derivatives exotic path dependent options

8/6/2019 Derivatives Exotic Path Dependent Options

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An Introduction to Exotic andPath-dependent Options

1 In this lecture. . .

Ghow to classify options according to important features

Ghow to think about derivatives in a way that makes it easy to com-

pare and contrast different contracts

Gthe names and contract details for many basic types of exotic op-

tions


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2 Discrete cashflows

Imagine a contract that pays the holder an amount 6 at time 96

. The

contract could be a bond and the payment a coupon. If we use ' 9

to denote the contract value and 9 >6

and 9 6

to denote just before and

just after the cashflow date then simple arbitrage considerations leadto

' 9

>

6

' 9

6

T

This is a jump condition. The value of the contract jumps by the

amount of the cashflow. The behaviour of the contract value acrossthe payment date is shown in the figure.


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3

0

5

10

15

20

25

30

0 0.5 1 1.5 2

drop in contract value

discrete cashflow

1.A discrete cashflow and its effect on a contract value.If the contract is contingent on an underlying variable so that we

have 9 6 W then we can accommodate cashflows that depend on

the level of the asset 6 i.e. we could have T 6 .


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3 Early exercise

Early exercise is a common feature. For example, the conversion of

convertible bonds is mathematically identical to the early exercise

of an American option.

G

The key point about early exercise is that the holder of this valuable

right should ideally act optimally, i.e. they must decide when to

exercise or convert.

In the partial differential equation framework that has been set up,

this optimality is achieved by solving a free boundary problem, with

a constraint on the option value, together with a smoothness condi-

tion.

American options are path dependent.


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4 Weak path dependence

GOptions whose value depends on the asset history, but can still be

written as 9 6 W are said to be weakly path dependent.

After early exercise, the next most common reason for weak path

dependence in a contract is a barrier. Barrier (or knock-in, or knock-

out) options are triggered by the action of the underlying hitting a

prescribed value at some time before expiry. For example, as long

as the asset remains below 150, the contract will have a call payoff

at expiry. However, should the asset reach this level before expiry

then the option becomes worthless; the option has knocked out.


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0

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

2.Two paths having the same value at expiry but with completelydifferent payoffs.


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5 Strong path dependence

Of particular interest, mathematical and practical, are the strongly

path-dependent contracts. These have payoffs that depend on some

property of the asset price path in addition to the value of the under-

lying at the present moment in time; in the equity option language,we cannot write the value as 9 6 W . The contract value is a function

of at least one more independent variable.

The Asian option has a payoff that depends on the average value of

the underlying asset from inception to expiry. We must keep track ofmore information about the asset price path than simply its present

position. The extra information that we need is contained in the run-

ning average. This is the average of the asset price from inception

until the present, when we are valuing the option.


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6 Time dependence

Here we are concerned with time dependence in the option contract.

We can add such time dependence to any of the features described

above. For example, early exercise might only be permitted on cer-

tain dates or during certain periods. This intermittent early exerciseis a characteristic of Bermudan options. Similarly, the position of

the barrier in a knock-out option may change with time. Every month

it may be reset at a higher level than the month before.

G These contracts are referred to as time inhomogeneous.


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7 Dimensionality

Dimensionality refers to the number of underlying independent vari-

ables.

GThe vanilla option has two independent variables, 6 and W , and is

thus two dimensional.

GThe weakly path-dependent contracts have the same number of

dimensions as their non-path-dependent cousins, i.e. a barrier call

option has the same two dimensions as a vanilla call.

For these contracts the roles of the asset dimension and the time

dimension are quite different from each other. This is because the

governing equation, the BlackScholes equation, contains a second

asset-price derivative but only a first time derivative.


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We can have two types of three-dimensional problem.

GThe first occurs when we have a second source of randomness,

such as a second underlying asset. We might, for example, have an

option on the maximum of two equities. In the governing equation

we will see a second derivative of the option value with respect to

each asset. We say that there is diffusion in both 6

and 6

.

GThe other type of problem that is also three dimensional is the

strongly path-dependent contract. The new independent variable

is a measure of the path-dependent quantity on which the optionis contingent. The new variable may be the average of the asset

price to date, say. In this case, derivatives of the option value with

respect to this new variable are only of the first order. Thus the

new variable acts more like another time-like variable.


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8 The order of an option

The basic, vanilla options are of first order. Their payoffs depend

only on the underlying asset, the quantity that we are directly mod-

eling. Other, path-dependent, contracts can still be of first order if

the payoff only depends only on properties of the asset price path.

G

Higher order refers to options whose payoff, and hence value, is

contingent on the value ofanother option.

The obvious first-order options are compound options, for exam-

ple, a call option giving the holder the right to buy a put option. The

compound option expires at some date 7

and the option on which it

is contingent, expires at a later time 7

. Technically speaking, such

an option is weakly path dependent.


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From a practical point of view, the compound option raises some

important modeling issues.

GThe payoff for the compound option depends on the market value

of the underlying option, and not on the theoretical price.

If you hold a compound option, and want to exercise the first optionthen you must take possession of the underlying option. High order

option values are very sensitive to the basic pricing model and should

be handled with care.


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9 Decisions, decisions

Holding an American option you are faced with the decision whether

and when to exercise your rights. The American option is the most

common contract that contains within it a decision feature. Other

contracts require more subtle and interesting decisions to be made.Well be seeing several examples of these later.

GFor example, the passport option is an option on a trading account.

You buy and sell some asset, if you are in profit on the expiry of

the option you keep the money, if you have made a loss it is writtenoff. The decisions to be made here are when to buy, sell or hold,

and how much to buy, sell or hold.


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10 Compounds and choosers

Compound and chooser options are simply options on options.

The compound option gives the holder the right to buy (call) or sell

(put) another option. Thus we can imagine owning a call on a put, for

example. This gives us the right to buy a put option for a specifiedamount on a specified date. If we exercise the option then we will

own a put option which gives us the right to sell the underlying.

This compound option is second order because the compound op-

tion gives us rights over another derivative.


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It is possible to find analytical formul for the price of basic com-

pound options in the BlackScholes framework when volatility isconstant. These formul involve the cumulative distribution func-

tion for a bivariate Normal variable.

GHowever, because of the second-order nature of compound options

and thus their sensitivity to the precise nature of the asset price

random walk, these formul are dangerous to use in practice.


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Chooser options are similar to compounds in that they give the

holder the right to buy a further option. With the chooser option theholder can choose whether to receive a call or a put, for example.


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11 Range notes

Range notes are very popular contracts, existing on the lognormal

assets such as equities and currencies, and as fixed-income products.

GIn its basic, equity derivative, form the range note pays at a rate of

/all the time that the underlying lies within a given range,

6

O

6 6

X

.

That is, for everyG W

that the asset is in the range you receive/ G W

.

Introducing, $

as the function taking the value 1 when$

O

T $ T

$

X

and zero otherwise, the range note satisfies

'

9

6

#

9

# 6

U 6

# 9

# 6

> U 9 / , 6


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Preliminary and IndicativeFor Discussion Purposes Only

6 Month In-Out Range Accrual Option on MXN/USD FX Rate

Settlement Date One week from Trade DateMaturity Date 6 months from Trade DateOption Premium USD 50,000+Option Type In MINUS Out Range Accrual on MXN/USD FX

rateOption Payment Date 2 business days after Maturity DateOption Payout USD 125,000 * IndexWhere Index

FX daily In The number of business days Spot MXN/USDExchange Rate is within Range

FX daily Out The number of business days Spot MXN/USDExchange Rate is outside Range

Range MXN/USD 7.7200-8.1300Spot MXN/USD Exchange

Rate

Official spot exchange rate as determined by the

Bank of mexico as appearing on Reuters pageBNMX at approximately 3:00 p.m. New York time.Current Spot MXN/USD 7.7800

This indicative termsheet is neither an offer to buy or sell securities or an OTC derivative product whichincludes options, swaps, forwards and structured notes having similar features to OTC derivative transactions,nor a solicitation to buy or sell securities or an OTC derivative product. The proposal contained in theforegoing is not a complete description of the terms of a particular transaction and is subject to change withoutlimitation.

zero)ofminimumato(subjectDaysBusinessTotal

OutdailyFXMINUSIndailyFX


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12 Barrier options

q Barrier options have a payoff that is contingent on the underlying

asset reaching some specified level before expiry.

The critical level is called the barrier, there may be more than one.

Barrier options are weakly path dependent.

Barrier options come in two main varieties, the in barrier option

(or knock-in) and the out barrier option (or knock-out). The for-

mer only have a payoff if the barrier level is reached before expiry

and the latter only have a payoff if the barrier is not reached be-

fore expiry. These contracts are weakly path dependent, meaning

that the price depends only on the current level of the asset and the

time to expiry. They satisfy the BlackScholes equation, with spe-

cial boundary conditions.


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14 Lookback options

qLookback options have a payoff that depends on the realized max-

imum or minimum of the underlying asset over some period prior

to expiry.

An extreme example, that captures the flavour of these contracts

is the option that pays off the difference between that maximum re-

alised value of the asset and the minimum value over the next year.

Thus it enables the holder to buy at the lowest price and sell at the

highest, every traders dream. Of course, this payoff comes at a price.And for such a contract that price would be very high.

Again the maximum or minimum can be calculated continuously

or discretely, using every realized asset price or just a subset.

derivatives exotic path dependent options

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