154
10 Quantum Finance and Path Integrals
x
), which proves that the basis of energy eigenstates as used in (10.31) is indeed
complete.
We immediately exploit this completeness relation to calculate the pricing kernel
p DO (x, τ ; x
) = =x|e
−τ H DO |x
p DO (x, τ ; x
) =
∞
σ 2 γ 2 /2
d E
2πσ 2
2E/σ 2 − γ 2
x|e
−τ H DO |EE|x
=
∞
0
dp
2π
e
−τ E
x|EE|x
,
(10.34)
where we replaced the integration over E by an integration over p with p =
2E/σ 2 − γ 2 , as before. Furthermore, we used e
−τ H DO |E = e
−τ E
|E, because
|E is an eigenstate of the Hamiltonian. We proceed by inserting the expressions for
the eigenfunctions x|E and E|x
and express the energy in the exponent through
p by E = ( p
2
+ γ
2
)σ
2
/2, such that after some further algebra, we get
p DO (x, τ ; x
) = e
−γ
2 τ σ
2 /2+α(x−x
)
∞
0
dp
2π
e
− p
2 τ σ
2 /2
(10.35)
×
e
i p(x−x
)
+ e
−i p(x−x
)
− e
i p(x+x
−2B)
− e
−i p(x+x
−2B)
= e
−γ
2 τ σ
2 /2+α(x−x
)
∞
−∞
dp
2π
e
− p
2 τ σ
2 /2
e
i p(x−x
)
− e
i p(x+x
−2B)
.
After evaluating the Gaussian integrals over p, we finally obtain
p DO (x, τ ; x
) =
1
√
2πσ 2 τ
exp
−
(σ
2
/2 + r f )
2
τ
2σ 2
+ α(x − x
)
(10.36)
×
exp
−
(x − x
)
2
2σ 2 τ
− exp
−
(x + x
− 2B)
2
2σ 2 τ
.
The first term in the curly braces, combined with the exponential forefactors, yields
the following result
(σ
2
/2 + r f )
2
τ
2σ 2
−
σ
2
/2 − r f
σ 2
(x − x
) +
(x − x
)
2
2σ 2 τ
(10.37)
=
(x − x
+ (r f − σ
2
/2)τ )
2
2σ 2 τ
+ r f τ ,
where we completed the square to absorb the term proportional to x − x
into the
quadratic term. The second term can be handled in much the same way, but we need