Simple Exotics

11
Simple Exotics
The purpose of this part of the book is to introduce the reader to the most important types
of equity derivatives and to illustrate the pricing techniques which have been introduced in
the last two parts. Exotic options can mostly be priced using classical statistical techniques,
although we will see in Part 4 of this book that some of the analysis can be simplified (or at least
rendered more elegant) using stochastic calculus.
There is no firm definition of an exotic option and we usually take it to mean anything that
is not a simple European or American put or call option. We start with a chapter on simple
extensions of the Black Scholes methodology, which should really be understood by anyone
involved with options, whether or not they have a specific interest in exotics.
11.1 FORWARD START OPTIONS
(i) Suppose we buy an option with maturity T which only starts running at time τ . If the strike
price is set now, pricing becomes fairly trivial: the price of a European option depends only on
the final stock price and the strike price so there is no difference whatsoever between “starting
now” and “starting at time τ ”; with an American option we must take into account the fact
that we cannot exercise the option between now and time τ , but this is easily accommodated
within a tree or a finite difference scheme. But the type of options considered here are those
that are at-the-money or 20% out-of-the-money at some future starting date.
(ii) Homogeneous Functions: This is an important mathematical property of option prices which
we use freely in the following chapters. The concept is so intuitive that most people use it
instinctively without placing a name to it.
If the reader is already working within a derivatives environment, it is quite likely that
his option model has the initial stock price preset to 100; this yields option prices directly
as a percentage of the stock price. We know that the initial stock price will not be 100 but
we also know that things move proportionately: if an option is priced at 5.5 on our preset
model with S
0
= 100 and strike price X = 120, we know immediately that the price of a
similar option with S
0

= 40 and X = 48 would be 2.2. It is immediately apparent to most
people that the strike price has to move in line with the stock price for this reasoning to work,
just as it is apparent to most that we should not change the time to maturity or the interest
rate.
An equally obvious conclusion is reached concerning the number of shares on which an
option is written. Suppose we own a call option on one share and the company suddenly
declares a 2 for 1 stock split. We know that the share price would fall in half, but we
would be kept whole if our call option were replaced by two options of the same matu-
rity, each on one of the new shares, and with the strike price equal to half the original strike
price.
11 Simple Exotics
Let f (nS
0
, nX) be the value of an option on n shares, where S
0
is the stock price and X is
the strike. The homogeneity condition just described may be written
f (nS
0
, nX) = S
0
f

n, n
X
S
0

= Xf


n
S
0
X
, n

= nf(S
0
, X) = nS
0
f

1,
X
S
0

(11.1)
It should be pointed out that this property holds true for most options we encounter, although
sometimes with modification: for example, barrier options are homogeneous in spot price,
strike price and barrier value. However there are exceptions such as power options which are
described later in this chapter.
(iii) Forward Start with Fixed Number of Shares: Consider an option starting in some time τ in
the future, maturing in time T and with a strike price equal to a predetermined percentage α of
the starting stock price. Using the homogeneity property, the value of this option in time τ is
f (S
τ
,αS
τ
, T − τ ) = S

τ
f (1,α,T − τ )
The term f (1,α,T − τ ) is non-stochastic and may be calculated immediately. If we buy
f (1,α,T − τ ) units of stock today at a cost of S
0
f (1,α,T − τ ), the value of this stock in
time τ will be S
τ
f (1,α,T − τ ); but this is the same as the future value of the forward starting
option. Today’s value of the forward starting option must therefore be
S
0
f (1,α,T − τ ) = f (S
0
,αS
0
, T − τ )
i.e. the valueof the forward starting option is the same as if the option started running today, with
the time to maturity set equal to the length of time between the start and maturity (Rubinstein,
1991c).
A further refinement is needed if the stock pays a dividend. Remember that if we hold a
packet of stock from now to time τ , we will receive a dividend but if we hold an option, we
will not. The adjustment to the formula can be made by the usual substitution S
τ
→ S
τ
e
−qτ
for continuous dividends to give
f

forward start
= e
−qτ
f (S
0
,αS
0
, T − τ )
(iv) Forward Start with Fixed Value of Shares: The last subparagraph dealt with a forward starting
option on a fixed number of shares. But suppose we were asked to price a forward starting
option on $1000 of shares. The value of this option in time τ will be
f (n
τ
S
τ
,αn
τ
S
τ
, T − τ ) = n
τ
S
τ
f (1,α,T − τ )
where n
τ
S
τ
= $1000 in our example. Therefore, the value of this option in time τ is completely
determinate (non-stochastic); today’s value is simply obtained by present valuing this sum:

f
forward start
= e
−rτ
f (S
0
,αS
0
, T − τ)
(v) The contrasting results of the last two subsections are well illustrated in the foreign exchange
market:
r
For an option to buy £1 for dollars, forward start means discounting back by the sterling
interest rate.
r
For an option to buy sterling for $1, forward start means discounting back by the dollar
interest rate.
146
11.2 CHOOSERS
(vi) Cliquets (Ratchets): As the name implies, this type of option was first used widely in France.
It is designed for an investor who likes the basic idea of a call option but is concerned that
the stock price might spend most of its time above the strike price, only to plunge just before
maturity. In such a case, the cliquet would capture the effect of the early price rise. It is really a
series of forward starting options strung together. The option has a final maturity T (typically
1 year) and a number of re-set dates τ
1
,τ
2
, ··· (typically quarterly). The payoff and re-set
sequence is as follows:

r
At τ
1
, the option pays max[0, S
τ
1
− S
0
].
r
At τ
2
, the option pays max[0, S
τ
2
− S
τ
1
], etc.
Clearly, each of these is the payoff of an at-the-money forward starting call option. The fair
value is therefore given by
f
cliquet
= C(S
0
, S
0
,τ
1
) + e

−qτ
1
C(S
0
, S
0
,τ
2
− τ
1
) +···+e
−qτ
n−1
C(S
0
, S
0
,τ
n
− τ
n−1
)
Common variations on the structure have the effective strikes slightly out-of-the-money, or
have the payouts rolled into a single payment at final maturity.
11.2 CHOOSERS
(i) The 1990 Kuwait invasion led to a jump in the price of crude oil. Speculators were then faced
with a dilemma: if a withdrawal were negotiated, the oil price would fall back; but a declaration
of war by the US would lead to a further jump upwards. A ready-made strategy for this situation
is the straddle, consisting of both an at-the-money put and an at-the-money call. This has a
positive payoff whichever way the oil price moves; but it has the great drawback of being very

expensive.
(ii) The Simple Chooser: This option has a strike X and a final maturity T. The owner of the option
has until time τ to declare whether he wants the option to be either a call option or a put option.
The chooser is sold as an option which has the benefits of a straddle, but at a much lower cost.
Clearly, at the limit τ = T the option becomes a straddle while at the limit τ = 0 it becomes
a put or call option.
The pricing of this option is surprisingly easy (Rubinstein, 1991b): at time τ , the holder of
the option will choose put or call depending on which is more valuable. The payoff at time τ
can therefore be written
Payoff
τ
= max[P(S
τ
, X, T − τ ), C(S
τ
, X, T − τ )]
Using the put–call parity relationship of Section 2.2(i) gives
Payoff
τ
= max

C(S
τ
, X, T − τ ) + X e
−r(T −τ )
− S
τ
e
−q(T −τ)
, C(S

τ
, X, T − τ )

= C(S
τ
, X, T − τ ) + e
−q(T −τ)
max

X e
−(r−q)(T −τ )
− S
τ
, 0

Taking these two terms separately, the instrument which has a value C(S
τ
, X, T − τ ) in time
τ when the stock price is S
τ
is obviously a call option maturing in time T; its value today is
C(S
0
, X, T ).
147
11 Simple Exotics
The form of the second term in the payoff is that of a put option maturing in time τ . Its value
today may be written e
−q(T −τ)
P(S

0
, X e
−(r−q)(T −τ )
,τ). Putting these together gives
f
simple chooser
= C(S
0
, X, T ) + e
−q(T −τ)
P

S
0
, X e
−(r−q)(T −τ )
,τ

(iii) Complex Chooser: The concept of the chooser can be very simply extended so that the put
and call options have different strike prices and maturities. Unfortunately, the mathematics of
the pricing does not extend so simply and we therefore defer this until Section 14.2.
11.3 SHOUT OPTIONS
(i) Like cliquets, these options are for investors who think that the underlying stock price might
peak at some time before maturity. The shout option is usually a call option, but with a
difference: at any time τ before maturity, the holder may “shout”. The effect of this is that he
is guaranteed a minimum payoff of S
τ
− X , although he will get the payoff of the call option
if this is greater than the minimum.
(ii) Payoffs: By definition, the final payoff of the option is max[0, S

τ
− X, S
T
− X ]. In practice,
S
τ
− X is always greater than zero; if not, we would have S
τ
< X which means that the holder
of the option had shouted at a time when the effect was to turn the shout option into a simple
European call option, for no economic benefit in exchange. The payoff at time T can therefore
be written
max[S
τ
− X, S
T
− X ] = S
τ
− X + max[0, S
T
− S
τ
]
At time τ if a shout is made, the value of this payoff is
e
−r(T −τ )
(S
τ
− X ) + C(S
τ

, S
τ
, T − τ)
(iii) Shout Pricing: This option is easily priced using a binomial model as we would for any
American option (Thomas, 1993). The final nodes in the tree are max[0, S
T
− X ] as they
would be for a call option, i.e. if we get as far as the final nodes, it means that no shout took
place.
At each node before the final column, the holder has the choice of shouting or not shouting. To
decide which, compare the value obtained by discounting back the values of the two subsequent
nodes with the time τ payoff produced by shouting, i.e. e
−r(T −τ )
(S
τ
− X ) + C(S
τ
, S
τ
, T − τ);
we enter whichever value is greater at that node.
In spirit this is the same as the binomial method for pricing American options which was
explained in Chapter 7; in that case we rolled back through the tree and at each node we selected
the greater of the payoff value or the calculated discounted average. The present procedure
calls for the Black Scholes value of a call option to be calculated at each node. However, with
the assumption of constant volatility, the Black Scholes formula only needs to be calculated
once for each time step. The homogeneity property described in Section 11.1(ii) says that the
price of an at-the-money option is proportional to the stock price, so that for an entire column
of nodes we only need to calculate the constant of proportionality C(1, 1, T − τ ) once.
148

11.4 BINARY (DIGITAL) OPTIONS
(iv) Put Shout: A precisely analogous put option with shout feature can be constructed. A gener-
alized payoff at time T can be written as
max[φ(S
τ
− X ),φ(S
T
− X )] = φ(S
τ
− X ) + max[0,φ(S
T
− S
τ
)]
where

φ =+1 for a call
φ =−1 for a put
At each node in the tree, a shouted value would be
e
−r(T −τ)
φ(S
τ
− X ) + Option(S
τ
, S
τ
, T − τ )
where the option is either a put or a call option.
(v) Strike Shout: The shout options described above locked in a minimum payout. Another version

of this type of option locks in a new strike price when the shout is made, and is even simpler
to price than the previous ones.
The payoff at time T for a calll option with strike shout is
max[0, S
T
− X, S
T
− S
τ
] = max[0, S
T
− S
τ
]
This is the same as in subsection (ii) above, but without the minimum payout. The rest of the
analysis is as before.
11.4 BINARY (DIGITAL) OPTIONS
(i) Recall the simple derivation of the Black Scholes formula which was given in Section 5.2. In
its simplest form, this may be written
C(S
0
, X, T ) =

∞
0
F(S
T
) max[0, S
T
− X ]dS

T
=

∞
X
F(S
T
)(S
T
− X )dS
T
= S
0
e
−qT
N[d
1
] − X e
−rT
N[d
2
]
where F(S
T
) is the (lognormal) probability distribution of S
T
. The two terms in this equa-
tion will now be interpreted separately, rather than together as they were before (Reiner and
Rubinstein, 1991b).
Cash

f
X
P
T
S
Figure 11.1 Cash or nothing option
(ii)
Cash or Nothing Option (Bet): In the term

∞
X
F(S
T
)X dS
T
, the factor X appears in two un-
related roles: as a constant multiplicative factor in
the integrand and again as the lower limit of inte-
gration. The first role is trivial and we will drop
X from the integrand. Then e
−rT

∞
X
F(S
T
)dS
T
=
e

−rT
N[d
2
] is the present value of the risk-neutral
probability that X < S
T
, and can be interpreted as
the arbitrage-free value of an option which pays out
$1 if S
T
is above X, and $0 otherwise (Figure 11.1).
This option is essentially a bet: “I will give you $1
if the stock price is over $100 in 6 months”. Its value is given by the second term in the Black
Scholes formula.
149