe asset price reached the barrier level, H, while anthe asset price reached the barrier level, H, while an upup--andand--in callin call is one that does not come into existenceis one that does not come into existenceunless the asset price hits the barrier.unless the asset price hits the barrier.

There are, of course, similar barrier options for puts.There are, of course, similar barrier options for puts.

Hull and McDonald both supply you with exact pricing formulas.Hull and McDonald both supply you with exact pricing formulas.

On an exam I would expect you to know what theseOn an exam I would expect you to know what thesebarrier options are, and be able to work through abarrier options are, and be able to work through abinomial example of them.binomial example of them.


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Barrier OptionsBarrier Options These can also be valued through a Binomial lattice, butThese can also be valued through a Binomial lattice, but

its trickier than it might at first appear.its trickier than it might at first appear.

Lets value a down and out option through the lattice weLets value a down and out option through the lattice wehave been using this evening.have been using this evening.

Usually its easiest to explicitly solve for one of the valuesUsually its easiest to explicitly solve for one of the valuesand then to use the relationship between a normal calland then to use the relationship between a normal calland the barrier option to determine the other barriersand the barrier option to determine the other barriersvalue. Lets begin with thevalue. Lets begin with the down_and_outdown_and_out call sincecall sincevaluing it requires a relatively minor change to yourvaluing it requires a relatively minor change to yourboundary conditions:boundary conditions:

T

( )

t

When t=T

max(0, ) i S and 0 other ise.

When t


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To value a down_and_in callTo value a down_and_in call i.e. one where the calli.e. one where the callonly activates upon hitting the lower valueonly activates upon hitting the lower value you againyou againjust adjust your standard boundary conditions.just adjust your standard boundary conditions.

You simply have to realize that at a node where the stock price is atYou simply have to realize that at a node where the stock price is at

or below the barrier, then the value of the barrier option is exactlyor below the barrier, then the value of the barrier option is exactlythe same as the value of a standard call at that node.the same as the value of a standard call at that node.

At nodes where the stock price is above the barrier value, then theAt nodes where the stock price is above the barrier value, then thevalue of the option is equal to the expected present value of thevalue of the option is equal to the expected present value of thebarrier option at the next time step!barrier option at the next time step!

T

T

T

-r dt

T

I t=T

max(0, ) i S

0 i S

i t


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To actually implement aTo actually implement a down_and_indown_and_inbinomial pricingbinomial pricingmodel, you really have to track two options through themodel, you really have to track two options through thelattice: a standard European option, and then the downlattice: a standard European option, and then the downand in option.and in option.

At a node where the barrier it touched (or lower), you set theAt a node where the barrier it touched (or lower), you set thevalue of your DI option to equal the European option (in thisvalue of your DI option to equal the European option (in thisway you take into account the payoffs when the DI is activeway you take into account the payoffs when the DI is activelater, but above the barrier.) If the stock is above the barrier,later, but above the barrier.) If the stock is above the barrier,you set the option equal to the discounted expected value of theyou set the option equal to the discounted expected value of the

future values of the DI option.future values of the DI option.


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Binary OptionsBinary Options

A Binary Option is basically a standard option, but one inA Binary Option is basically a standard option, but one inwhich the payout is altered such that it only pays a fixedwhich the payout is altered such that it only pays a fixedamount if the option ends up in the money.amount if the option ends up in the money.

In aIn a cash or nothingcash or nothingbinary, the amount is a lump sum,binary, the amount is a lump sum,

denoted Q.denoted Q. In the standard BlackIn the standard Black--Scholes equation, N(dScholes equation, N(d22) is the (risk) is the (risk--neutral)neutral)

probability of the stock price being greater than strike price at timeprobability of the stock price being greater than strike price at timeT. Thus the value of the Binary option, with strike price K, is givenT. Thus the value of the Binary option, with strike price K, is givenby:by:

cash or nothing 2

2

0

2

( )

ln2

rtC Qe N d

where

Sr T

Kd

T

W

W

!

- !


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Similarly, an asset or nothing put pays off the asset priceSimilarly, an asset or nothing put pays off the asset priceif it ends up in the money, its value is given by:if it ends up in the money, its value is given by:

Example: A stock is currently priced at $20, has volatility ofExample: A stock is currently priced at $20, has volatility of30%, and pays no dividend. You hold an30%, and pays no dividend. You hold an assetor nothingassetor nothing callcallwith a strike of 22, that expires in 6 months. Assuming the riskwith a strike of 22, that expires in 6 months. Assuming the risk

free rate is 5%, what is the value of this option?free rate is 5%, what is the value of this option? $6.62.$6.62.

asset or nothing 0 2( )qt p S e d

!


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Obviously it is a trivial modification of the standard callObviously it is a trivial modification of the standard callor put to implement these in a binomial model. In fact,or put to implement these in a binomial model. In fact,all you really have to do is modify the terminal boundaryall you really have to do is modify the terminal boundaryconditions for a European call or putconditions for a European call or put otherwise theyotherwise they

are priced exactly the same in a binomial!are priced exactly the same in a binomial!

T T_ _ _ _

T T

T T_ _ _ _

T T

0 if if

if 0 if

0 if if

if 0 if

cash or nothing cash or nothing

T T

Tasset or nothing asset or nothing

T T

T

K Q K

c p

Q K K

K S K

c p

S K K

! ! "! "!

! ! "! "!


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Lookback OptionsLookback Options

A Lookback option is one in which the payout is aA Lookback option is one in which the payout is afunction of both the terminal stock valuefunction of both the terminal stock value andand thethemaximum or minimum value the stock achieves duringmaximum or minimum value the stock achieves duringthe life of the option.the life of the option.

For a EuropeanFor a European--style lookback call, the payoff is the greater ofstyle lookback call, the payoff is the greater ofzero or the difference between the final asset price and thezero or the difference between the final asset price and theminimum value of the asset during the life of the instrument:minimum value of the asset during the life of the instrument:

cclookbacklookback=max(0,S=max(0,STT--SSminimumminimum))

For a EuropeanFor a European--style lookback put, the payoff is the greater ofstyle lookback put, the payoff is the greater of

zero or the difference between the maximum value the assetzero or the difference between the maximum value the assetreaches during the life of the option and the final asset price.reaches during the life of the option and the final asset price.


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Lookback OptionsLookback Options

There are valuation formulas for lookbacks.There are valuation formulas for lookbacks.

Note that SNote that Sminmin is the minimum price achieved at the timeis the minimum price achieved at the timeof valuationof valuation at time t=0, then Sat time t=0, then Sminmin=S=S0.0.

1

2 2

0 1 0 1 min 2 3

2

0 min1

2 1

2

0 min3

2

0 min1 2

( ) ( ) ( ) ( )2 1 2

here

ln( / ) ( / 2)a

ln( / ) ( / 2)

2( / 2) ln( / )

YqT qT rT

lookbackc S e a S e a S e a e ar r q

S S r q T

T

a a T

S S r q Ta

T

r q S S Y

W W

W

W

W

W

W

W

W

!

!

!

!

!


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Shout OptionsShout Options

A shout option is one in which the long party can shoutA shout option is one in which the long party can shoutat the short party one time during the life of the option,at the short party one time during the life of the option,which sets a sort of lower payoff level.which sets a sort of lower payoff level.

At maturity, the holder receives either the intrinsic valueAt maturity, the holder receives either the intrinsic value

at the time of the shout, or the payoff to a usualat the time of the shout, or the payoff to a usualEuropean call/put (depending upon the type of optionEuropean call/put (depending upon the type of optionthat it is.)that it is.)

Assume that you shout at time tau, then your payoutAssume that you shout at time tau, then your payoutat maturity would be (for a call):at maturity would be (for a call):

Essentially a shout allows you lock in a payout withoutEssentially a shout allows you lock in a payout withoutforcing you to give up your ability to earn a higherforcing you to give up your ability to earn a higherpayout if the price increases.payout if the price increases.

max(0, , ) shout T c S K S K X!


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Consider if you had a shout option with 6 months toConsider if you had a shout option with 6 months tomaturity. The stock price is $20, and the strike is $22.maturity. The stock price is $20, and the strike is $22.Lets say that after 1 month the stock price had risen toLets say that after 1 month the stock price had risen to$30. You could shout at that point and you would$30. You could shout at that point and you would

guarantee that you would earn at least (30guarantee that you would earn at least (30--22)=$8 at22)=$8 atmaturity.maturity.

If subsequently the stock price fell to $28 at maturity,If subsequently the stock price fell to $28 at maturity,you would still earn a payout based on your shoutedyou would still earn a payout based on your shoutedstock price of $30, so you would get $8.stock price of $30, so you would get $8.

If, however, the stock price rose to finish at $40, youIf, however, the stock price rose to finish at $40, youwould receive (40would receive (40--22) = $18.22) = $18.

Notice that you can write the terminal payout as:Notice that you can write the terminal payout as:

max(0, ) ( )TS S S K X X


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Shout options are very similar to American options, andShout options are very similar to American options, andthey are normally valued through a lattice typethey are normally valued through a lattice typearrangement.arrangement. You can value them as you work backwards: you simplyYou can value them as you work backwards: you simply

compare at each node the value of the option if you shout orcompare at each node the value of the option if you shout ornot at that node, and assume the option is worth the morenot at that node, and assume the option is worth the morevaluable of the two.valuable of the two.

This is somewhat trickier that it may at first appear. Note thatThis is somewhat trickier that it may at first appear. Note thatyou have to determine the value of the option if you shout atyou have to determine the value of the option if you shout atthat pointthat point not the intrinsic value of the option. This means younot the intrinsic value of the option. This means you

have to go back to the end of the lattice and revalue the optionhave to go back to the end of the lattice and revalue the optionknowing that you shouted at that node!knowing that you shouted at that node!

This is not too difficult to do in software, since you can simplyThis is not too difficult to do in software, since you can simplycall a function recursively, but it is challenging to illustratecall a function recursively, but it is challenging to illustrategraphically.graphically.


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To get a sense of whatI mean by this, consider a simpleTo get a sense of whatI mean by this, consider a simpleexample. Assume you have a call shout option on aexample. Assume you have a call shout option on astock with a price of $20, 3 months to maturity, volatilitystock with a price of $20, 3 months to maturity, volatilityof 30%, a risk free rate of 5%, and a time step of 1of 30%, a risk free rate of 5%, and a time step of 1

month. The strike is 22.month. The strike is 22. From the CRR binomial model, the relevant parameters are:From the CRR binomial model, the relevant parameters are:

u=1.090, d=.917, and p = .5024u=1.090, d=.917, and p = .5024

The lattice this generates is:The lattice this generates is:


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Shout Options:Shout Options:

20

21.81

23.78

25.93

21.81

20

18.3418.34

16.82

16.82


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20

21.81

23.78

25.93Max(0,25.93-22)= 3.93

21.810

20

18.3418.340

16.82

16.820


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4444


We can then begin to work backwards through theWe can then begin to work backwards through thelattice. At each node of the lattice we assume that thelattice. At each node of the lattice we assume that theoption value is the greater of two conditional values:option value is the greater of two conditional values:

1.1. The value of the option if we do not shout at this node.The value of the option if we do not shout at this node.

2.2. The value of the option if we do shout at this node.The value of the option if we do shout at this node.

Calculating the first one is relatively simple, justCalculating the first one is relatively simple, justdetermine the discounted expected value of the optiondetermine the discounted expected value of the optionat the next time stepat the next time step the procedure we normally usethe procedure we normally use

when valuing an option.when valuing an option. The next slide shows this value for each node at time step 2:The next slide shows this value for each node at time step 2:


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The second step is a little trickier, we have to determineThe second step is a little trickier, we have to determinethe value of the option if we shout at this node.the value of the option if we shout at this node. This will not matter at nodes (2,0) and (2,1), since the stockThis will not matter at nodes (2,0) and (2,1), since the stock

price is less than the strike, we would not shout here, and theprice is less than the strike, we would not shout here, and thevalue is 0.value is 0.

At node (2,2), however, if we shout, we have to figure out theAt node (2,2), however, if we shout, we have to figure out thevalue of the option under the assumption that we shouted at thisvalue of the option under the assumption that we shouted at thisnode.node.

If we shout at (2,2), the minimum payout would be (23.78If we shout at (2,2), the minimum payout would be (23.78--22)=$1.78 at each of the nodes that we can still reach on level22)=$1.78 at each of the nodes that we can still reach on level

3.3. This wont affect node (3,3), since the payout is higher, but it wouldThis wont affect node (3,3), since the payout is higher, but it would

affect node (3,2), where Saffect node (3,2), where Stt=1.81, and were the payout was=1.81, and were the payout wasoriginally 0.originally 0.


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20

21.81

23.78

1.98 no shout(.502*3.93+.498(1.78))e -.05/12

=2.87 with Shout

25.933.933.93 stays the same!

21.8101.78 because of theshout!20

0

18.3418.340

16.82

016.820

Determine the value of the option assuming that we shouted at node(2,2).


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We then assume then replace the value of the option atWe then assume then replace the value of the option atthat node with the $2.87 value we calculated.that node with the $2.87 value we calculated.


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20

21.81

23.78

2.87

25.933.93

21.810

200

18.3418.340

16.82

016.820


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We can then work on time step 1.We can then work on time step 1.

In this case we are fortunate in that the stock price at bothIn this case we are fortunate in that the stock price at bothnodes is below the strike of $22, so we can simply work our waynodes is below the strike of $22, so we can simply work our waybackwards as we normally would.backwards as we normally would.


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20(.502*1.45)e-.05/12

=0.72

21.81(.502*2.87)e-.05/12

=1.45

23.78

2.87

25.933.93

21.810

200

18.3418.340

16.82

016.820


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5252


We can then work on time step 1.We can then work on time step 1.

If, however, the stock price were above the strike at a node, sayIf, however, the stock price were above the strike at a node, say(1,1), we would have to determine the value of the option at(1,1), we would have to determine the value of the option atnode (1,1) assuming that we shouted at 1,1 (we would ignorenode (1,1) assuming that we shouted at 1,1 (we would ignore

any shouting at (2,2) or (2,1) when we did so).any shouting at (2,2) or (2,1) when we did so). In other words, as you work your way backwards through theIn other words, as you work your way backwards through the

lattice, at any node where you might possibly shout, you have tolattice, at any node where you might possibly shout, you have towork out a miniwork out a mini--lattice to determine the option valuelattice to determine the option valueconditional on your shouting at that node.conditional on your shouting at that node.

Again, this can be done through a recursive procedure, which isAgain, this can be done through a recursive procedure, which isrelatively easy to implement in VBA or C++ code, but is muchrelatively easy to implement in VBA or C++ code, but is muchmore tedious to do by hand in a spreadsheet.more tedious to do by hand in a spreadsheet.


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5353

Asian OptionsAsian Options

The termThe term Asian OptionAsian Option means an option where themeans an option where thepayoff to the option is a function of the average price ofpayoff to the option is a function of the average price ofthe underlying for some portion of the life of the option.the underlying for some portion of the life of the option. There are actually many variants of Asian options available.There are actually many variants of Asian options available.

Average price call: max(0,SAverage price call: max(0,Saveave

--K)K)

Average price put: max(0,KAverage price put: max(0,K--SSaveave))

Average strike call: max(0,SAverage strike call: max(0,STT--SSaveave))

Average strike put: max(0,SAverage strike put: max(0,Saveave--SSTT))

The method for calculating the average can vary.The method for calculating the average can vary. Usually it is the arithmetic average, but can be the geometricUsually it is the arithmetic average, but can be the geometric

average.average. If arithmetic average, usually no closed form solution exists,If arithmetic average, usually no closed form solution exists,

although Hull presents some approximations. Typically will relyalthough Hull presents some approximations. Typically will relyupon Monte Carlo procedures to determine the value.upon Monte Carlo procedures to determine the value.


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Asian OptionsAsian Options

The period of from which the average is taken can vary tremendously:The period of from which the average is taken can vary tremendously:

Daily over life of the option.Daily over life of the option.

Daily over last n days of options life.Daily over last n days of options life.

Daily over first n days of options life.Daily over first n days of options life.

Weekly, monthly, etc.Weekly, monthly, etc.

From specific days during the life of the option (perhaps on the 15From specific days during the life of the option (perhaps on the 15thth of everyof everymonth.)month.)

Notice that frequently its not practical to use backwards pricing for anNotice that frequently its not practical to use backwards pricing for anAsian option.Asian option.

To see this, consider that at each terminal node, you would have multipleTo see this, consider that at each terminal node, you would have multipleaverages, depending upon which path you followed to that node.averages, depending upon which path you followed to that node.

You would have to evaluate the terminal conditions for each path availableYou would have to evaluate the terminal conditions for each path availablein the lattice.in the lattice.

How many paths are there? 2How many paths are there? 2TT, so if you had a 20 time step lattice, you, so if you had a 20 time step lattice, youwould have to evaluate 2would have to evaluate 22020 = 1,048,576 paths.= 1,048,576 paths.

This simply is not practical, its faster to use Monte Carlo to arrive at anThis simply is not practical, its faster to use Monte Carlo to arrive at anapproximate answer.approximate answer.

topic 13 exotic options

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