Slides ExoticOptionsI

Rangarajan K. Sundaram Derivatives: Principles & Practice 1

Chapter 18. Exotic Options I:

Path-Independent Options

Rangarajan K. Sundaram

Stern School of Business

New York University

Copyright 2011 by The McGraw-Hill Companies, Inc. All

rights reserved.


Outline

What are "Exotic" Options? Path-Independent Options Digital Options Compound Options Chooser Options Forward Starts Exchange Options Quantos Other PI Options


Introduction


Exotic Options

Any option different from vanilla options is called "exotic."

Exotic options are not necessarily more complex in form than vanilla options.

Some, such as digitals, have very simple structures.

But many, such as barriers, Asians, and quantos, are certainly more

complex.

What do exotics bring to the table?

Primarily, richer payoff patterns/costs than we can obtain with vanillas

(essentially,insurance contracts with greater flexibility than the

boilerplate.)

Examples: Compounds, Barriers, Asians, ...


Categorization of Exotic Options

There is a huge variety of exotic options, so to fix ideas, some sort of

categorization helps.

One particularly useful categorization is based on how payoffs are defined:

Path-independent exotics are those whose payoff at maturity

depends only on the price of the underlying at maturity and not on how

that price was reached.

Path-dependent exotics are those whose payoffs may depend on

some or all of the path taken by prices over the life of the contract.

Other categorizations too are sometimes used, such as on how many

different variables affect option value (the "dimension" of an option) and the

manner in which the option payoff depends on the price path (the "order" of

an option)


The Black-Scholes Setting

In many cases, exotic prices can be expressed in closed-form in a Black-

Scholes setting.

In such cases, we will discuss the results using the usual Black-Scholes

notation:

S : price of underlying.

K : Strike price of option.

T : Time-to-maturity.

: Volatility of underlying.

r : risk-free interest rate.

: dividend rate on underlying.


Vanilla Option Pricing Formulae

The Black-Scholes prices of European call and put options with strike K and

maturity T are given by

C = e-TS N (d1) PV (K ) N (d2)

P = PV (K) N (d2) eTS N (d1)

where


The Components of the Black-Scholes Formula

For a call:

The delta of the call option is given by eT N (d1).

The (risk-neutral or risk-adjusted) probability that the call finishes in-

the-money (i.e., that ST K ) is N (d2).

For a put:

The delta of the put option is given by eT N (d1).

The (risk-neutral or risk-adjusted) probability that the put finishes in-

the-money (i.e., that ST K ) is N (d2).


Behavior of Vanilla Option Prices


Behavior of Vanilla Option Deltas


A Common Binomial Framework

Where closed-forms are not available, or where we wish to illustrate special

points, we use a two-period binomial model.

The parameters are: S = 100, u = 1.10, d = 0.90, R = 1.02.

The stock price tree is presented on the next page.

For future reference, note that the risk-neutral probability is:


Binomial Example: Stock Price Evolution


Option Prices in the Binomial Tree

Consider, in this setting, a two-period European call with a strike of K = 100.

The initial price of the call is 7.27.

After one period, it's possible values are Cu = 12.35 and Cd = 0.

After two periods, the value of the call is Cuu = 21; Cud = Cdu = 0 and

Cdd = 0.

The call price tree is presented on the next page.

The corresponding numbers for a two-period European put are:

3.38, (0.39, 8.04), and (0, 1, 1, 19).


European Call Prices


Path-Independent Options

Path-independent options are those whose payoffs at T depend only on ST and not on St for t < T. Note that vanilla options are path-independent.

We examine several classes of path-dependent options:

Binary (or digital) options.

Chooser options.

Compound options.

Forward start options.

Exchange options.

Quanto options.

Others.


Order of Analysis

In each case, we follow a four-step process.

Definition: How are payoffs determined?

"Purpose:" What do we obtain beyond vanilla options?

Valuation: In particular, are closed-forms available?

Hedging: The behavior of its delta and other greeks.

Common theme in this process:

Pricing involves no surprises (but may be computationally hard).

Behavior of the option delta and other greeks may depart in sharp

ways from vanilla options.


Digital Options


Binary/Digital Options

Any option with a discontinuous payoff pattern is called a binary or

digital option.

Canonical example: cash-or-nothing options. Here, the option holder

receives a flat payment $M if the option finishes in the money (with respect

to a specified strike), and nothing otherwise.

Many other kinds of digitals:

Asset-or-nothing option: Holder receives one unit of the underlying if

the option finishes in-the-money, nothing otherwise.

Embedded in structured products.

Why binary options? Straight bets on the market.


Examples: Structured Products with Embedded Binary

Options

4-year principal-protected equity-linked investment. Coupon received in

year k:

6.50%, if the returns on the S&P 500 and Eurostoxx 50 indices in

year k are both greater than 12%.

1.5%, if the returns on one (or both) of the indices is less than 12%.

24-month principal-protected note linked to commodity prices (oil, gas,

nickel, zinc, silver, corn). Annual coupon:

15%, if no commodity declines by more than 25%.

0%, otherwise.


Examples: Exchange-Traded Binary Options

The Chicago Board of Options Exchange (CBOE) offers binary options on the

S&P 500 index and on the VIX, a volatility index based on the S&P 500.

On the S&P 500 index (Option Ticker: BSZ):

On the VIX:

Ticker: BVZ

Payoffs at T : Similar to BSZ.


Binary Payoffs: Cash-or-Nothing Options


Binary Payoffs: Asset-or-Nothing Options


Pricing Cash-or-Nothing Options

Consider a binary call option that pays M whenever the stock price at

maturity satisfies ST > K.

In the Black-Scholes setting, the price of this option is just the discounted

value of M times the risk-neutral probability of the option finishing in-the-

money, i.e.,

C C-or-N = erT M x N (d2),

where, as in the Black-Scholes model,


Cash-or-Nothing Option Prices


Binary Greeks

Binary cash-or-nothing options offer the clearest illustration of how different

exotic greeks can be from the vanilla greeks.

Cash-or-nothing calls are "like" vanilla calls in that they pay off when the

stock price is high but pay nothing if it is low, but the behavior of the greeks

is wildly different.

The figures on the next several pages illustrate the cash-or-nothing

Delta (increases, then decreases).

Gamma (can be positive or negative)

Theta (can be positive or negative)

Vega (can be positive or negative)

Rho (can be positive or negative).


Cash-or-Nothing Delta


Cash-or-Nothing Gamma


Cash-or-Nothing Theta


Cash-or-Nothing Vega


Cash-or-Nothing Rho


Compound Options


Compound Options

A compound option is an option written on an option.

The strike price of the compound option is the price at which under lying

option can be purchased or sold.

This price is often called the "front fee" to distinguish it from the strike price

of the underlying option (which is referred to as the "back fee").

For notational purposes, we will use k and t to denote the strike and

maturity of the compound option, and K and T for the corresponding

parameters of the underlying option.

There are four basic kinds of compound options:

Call on call.

Call on put.

Put on call.

Put on put.


Why Compound Options?

Locking-in factor.

Suppose you are looking to buy a put for protection against falling prices.

If you find current put prices high and decide to wait, then if prices do

fall, the puts will also become more expensive.

Buying a call on a put today involves a lower current expense and allows

you to buy the put later if prices do actually decline.

Installment options.

In an installment option, the premium is made over several payments.

The buyer has the right to stop making premium payments and walk

away from the deal.

This is effectively a compound option in which the payment of each

installment gives you the right to proceed to the next installment.


Payoffs from Compound Options

In an important sense, compound options are like vanilla options on the

under lying but with complicated payoff structures.

To illustrate, consider a call-on-a-call.

Recall that k and t denote the strike and maturity respectively, of the

compound call, and K and T denote those of the underlying call.

At time t when the compound call matures, the underlying call has T t

years to maturity. Denote its value at this point by C (St, K, T t ).


Call-on-Call Payoffs

For the compound call to be in-the-money at maturity, we must have

C (St, K, T t ) k. In this case, the compound call payoff is

C (St, K, T t ) k.

Now, the value of the underlying call increases as St increases. Therefore,

there is some critical value S*t such that C (St, K, T t ) k if and only if

St St*.

This means the payoff of the compound call at time t is


Call-on-Call Payoffs


Put-on-Call Payoffs

Next, consider a compound put-on-call, again with strike k and maturity t.

For the compound put to finish in-the-money, the underlying call has to be

worth less than k, which means that St has to finish below St*.

If the compound put finishes in-the-money, its payoff is

k C (St, K, T t ).

Thus, the payoff of the compound put-on-call is


Put-on-Call Payoffs


Call-on-Put and Put-on-Put Payoffs

Analogously, we can derive the payoffs of a compound call-on-put and a

compound put-on-put.

If the strike of the compound option is k, then:

for the call-on-put to finish in-the-money, the underlying put must be

worth more than k.

for the put-on-put to finish in-the-money, the underlying put must be

worth less than k.

Since the value of the underlying put decreases as St increases, there is

a critical value St** such that the put is worth more than k if and only St <

St * .

From this, it is simple to derive the payoffs on the following pages.


Call-on-Put Payoffs


Put-on-Put Payoffs


Pricing Compound Options

The "true" driver of compound option values is the underlying asset. Thus,

to price compound option, we begin by modeling the price process of the

underlying and derive compound prices off that.

When the underlying price follows a lognormal diffusion as in Black-

Scholes, closed-form solutions for compound option prices are available.

However, these are messy expressions involving the bivariate normal

distribution, so we do not replicate them here.

Compound options can also be priced in a binomial setting.

For a simple illustration, consider the two-period put option in the

binomial example introduced earlier.

Consider a call on the put. Suppose the call has a maturity of one

period and a strike of k = 4. What is its initial value?


Compounds: A Binomial Example

At the end of one period, the value of the underlying put is

Thus, the payoff of the call-on-put at maturity is


Compounds: A Binomial Example

Using the risk-neutral probability of 0.60 of an up move, the initial value of

the compound call-on-put is, therefore,

Of course, we could have also priced the compound by replicating it with a

portfolio of units of the underlying and cash. The replicating portfolio is

A short position in the underlying of 0.202 units.

Investment of 21.78 for one period.


Hedging Compound Options

Since the "true" driver of the compound option values is the underlying, the

compound can be delta hedged by taking positions in the underlying.

The binomial example showed that the delta of a call-on-put was negative. In

the example, the call-on-put is equivalent to being short 0.202 units of the

underlying.

Why is it negative?

Analogously, the delta of a

Call-on-call and put-on-put is positive.

Put-on-call is negative.


Chooser Options


Chooser Options

A chooser option (or U-Choose option) is one where you buy the option

today, but you get to decide whether it is to be a call or a put at a later date.

Contract specifies three things:

Strike price K.

Choice date Tc.

Maturity date T.

Why use chooser options?

A chooser is a purchase of volatility today with a directional choice made

later.

Thus, a chooser is like a straddle, but cheaper because an irrevocable

directional commitment is made before maturity.


Decomposing Choosers

It can be shown using put-call parity that the chooser is identical to a

portfolio consisting of

A call with strike K and maturity T.

e (TTc ) puts with strike e(r)(TTc ) K and maturity Tc.

In particular, when = 0, the chooser is equivalent to

A call with strike K and maturity T.

A put with strike er (TTc ) K and maturity Tc.

Thus, closed-form solutions are easy to derive for choosers in a Black-

Scholes setting.


Choosers and Straddles Compared

A straddle consists of a call with strike K and maturity T, and a put with

strike K and maturity T.

Comparing the straddle with the chooser, we see that:

the calls in the two are identical, but

the put in the chooser has a lower strike and lower maturity; and

if > 0 there are also fewer puts in the chooser.

Thus, the decomposition of the chooser enables us to pinpoint exactly the

difference between a straddle and a chooser.


Chooser versus Straddle: Prices


Hedging Choosers

Hedging the chooser is simply a matter of hedging the implied call and put

in the chooser.

Thus, the chooser's greeks are simply the sum of the greeks of the call and

put that constitute the chooser.

So, for example:

At very low values of S, the delta of the chooser goes towards 1,

while at very high values of S, the delta goes towards +1.

The gamma of the chooser is always positive since the chooser is the

sum of a long call and long put, etc.


Chooser Deltas


Forward Starts


Forward-Start Options

A forward-start option is one that comes to life at a specified point T * in

the future and has a life of years (measured from T *).

The strike price of the forward-start is not specified at maturity but is

determined at T * as K = ST *, where ST * is the price of the under lying at

T *, and > 0 is a parameter specified in the contract.

The most popular choice is = 1, i.e., the forward start is at-the-money

when it comes to life.

Why forward starts? What do you lock-in today?


The Sprint Forward-Start

In the 1990s, a number of firms reacted to declines in their stock prices by

"repricing" existing employee stock options, i.e., by reducing the strike prices

on these options to make them at-the-money again.

Investor groups protested that such actions destroyed the incentives the

options were supposed to be providing.

FASB responded to investor complaints by making it expensive for companies

to lower the strike prices on existing options grants.

FASB's new rules in fact made it expensive for companies to cancel existing

stock option grants and replace them with new option grants with lower strike

prices within six months of the cancellation date.


The Sprint Forward-Start

So in November 2000, Sprint Corporation offered its employees a plan to

cancel their existing stock options and replace them with new options that

would be at-the-money when they came to life six months and one day

after the cancellation.

Essentially Sprint replaced its existing options with forward starts in which

= 1

T* = 6 months plus one day.

T = original maturity minus (6 months plus one day).


Pricing Forward Starts

Let a forward start with parameters T *, and be given.

The price of the forward start is just the price of e-qT * vanilla options with

strike price equal to times the current price of the underlying.

maturity equal to years.

In notational terms,

C FS (S; , T *, ) = e-qT* C Vanilla(S, S, ).

If q = 0:

C FS (S; , T *, ) = C Vanilla(S, S, ).


Pricing Forward Starts: An Example

Assume a Black-Scholes setting.

Suppose you hold a vanilla call option with strike K = 20 and with 5 years

left to maturity.

Suppose the current price of the stock is 14 and the stock volatility is

40%.

Finally, suppose that the risk-free interest rate is 4%.

Consider a Sprint-like offer: You may trade in your current option for a

forward start that

comes to life in 6 months;

will be at-the-money when it comes to life; and

will have a maturity of 4 years from that point.

Should you take the offer?


Pricing Forward Starts: An Example

Value of the option you currently hold:

C Black-Scholes (S = 14, K = 20, T t = 5) = 4.13.

Value of the forward-start: C FS(S = 14; T * = , = 1, = 4 ) =

C Black-Scholes (S = 14, K = 14, T t = 4 ) = 5.46.

So, yes, you should take the offer.


Vanilla Options: Homogeneity of Degree 1

Let C (S, K, T ) denote the price of a vanilla option on a stock given a current

stock price of S, a strike price of K, and a time-to-maturity of T years.

Then, for any m > 0, we have

C (mS, mK, T ) = m x C (S, K, T ).

This property is called "Homogeneity of Degree 1" in (S, K ).


Pricing Forward Starts

Value of the forward start at T : C (S T ,S T,).

By the homogeneity property,

C (S T, S T, ) = S

T X C (1, , ).

So initial value of forward start:

C FS (S, T , , ) = PV (S T)C (1, , ) = e T *SC (1, , )

Invoking homogeneity again (this time in reverse),

C FS (S, T , , ) = eT * C ( S, S, ).

In words, this says the price of a forward start is the same as the price of

eT* units of a vanilla option with the same characteristics as the forward

start, i.e., with

a maturity of years, and

strike price equal to times the current stock price.


Forward Start Pricing and Hedging

Greeks of the forward start?

Just the greeks of eTS C (1, , )!

In particular, the delta of a forward start is just eT* C (1, , ), a

constant, and the gamma of the forward start is zero.

The other greeks are given by eqT*S times the relevant greek for

the C (1, , ) option.


Exchange Options


Exchange Options

Exchange options (also known as spread options, out performance options,

or "Margrabe options" after the first person to investigate them) are options

where the investor has the right to exchange one asset for another at the

end of the contract.

If the prices of the two assets are denoted S1 and S2 respectively, then the

payoff at time T for an investor who has the right to exchange asset 2 for

asset 1 is given by

max{S1T S2T, 0}.

Exchange options are a natural generalization of vanilla options. In a

standard vanilla call, the right is to exchange cash worth K for the underlying

asset. Cash is just a specific kind of asset with no volatility and a yield equal

to the risk-free rate.


Pricing Exchange Options: Notation

Suppose that the prices of assets 1 and 2 both evolve according to

lognormal diffusions. Let

1 and 2 be the respective yields of the assets.

1 and 2 their respective volatilities,

be their correlation.

The Black-Scholes model corresponds to the special case of this set-up

where S2t = K, 2 = 0, and 2 = r.


The Pricing Formula

Margrabe (1978) shows that the price of the exchange option is the natural

generalization of the Black-Scholes formula:

C Exchange = e1T S1 N (d1) e 2T S2 N (d2),

where

The Black-Scholes formula corresponds to the special case where S2 K,

2=r, and 2=0.


Hedging Exchange Options

The replicating portfolio for an exchange option consists of positions in both

assets.

Thus, there are two deltas in an exchange option, one with respect to each

asset:

1 = e-1T N (d1) 2 = e

-2T N (d2)

These deltas identify the complete replicating portfolio since there is no

position in cash required (recall that the second asset plays the role of cash).

As S1 increases relative to S2, both deltas move towards 1 (in absolute

value); as S1 declines relative to S2, both move towards zero.


Quantos


Quanto Options

A quanto option (also sometimes called a "wrong-currency option") is an

option in which the underlying asset trades in one currency but the payoffs

to the option holder are converted into another currency at a fixed pre-

specified exchange rate.

As motivation, think of a UK-based investor who wishes to buy a call option

on a US company whose shares trade in NY in USD.

If the investor buys the option in the stock's local currency (USD) and

converts any payoffs at maturity back into GBP, there is currency risk

in the transaction in addition to the stock price risk.

In a quanto, this risk is eliminated by converting the gains back to GBP

at a fixed rate.


Payoffs from a Quanto

Use a superscript f to denote amounts in the foreign currency; no

superscript means the domestic currency.

Suppose the quanto is a call with strike Kf. Then, the payoff at maturity of

the option, measured in the foreign currency is

max{ SfT Kf, 0}.

Let denote the fixed exchange rate (units of domestic currency per unit

of foreign currency) at which this is converted back into the domestic

currency.

The payoff at maturity received by the holder of the quanto call is then

x max{ SfT Kf, 0}.


An Example: The NYSE Arca Japan Index Option

Index value: computed using the yen prices of 210 stocks trading in Tokyo.

At maturity, the holder of the option receives $100 times the depth-in-the-

money of the option.

For example, if the strike price is 120, and the index level at maturity is 128,

the holder of a call receives

100 x max(128 120,0) = 800.

As the example shows, the rate need have no relationship to the actual

exchange rate.


Quantos as Exchange Options

Let Xt be the inverse of the time-t exchange rate, i.e., the no. of units of

the foreign currency per unit of the domestic currency.

Convert the payoffs of the quanto back into the foreign currency:

XT x max (S fT K

f, 0).

Rewrite this expression as

max (XT S fT XT K

f, 0),

Define AT = XT S fT, BT = XT K

f. Then, this is the same thing as

max (AT BT, 0).

This is just the payoff from units of an option to exchange BT for AT.


Pricing Quantos

We can identify the properties of AT and BT from the properties of XT and

SfT.

For example, if both XT and SfT are lognormal, so is their product AT.

Using this and Margrabe's formula, we can price the quanto.

Of course, this is the current price of the quanto in the foreign currency

since the payoffs were converted to that currency when we multiplied

through by XT.

To obtain the current price in the domestic currency, we divide through by

the current value of Xt.


Other PI Options


Other Path-Independent Options

A vast variety of other path-independent options exists. Some examples

are:

Maximum of Two-Assets: Payoff = max(S1T, S2T).

Minimum of Two Assets: Payoff = min(S1T, S2T).

Options on the Max: Payoff = max{0, max(S1T, S2T) K}.

And so on ...

In general, since path-independent options' payoffs only depend on the

distribution of ST, they are relatively easy to price.

This does not necessarily mean that hedging them in practice is easy (think

of the pin risk in delta-hedging a digital option that is at-the-money close

to maturity).

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