Barriers - Advanced Options

16
Barriers: Advanced Options
The last chapter laid out the principles of European barrier option pricing. This chapter con-
tinues the same analysis, applied to more complicated problems. The integrals get a bit larger,
but the underlying concepts remain the same. Lack of space prevents each solution being given
explicitly; but the reader should by now be able to specify the integrals corresponding to each
problem, and then solve them using the results of Appendix A.1.
16.1 TWO BARRIER OPTIONS
These are options which knock out or in when either a barrier above or a barrier below the
starting stock price is crossed. The analysis is completely parallel to what we have seen for a
single barrier option (Ikeda and Kunitomo, 1992).
(i) In the notation of this chapter, F
0
(x
T
, T ) is the normal distribution function for a particle
starting at x
0
= 0. The function is given explicitly in Section 15.1(i). F
non-abs
is the probabil-
ity distribution function for particles starting at x
0
= 0 which have not crossed either barrier
before time T. Two expressions have been derived for this function, which are given by equa-
tions (A8.9) and (A8.10) of the Appendix. They are both infinite series although there is no
correspondence between individual terms of the two series:
1. F
non-abs
(x
T

, T ) =
1
σ
√
2π T
+∞

n=−∞

exp

+
mu
n
σ
2

exp

−
1
2σ
2
T
(x
T
− mT − u
n
)
2


− exp

+
mv
n
σ
2

exp

−
1
2σ
2
T
(x
T
− mT − v
n
)
2

2. F
non-abs
(x
T
, T ) = exp

mx

T
σ
2

∞

n=1

a
n
e
−b
n
T
sin
nπ
L
(x
T
+ b)

L = a + b; u
n
= 2Ln; v
n
= 2(Ln − b)
a
n
=
2

L
sin
nπ b
L
; b
n
=
1
2

µ
σ

2
+

nπσ
L

2

(ii) The reasoning of Appendix A.8(iv) and (v) demonstrates that the distribution functions of
particles which start at x
0
= 0, then cross either the barrier at −bor+a, and then return to the
region −b to +a can be written
F
return
= F
0

− F
non-abs
16 Barriers: Advanced Options
-b
X
a
return
F
0
F
0
F
0
S
Figure 16.1 Double barrier-and-in call
The total probability distribution function for all
those particles that cross one of the barriers can now
be written
F
crossers
=





F
0
(x
T

, T ) x
T
< −b
F
return
(x
T
, T ) −b < x
T
< a
F
0
(x
T
, T ) a < x
T
(iii) As an example, we will look at the knock-in call op-
tion shown in Figure 16.1. We do not really need to
worry about whether we move up or down to the barrier:
C
ki
= e
−rT

+∞
0
(S
T
− X )
+

F
crossers
dS
T
= e
−rT

+∞
X
(S
T
− X )F
crossers
dS
T
= e
−rT

a
X
(S
T
− X )F
return
dS
T
+ e
−rT

∞

a
(S
T
− X )F
0
dS
T
(16.1)
The second integral is completely standard. The first depends on which form of series is used
for F
non-abs
. The sine series is completely straightforward to integrate while the other alternative
is handled using the procedures of Section 15.1.
(iv) The question of which of the two series to use and how many terms to retain is best handled
pragmatically. Set up both series and see how fast convergence takes place in each case. Both
series dampen off regularly, so it is for us to choose how accurate the answer needs to be. We
should expect to perform the calculation with one series or the other within four to six terms,
and often less.
16.2 OUTSIDE BARRIER OPTIONS
The barrier options described so far have been European options which are knocked in or
knocked out when the price of the underlying variable crosses a barrier. An extension of this is
a European option which knocks in when the price of a commodity other than the underlying
stock crosses a barrier. For example, an up-and-in call on a stock which knocks in when a foreign
exchange rate crosses a barrier. These options are called outside barrier options, as distinct
from inside barrier options, where the barrier commodity and the commodity underlying the
European option are the same. The reason for the terminology is anybody’s guess (Heynen
and Kat, 1994a).
We could repeat most of the material presented so far in this chapter, adapted for outside
barriers rather than inside barriers. However, these options are relatively rare so we will simply
describe a single-outside-barrier up-and-in call option; the reader should be able to generalize

this quite easily to any of the other options in this category.
(i) Outside Barrier, Up and In: The general principle remains as before; it is merely the form of
some of the distributions that is different. The price of the option is the present value of the
risk-neutral expectation of the payoff (Figure 16.2):
C
outside
u−i
= e
−rT

+∞
S
T
=0

+∞
Q
T
=0
(S
T
− X )
+
F
jo int
dS
T
dQ
T
(16.2)

190
16.2 OUTSIDE BARRIER OPTIONS
0
S
X
Q
K
return
F
0
F
0
Figure 16.2 Outside barrier, up-and-in call
where S
T
is the maturity value of the stock un-
derlying the call option and Q
T
is the maturity
value of the barrier commodity. The form of this
is the same as for inside barriers; but we need
to find an expression for F
jo int
which is the joint
probability distribution for the two price vari-
ables. The large topic of derivatives which de-
pend on the prices of two underlying assets is at-
tacked in Chapter 12. The material of that chapter
and of Appendix A.1 is used to solve equation
(16.2).

(ii) Separate Distributions of the Two Variables: As
before, we transform to the logs of prices: x
T
=
ln(S
T
S
0
); y
T
= ln(Q
T
/Q
0
). The distribution of x
T
is normal and the variable z
T
= [ln(S
T
/S
0
)− mT]/
σ
√
T is a standard normal variate (mean 0, variance 1); σ is the volatility of the stock and
m − r − q −
1
2
σ

2
.
The variate y
T
has a more complex distribution. As explained in Section 13.1(i), y
T
is
distributed as F
crossers
(y
T
, T ) which has different forms above and below the barrier at Q
T
= K
or y
T
= ln(K /Q
0
) = b.
b < y
T
: F
crossers
= F
0
(y
T
, T ) which is the distribution at time T of a particle which started at
y
0

= 0 and has drift m
Q
= r − q
Q
−
1
2
σ
2
Q
and variance σ
2
Q
. The variable
w
T
=
ln(Q
T
/Q
0
) − m
Q
T
σ
Q
√
T
is a standard normal variate.
y

T
< b: F
crossers
= F
return
= AF
0
(y
T
− 2b, T ) where A = exp(2m
Q
b/σ
2
Q
) = (K /Q
0
)
2m
Q
/σ
2
Q
and F
0
(y
T
− 2b, T ) is the distribution function for a particle which started at y
0
= 2b
and has drift m

Q
. The variable
w

T
=
ln(Q
T
/Q
0
) − m
Q
T − 2b
σ
Q
√
T
is therefore a standard normal variate.
(iii) Equation (16.2) may be rewritten
C
outside
u−i
= e
−rT


+∞
S
T
=X


K
Q
T
=0
A(S
T
− X )F
1joint
dQ
T
dS
T
+

+∞
S
T
=X

+∞
Q
T
=K
(S
T
− X )F
2joint
dQ
T

dS
T

and transforming to the variables Z
T
, w
T
and w

T
, this last equation can be written more
191
16 Barriers: Advanced Options
precisely as
C
outside
u−i
= e
−rT

A

+∞
Z
X

W

K
−∞

(S
0
e
mT+σ
√
Tz
T
− X )n
2
(z
T
,w

T
; ρ)dz
T
dw

T
+

+∞
Z
X

+∞
W
K
(S
0

e
mT+σ
√
Tz
T
− X )n
2
(z
T
,w
T
; ρ)dz
T
dw
T

Z
X
=
ln(X/S
0
) − mT
σ
√
T
; W

K
=
ln(K /Q

0
) − m
Q
T − 2b
σ
Q
√
T
; W
K
=
ln(K /Q
0
) − m
Q
T
σ
Q
√
T
n
2
(z
T
,w

T
; ρ) is the standard bivariate normal distribution describing the joint distribution of
the two standard normal variates z
t

and w

t
, which have correlation ρ. n
2
(z
T
,w
T
; ρ)isthe
standard bivariate normal distribution describing the joint distribution of the two standard
normal variates z
t
and w
t
, which have correlation ρ.
Note that the correlations between z
t
and w

t
are the same as between z
t
and w
t
; w

T
and w
T

essentially refer to the same random variable Q
T
, and differ only in their means, which does
not affect the correlations.
Using the results of equations (A1.20) and (A1.21), this last integral is evaluated as follows:
C
outside
u−i
= A

S
0
e
−qT
N[(σ
√
T − Z
X
)] − X e
−rT
N[−Z
X
]
−

S
0
e
−qT
N

2
[−(σ
√
T − Z
X
),−(ρσ
√
T − W

X
); ρ]−X e
−rT
N
2
[−Z
X
,−W

K
; ρ]

+

S
0
e
−qT
N
2
[−(σ

√
T − Z
X
),−(ρσ
√
T − W

X
); ρ] − X e
−rT
N
2
[−Z
X
,−W
K
; ρ]

(16.3)
16.3 PARTIAL BARRIER OPTIONS
In the foregoing it was always assumed that a barrier is permanent. However, the barrier could
be switched on and off throughout the life of the option. Such a pricing problem is usually
handled numerically, but the simplest case can be solved analytically using the techniques of
the last section (Heynen and Kat, 1994b).
This is an option on a single underlying stock at two different times, as described in
Chapter 14. The specific case we consider is an up-and-in call of maturity T, which knocks in
if the barrier is crossed before time τ , i.e. the barrier is switched off at time τ . Its value can be
written analytically as
C
partial

u−i
= e
−rT


+∞
S
T
=0

+∞
S
τ
=0
(S
T
− X )
+
F
jo int
dS
τ
dS
T
F
jo int
is the joint probability distribution of two random variables S
τ
and S
T

, where S
τ
is subject
to an absorbing barrier. This problem is almost precisely the same as the outside barrier option
problem solved in the last section. The formula given in equation (16.3) can therefore be
applied directly, with the following modifications:
r
Q
0
→ S
0
, σ
Q
→ σ and m
Q
→ m.
r
T → τ in the formulas for w

K
and w
K
.
r
The correlation between S
τ
and S
T
is shown in Appendix A.1(vi) to be ρ =
√

τ/T .
192
16.4 LOOKBACK OPTIONS
16.4 LOOKBACK OPTIONS
These are probably the most discussed and least used of the standard exotic options. The
problem is that on the one hand they have immense intuitive appeal and pricing presents some
interesting intellectual challenges; but on the other hand they are so expensive that no-one
wants to buy them. However, this book would not be complete without an explanation of how
to price them (Goldman et al., 1979).
0
S
max
S
min
S
T
X
T
S
Figure 16.3 Notation for lookbacks
(i) Floating Strike Lookbacks: Lookback options are quoted in two ways. The most common
way is with a floating strike, where the payoffs are defined as follows:
Payoff of C
fl str
= (S
T
− S
min
)
Payoff of P

fl str
= (S
max
− S
T
)
The lookback call gives the holder the right to buy stock at maturity at the lowest price achieved
by the stock over the life of the option. Similarly, the lookback put allows the holder to sell
stock at the highest price achieved.
The form of the payoff is unusual in that it does not involve an expression of the form
max[0,...], since (S
T
− S
min
) can never be negative; it has therefore been suggested that this
is not really an option at all, although this is largely a matter of semantics. However, it does
make the pricing formula straightforward to write out: risk neutrality gives
C
fl str
= e
−rT
{ES
T
−ES
min
}
= e
−rT
{F
0T

− v
min
} (16.4)
P
fl str
= e
−rT
{v
max
− F
0T
}
where F
0T
is the forward price.
(ii) Fixed Strike Lookbacks: As the name implies, these options have a fixed strike X. Referring
to Figure 16.3, the payoffs of the fixed strike call and put are given by
Payoff of C
fix str
= max[0, S
max
− X ]
Payoff of P
fix str
= max[0, X − S
min
]
These are sometimes referred to as lookforward options. They give the option holder the right
to exercise not at the final stock price, but at the most advantageous price over the life of
the option. The payoffs look more like normal option payoffs, containg the familiar “max”

function. However, in practice, the payoff can be further simplified, since the options are usually
193