146
10 Quantum Finance and Path Integrals
mechanical inability to predict the definite state of a system at a later time suggest
the possibility to find a common mathematical framework to describe both. Note
that in this framework x = log(S/S 0 ) assumes the role of the position coordinate in
a quantum mechanical system.
Let us start with a short refresher of the basics of quantum mechanics, but focus
on the points pertinent to the later discussion. We will then point out the similarities,
but also the differences, of how these concepts are used in finance. For a complete
treatment of quantum mechanics, the reader is referred to, for example [1].
10.1 Quantum Mechanics
We will extensively use the notation, first introduced by Dirac, to describe quantummechanical states as bra x| and ket |ψ vectors, where x| and |ψ are elements of
dual vector spaces. Furthermore, the scalar product between vectors in these dual
vector spaces is denoted by x|ψ = ψ(x), where ψ(x) is the wave function of
state |ψ. If we exchange the order of entries, we obtain the complex conjugate
ψ|x = =x|ψ
∗
= ψ(x)
∗ of the scalar product, where the asterisk denotes the complex conjugate. The wave function ψ(x) has the interpretation that the probability of
finding a particle in the range between x and x + dx is given by the ψ(x)ψ(x)
∗ dx.
Moreover, the states |x form a complete basis for the vector space, which is a
consequence of
1 =
dx|xx| =
dx ˆ
P(x) ,
(10.3)
where ˆ
P(x) = |xx| is the projection operator onto state |x. The integral simply
states that the basis comprising all position vectors |x is complete. The scalar product
of two basis vectors x
| and |x is given by x
|x = δ(x
− x), where δ(y) is Dirac’s
delta-function. Using the representation of the delta-function as an integral over a
complex exponential, we can write
x
|x = δ(x
− x) =
1
2π
∞
−∞
e
ik(x
−x) dk =
1
2π
∞
−∞
x
|kk|xdk
(10.4)
with k|x = e
−ikx
. Note that k is related to the momentum p of a particle by k = p/
with = h/2π, where h is Planck’s constant. The last equation indicates that also
the vectors |k form a complete basis and we have
1 =
1
2π
dk|kk| .
(10.5)
Here |kk| is a projector onto a state with momentum p = k.
10 Quantum Finance and Path Integrals
mechanical inability to predict the definite state of a system at a later time suggest
the possibility to find a common mathematical framework to describe both. Note
that in this framework x = log(S/S 0 ) assumes the role of the position coordinate in
a quantum mechanical system.
Let us start with a short refresher of the basics of quantum mechanics, but focus
on the points pertinent to the later discussion. We will then point out the similarities,
but also the differences, of how these concepts are used in finance. For a complete
treatment of quantum mechanics, the reader is referred to, for example [1].
10.1 Quantum Mechanics
We will extensively use the notation, first introduced by Dirac, to describe quantummechanical states as bra x| and ket |ψ vectors, where x| and |ψ are elements of
dual vector spaces. Furthermore, the scalar product between vectors in these dual
vector spaces is denoted by x|ψ = ψ(x), where ψ(x) is the wave function of
state |ψ. If we exchange the order of entries, we obtain the complex conjugate
ψ|x = =x|ψ
∗
= ψ(x)
∗ of the scalar product, where the asterisk denotes the complex conjugate. The wave function ψ(x) has the interpretation that the probability of
finding a particle in the range between x and x + dx is given by the ψ(x)ψ(x)
∗ dx.
Moreover, the states |x form a complete basis for the vector space, which is a
consequence of
1 =
dx|xx| =
dx ˆ
P(x) ,
(10.3)
where ˆ
P(x) = |xx| is the projection operator onto state |x. The integral simply
states that the basis comprising all position vectors |x is complete. The scalar product
of two basis vectors x
| and |x is given by x
|x = δ(x
− x), where δ(y) is Dirac’s
delta-function. Using the representation of the delta-function as an integral over a
complex exponential, we can write
x
|x = δ(x
− x) =
1
2π
∞
−∞
e
ik(x
−x) dk =
1
2π
∞
−∞
x
|kk|xdk
(10.4)
with k|x = e
−ikx
. Note that k is related to the momentum p of a particle by k = p/
with = h/2π, where h is Planck’s constant. The last equation indicates that also
the vectors |k form a complete basis and we have
1 =
1
2π
dk|kk| .
(10.5)
Here |kk| is a projector onto a state with momentum p = k.