10.3 Pricing Kernel
151
p BS (x, τ ; x
) =
e
−r f τ
√
2πσ 2 τ
exp
−
x − x
+ (r f − σ
2
/2)τ
2
2σ 2 τ
.
(10.22)
It can be shown that the pricing kernel actually equals the Green’s function from
(5.11), where we have to keep in mind that some of the variables are named differently.
For clarity we repeat the previously-made statement that the pricing kernel acts like
a propagator in quantum mechanics, which gives the transition probability between
initial and final states. This gives a rather intuitive picture of how options are priced.
Since the kernel embeds all the dynamics of the stock market fluctuations, it enables
us to calculate the value of any option with any given payoff function g(x) by simply
convoluting with the pricing kernel.
Instead of repeating the calculations from Sect. 5.2 to determine the pricing formulae for previously calculated options, we will use the new formalism to calculate
the pricing of barrier options, which is difficult using the methods from Sect. 5.2.
10.4 Barrier Options
Here we consider the down-and-out barrier option, which becomes void, once the
stock price passes a lower limit S DO /S 0 = e
−B
. We incorporate this property in the
Hamiltonian by introducing a potential V (x) that is infinite for x ≤ B and thus forces
the wave function to become zero in that region. The lower limit thus acts like barrier
for the wave function, hence the name. The Hamiltonian is then given by
H DO = H BS + V (x) = −
σ
2
2
∂
2
∂ x 2 +
σ
2
2
− r f
∂
∂ x
+ r f + V (x) ,
(10.23)
where V (x) = ∞ for x ≤ B and V (x) = 0 for x > B.
In order to calculate the pricing kernel x|e
−τ H DO |x
, we need to find the eigenfunctions and eigenvalues for H DO . We already know that they are zero in the region
x ≤ B. Conversely, in the region x > B, where the potential is zero, we can assume
that the eigenfunctions resemble those of the unperturbed Black-Scholes Hamiltonian H BS . Moreover, they must vanish at the boundary x = B. We therefore write
the eigenfunctions ψ E (x) with eigenvalue E in the following form
x|E = ψ E (x) = e
(α+i p)(x−B)
− e
(α−i p)(x−B)
= 2ie
α(x−B) sin ( p(x − B))
(10.24)
with unknown parameters α and p. By construction, ψ E (x) vanishes at x − B and
we assume that it is only defined for x > B.
We now use ψ E (x) as an Ansatz for the eigenfunction of H DO and therefore need
to calculate the derivatives of ψ(x) with respect to x. Differentiating twice, we find
151
p BS (x, τ ; x
) =
e
−r f τ
√
2πσ 2 τ
exp
−
x − x
+ (r f − σ
2
/2)τ
2
2σ 2 τ
.
(10.22)
It can be shown that the pricing kernel actually equals the Green’s function from
(5.11), where we have to keep in mind that some of the variables are named differently.
For clarity we repeat the previously-made statement that the pricing kernel acts like
a propagator in quantum mechanics, which gives the transition probability between
initial and final states. This gives a rather intuitive picture of how options are priced.
Since the kernel embeds all the dynamics of the stock market fluctuations, it enables
us to calculate the value of any option with any given payoff function g(x) by simply
convoluting with the pricing kernel.
Instead of repeating the calculations from Sect. 5.2 to determine the pricing formulae for previously calculated options, we will use the new formalism to calculate
the pricing of barrier options, which is difficult using the methods from Sect. 5.2.
10.4 Barrier Options
Here we consider the down-and-out barrier option, which becomes void, once the
stock price passes a lower limit S DO /S 0 = e
−B
. We incorporate this property in the
Hamiltonian by introducing a potential V (x) that is infinite for x ≤ B and thus forces
the wave function to become zero in that region. The lower limit thus acts like barrier
for the wave function, hence the name. The Hamiltonian is then given by
H DO = H BS + V (x) = −
σ
2
2
∂
2
∂ x 2 +
σ
2
2
− r f
∂
∂ x
+ r f + V (x) ,
(10.23)
where V (x) = ∞ for x ≤ B and V (x) = 0 for x > B.
In order to calculate the pricing kernel x|e
−τ H DO |x
, we need to find the eigenfunctions and eigenvalues for H DO . We already know that they are zero in the region
x ≤ B. Conversely, in the region x > B, where the potential is zero, we can assume
that the eigenfunctions resemble those of the unperturbed Black-Scholes Hamiltonian H BS . Moreover, they must vanish at the boundary x = B. We therefore write
the eigenfunctions ψ E (x) with eigenvalue E in the following form
x|E = ψ E (x) = e
(α+i p)(x−B)
− e
(α−i p)(x−B)
= 2ie
α(x−B) sin ( p(x − B))
(10.24)
with unknown parameters α and p. By construction, ψ E (x) vanishes at x − B and
we assume that it is only defined for x > B.
We now use ψ E (x) as an Ansatz for the eigenfunction of H DO and therefore need
to calculate the derivatives of ψ(x) with respect to x. Differentiating twice, we find