156
10 Quantum Finance and Path Integrals
Fig. 10.1 The left-hand
image illustrates the use of
image charges in order to
satisfy the boundary
conditions on a conducting
surface and the right-hand
side shows the use of image
sources for diffusive
problems
Image sources can always be used, if the underlying partial differential equations
are linear, which is also the case for the Black-Scholes equation with its underlying
diffusive process as defined by the diffusion equation (4.15). Placing an anti-source,
equally strong as the original source, on the other side of an absorbing boundary
forces the solution to be zero on the boundary. This is illustrated in the sketch on
the right-hand side in Fig. 10.1. The two terms in (10.41) thus correspond to a pair
of diffusion processes that ensure that the boundary condition p DO (x, τ, x
) = 0 on
the absorbing boundary at x = B is satisfied. The additional forefactor from (10.40)
is needed, because the Black-Scholes equation includes an additional drift term and
uses the log-normal form of the distribution function, rather than a plain Gaussian.
After this digression on image charges, let us now have a look at another advanced
concept—path integrals. First we review their use in quantum mechanics before
addressing applications in finance.
10.5 Path Integrals in Quantum Mechanics
Path integrals provide yet another, and quite remarkable, way to characterize the
dynamics of quantum systems by Green’s functions K (x b , t b ; x a , t a ), which describe
how wave functions ψ(x, t) evolves from time t a to time t b
ψ(x b , t b ) =
∞
−∞
K (x b , t b ; x a , t a )ψ(x a , t a )dx a .
(10.42)
Due to Feynman’s tremendous intuition, we now know that we can write the Green’s
function K , also referred to as propagator, as the sum over all paths [3] that start at
time t a at location x a and end at time t b at location x b . But we need to put a weight to
each possible path and Feynman conjectured that this weight depends on the classical
action
S(b, a) =
t b
t a
L(x, ˙
x)dt with x(t a ) = x a and x(t b ) = x b ,
(10.43)
10 Quantum Finance and Path Integrals
Fig. 10.1 The left-hand
image illustrates the use of
image charges in order to
satisfy the boundary
conditions on a conducting
surface and the right-hand
side shows the use of image
sources for diffusive
problems
Image sources can always be used, if the underlying partial differential equations
are linear, which is also the case for the Black-Scholes equation with its underlying
diffusive process as defined by the diffusion equation (4.15). Placing an anti-source,
equally strong as the original source, on the other side of an absorbing boundary
forces the solution to be zero on the boundary. This is illustrated in the sketch on
the right-hand side in Fig. 10.1. The two terms in (10.41) thus correspond to a pair
of diffusion processes that ensure that the boundary condition p DO (x, τ, x
) = 0 on
the absorbing boundary at x = B is satisfied. The additional forefactor from (10.40)
is needed, because the Black-Scholes equation includes an additional drift term and
uses the log-normal form of the distribution function, rather than a plain Gaussian.
After this digression on image charges, let us now have a look at another advanced
concept—path integrals. First we review their use in quantum mechanics before
addressing applications in finance.
10.5 Path Integrals in Quantum Mechanics
Path integrals provide yet another, and quite remarkable, way to characterize the
dynamics of quantum systems by Green’s functions K (x b , t b ; x a , t a ), which describe
how wave functions ψ(x, t) evolves from time t a to time t b
ψ(x b , t b ) =
∞
−∞
K (x b , t b ; x a , t a )ψ(x a , t a )dx a .
(10.42)
Due to Feynman’s tremendous intuition, we now know that we can write the Green’s
function K , also referred to as propagator, as the sum over all paths [3] that start at
time t a at location x a and end at time t b at location x b . But we need to put a weight to
each possible path and Feynman conjectured that this weight depends on the classical
action
S(b, a) =
t b
t a
L(x, ˙
x)dt with x(t a ) = x a and x(t b ) = x b ,
(10.43)