10.4 Barrier Options
155
to add and subtract terms proportional to 2B − 2x in order to be able to express the
quadratic term in the form discussed below. For the sum of the terms in the exponent,
we have
(σ
2
/2 + r f )
2
τ
2σ 2
−
σ
2
/2 − r f
σ 2
(x − x
) +
(2B − x − x
)
2
2σ 2 τ
(10.38)
=
2B − x − x
+ (r f − σ
2
/2)τ
2
2σ 2 τ
+ r f τ + 2
r f − σ
2
/2
σ 2
(x − B) .
Collecting the two terms and inserting in (10.36) the pricing kernel p DO (x, τ ; x
)
now reads
p DO (x, τ ; x
) =
1
√
2πσ 2 τ
exp
−
(x − x
+ τ (r f − σ
2
/2))
2
2σ 2 τ
− r f τ
−
1
√
2πσ 2 τ
exp
−
(2B − x − x
+ τ (r f − σ
2
/2))
2
2σ 2 τ
− r f τ
× exp
−2
r f − σ
2
/2
σ 2
(x − B)
.
(10.39)
Comparison with (10.22) reveals that the first term is equal to the plain Black-Scholes
pricing kernel p BS (x, τ ; x
) and the second term is given by p BS (2B − x, τ ; x
) with
an additional forefactor
exp
−2
r f − σ
2
/2
σ 2
(x − B)
=
e
x
e B
−
2(r f −σ 2 /2)
σ 2
(10.40)
Thus the pricing kernel for our barrier option can finally be written as
p DO (x, τ ; x
) = p BS (x, τ ; x
) −
e
x
e B
−
2(r f −σ 2 /2)
σ 2
p BS (2B − x, τ ; x
) (10.41)
with p BS (x, τ ; x
) given by (10.22). All prices for dropout options with any given
payoff function g(x) can now be calculated by convoluting the payoff with the pricing
kernel from (10.41). All information about the dynamics of the process, such as r f and
σ , but also what happens along the way—the absorbing boundary—is ecapsulated in
p DO (x, τ, x
). That’s why different dropout options use the same kernel, but depend
on the specific payoff function. Before moving on to path integrals, let us briefly
reflect on the way that the dropout kernel p DO is assembled from two Black-Scholes
kernels p BS .
The difference of the two Black-Scholes kernels in (10.41) is constructed in such
a way that p DO (x, τ, x
) = 0 at x = B. This follows the same spirit as placing
image charges in electrostatic problems in order to satisfy the boundary conditions
on, for example, a conducting plane, as shown on the left-hand side in Fig. 10.1.
155
to add and subtract terms proportional to 2B − 2x in order to be able to express the
quadratic term in the form discussed below. For the sum of the terms in the exponent,
we have
(σ
2
/2 + r f )
2
τ
2σ 2
−
σ
2
/2 − r f
σ 2
(x − x
) +
(2B − x − x
)
2
2σ 2 τ
(10.38)
=
2B − x − x
+ (r f − σ
2
/2)τ
2
2σ 2 τ
+ r f τ + 2
r f − σ
2
/2
σ 2
(x − B) .
Collecting the two terms and inserting in (10.36) the pricing kernel p DO (x, τ ; x
)
now reads
p DO (x, τ ; x
) =
1
√
2πσ 2 τ
exp
−
(x − x
+ τ (r f − σ
2
/2))
2
2σ 2 τ
− r f τ
−
1
√
2πσ 2 τ
exp
−
(2B − x − x
+ τ (r f − σ
2
/2))
2
2σ 2 τ
− r f τ
× exp
−2
r f − σ
2
/2
σ 2
(x − B)
.
(10.39)
Comparison with (10.22) reveals that the first term is equal to the plain Black-Scholes
pricing kernel p BS (x, τ ; x
) and the second term is given by p BS (2B − x, τ ; x
) with
an additional forefactor
exp
−2
r f − σ
2
/2
σ 2
(x − B)
=
e
x
e B
−
2(r f −σ 2 /2)
σ 2
(10.40)
Thus the pricing kernel for our barrier option can finally be written as
p DO (x, τ ; x
) = p BS (x, τ ; x
) −
e
x
e B
−
2(r f −σ 2 /2)
σ 2
p BS (2B − x, τ ; x
) (10.41)
with p BS (x, τ ; x
) given by (10.22). All prices for dropout options with any given
payoff function g(x) can now be calculated by convoluting the payoff with the pricing
kernel from (10.41). All information about the dynamics of the process, such as r f and
σ , but also what happens along the way—the absorbing boundary—is ecapsulated in
p DO (x, τ, x
). That’s why different dropout options use the same kernel, but depend
on the specific payoff function. Before moving on to path integrals, let us briefly
reflect on the way that the dropout kernel p DO is assembled from two Black-Scholes
kernels p BS .
The difference of the two Black-Scholes kernels in (10.41) is constructed in such
a way that p DO (x, τ, x
) = 0 at x = B. This follows the same spirit as placing
image charges in electrostatic problems in order to satisfy the boundary conditions
on, for example, a conducting plane, as shown on the left-hand side in Fig. 10.1.