10.2 Black-Scholes Hamiltonian
149
where we use partial integration to shift the derivative from f to g and assume that
f and g vanish at the integral boundaries. Equation 10.12 implies that the derivative
operator is anti-hermitian
∂
∂ x
†
= −
∂
∂ x
.
(10.13)
In a similar fashion, we can show that the position operator x is hermitian. This
mixture of hermitian and anti-hermitian operators requires us to pay extra attention
in the calculations.
Third, we found in Sect. 5.2 and, in particular, in (5.13) that the value of an
option at a time τ before maturity can be written as the convolution of the payoff
function and the Green’s function. This concept resembles that of an operator—a
propagator—pushing a quantum mechanical state forward in time, as illustrated in
(10.11). Having established this correspondence, we now develop methods to pursue
this analogy of the quantum-mechanical description to the stochastic description
further. We will loosely base the discussion on [2].
Let us use the Hamiltonian H BS to find the temporal evolution of the option c
by integrating (10.1), which reads ∂c/∂t = H BS c. Using arguments, similar to those
that led to (10.11), we write
c(x, t) = e
t H BS c(x, 0) or |c(t) = e
t H BS |c(0) ,
(10.14)
where we recover the equation on the left-hand side by multiplying with x| from the
left. At time τ = T − t before maturity, where the value of the option is the payoff
function g(x), we can write |c(T ) = e
T H BS |g, such that
|c(t) = e
−(T −t)H BS |g = e
−τ H BS |g .
(10.15)
This equation has a rather intuitive interpretation of the Black-Scholes Hamiltonian
H BS mapping the payoff function g backwards in time to the present time τ =
T − t prior to maturity. Different options are characterized by their individual payoff
functions g(x), but the dynamics of mapping the value back to the present time is
common to all options and is determined by the Hamiltonian H BS .
10.3 Pricing Kernel
In order to evaluate (10.15), we need to choose a basis and therefore multiply with x|
from the left-hand side and also insert the identity from (10.3) between the operator
e
−τ H BS and |g. This results in
149
where we use partial integration to shift the derivative from f to g and assume that
f and g vanish at the integral boundaries. Equation 10.12 implies that the derivative
operator is anti-hermitian
∂
∂ x
†
= −
∂
∂ x
.
(10.13)
In a similar fashion, we can show that the position operator x is hermitian. This
mixture of hermitian and anti-hermitian operators requires us to pay extra attention
in the calculations.
Third, we found in Sect. 5.2 and, in particular, in (5.13) that the value of an
option at a time τ before maturity can be written as the convolution of the payoff
function and the Green’s function. This concept resembles that of an operator—a
propagator—pushing a quantum mechanical state forward in time, as illustrated in
(10.11). Having established this correspondence, we now develop methods to pursue
this analogy of the quantum-mechanical description to the stochastic description
further. We will loosely base the discussion on [2].
Let us use the Hamiltonian H BS to find the temporal evolution of the option c
by integrating (10.1), which reads ∂c/∂t = H BS c. Using arguments, similar to those
that led to (10.11), we write
c(x, t) = e
t H BS c(x, 0) or |c(t) = e
t H BS |c(0) ,
(10.14)
where we recover the equation on the left-hand side by multiplying with x| from the
left. At time τ = T − t before maturity, where the value of the option is the payoff
function g(x), we can write |c(T ) = e
T H BS |g, such that
|c(t) = e
−(T −t)H BS |g = e
−τ H BS |g .
(10.15)
This equation has a rather intuitive interpretation of the Black-Scholes Hamiltonian
H BS mapping the payoff function g backwards in time to the present time τ =
T − t prior to maturity. Different options are characterized by their individual payoff
functions g(x), but the dynamics of mapping the value back to the present time is
common to all options and is determined by the Hamiltonian H BS .
10.3 Pricing Kernel
In order to evaluate (10.15), we need to choose a basis and therefore multiply with x|
from the left-hand side and also insert the identity from (10.3) between the operator
e
−τ H BS and |g. This results in