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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.
Rangarajan K. Sundaram Derivatives: Principles & Practice 2
Outline
What are "Exotic" Options? Path-Independent Options Digital Options Compound Options Chooser Options Forward Starts Exchange Options Quantos Other PI Options
Rangarajan K. Sundaram Derivatives: Principles & Practice 3
Introduction
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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, ...
Rangarajan K. Sundaram Derivatives: Principles & Practice 5
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)
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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.
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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
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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).
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Behavior of Vanilla Option Prices
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Behavior of Vanilla Option Deltas
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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:
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Binomial Example: Stock Price Evolution
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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).
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European Call Prices
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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.
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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.
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Digital Options
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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.
Rangarajan K. Sundaram Derivatives: Principles & Practice 19
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.
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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.
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Binary Payoffs: Cash-or-Nothing Options
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Binary Payoffs: Asset-or-Nothing Options
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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,
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Cash-or-Nothing Option Prices
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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).
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Cash-or-Nothing Delta
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Cash-or-Nothing Gamma
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Cash-or-Nothing Theta
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Cash-or-Nothing Vega
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Cash-or-Nothing Rho
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Compound Options
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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.
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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.
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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 ).
Rangarajan K. Sundaram Derivatives: Principles & Practice 35
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
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Call-on-Call Payoffs
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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
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Put-on-Call Payoffs
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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.
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Call-on-Put Payoffs
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Put-on-Put Payoffs
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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?
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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
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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.
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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.
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Chooser Options
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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.
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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.
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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.
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Chooser versus Straddle: Prices
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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.
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Chooser Deltas
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Forward Starts
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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?
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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.
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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).
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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, ).
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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?
Rangarajan K. Sundaram Derivatives: Principles & Practice 59
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.
Rangarajan K. Sundaram Derivatives: Principles & Practice 60
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 ).
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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.
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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.
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Exchange Options
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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.
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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.
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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.
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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.
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Quantos
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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.
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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}.
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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.
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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.
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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.
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Other PI Options
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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).