Chapter 10
Quantum Finance and Path Integrals
Abstract Stimulated by the resemblance of the Black-Scholes equation and the
Schrödinger equation, this chapter uses quantum-mechanical methods to determine
the pricing kernel, which turns out to be equivalent to the Green’s function found in
earlier chapters. Using the quantum-mechanical methods the down-and-out barrier
option is treated in some detail. After covering Feynman’s description of quantum
mechanics in terms of path integrals, they are used to re-derive the pricing kernel, found earlier. Since path integrals are very amenable to numerical evaluations,
we introduce Monte-Carlo methods, including the Metropolis-Hasting algorithm, to
evaluate the multi-dimensional integrals.
At first sight it appears strange to relate finance and option pricing to quantum
mechanical concepts, but we will see that there are several common points. First
note that near the end of Sect. 5.1 we derived the Black-Scholes equation in (5.6).
Now we observe that the substitution S/S 0 = e
x transforms it into the form of a
quasi-Schrödinger equation
∂c
∂t
= −
σ
2
2
∂
2 c
∂ x 2 +
σ
2
2
− r f
∂c
∂ x
+ r f c = H BS c
(10.1)
with the Black-Scholes Hamiltonian H BS
H BS = −
σ
2
2
∂
2
∂ x 2 +
σ
2
2
− r f
∂
∂ x
+ r f .
(10.2)
Thus, it appears that there is at least a formal resemblance between the Schrödinger
equation and the Black-Scholes equation, which we earlier found to be related to the
diffusion equation in Sect. 5.2. This is no surprise, because the stochastic meandering of stock prices—the inability to predict them at a later time—and the quantum
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_10
145
Quantum Finance and Path Integrals
Abstract Stimulated by the resemblance of the Black-Scholes equation and the
Schrödinger equation, this chapter uses quantum-mechanical methods to determine
the pricing kernel, which turns out to be equivalent to the Green’s function found in
earlier chapters. Using the quantum-mechanical methods the down-and-out barrier
option is treated in some detail. After covering Feynman’s description of quantum
mechanics in terms of path integrals, they are used to re-derive the pricing kernel, found earlier. Since path integrals are very amenable to numerical evaluations,
we introduce Monte-Carlo methods, including the Metropolis-Hasting algorithm, to
evaluate the multi-dimensional integrals.
At first sight it appears strange to relate finance and option pricing to quantum
mechanical concepts, but we will see that there are several common points. First
note that near the end of Sect. 5.1 we derived the Black-Scholes equation in (5.6).
Now we observe that the substitution S/S 0 = e
x transforms it into the form of a
quasi-Schrödinger equation
∂c
∂t
= −
σ
2
2
∂
2 c
∂ x 2 +
σ
2
2
− r f
∂c
∂ x
+ r f c = H BS c
(10.1)
with the Black-Scholes Hamiltonian H BS
H BS = −
σ
2
2
∂
2
∂ x 2 +
σ
2
2
− r f
∂
∂ x
+ r f .
(10.2)
Thus, it appears that there is at least a formal resemblance between the Schrödinger
equation and the Black-Scholes equation, which we earlier found to be related to the
diffusion equation in Sect. 5.2. This is no surprise, because the stochastic meandering of stock prices—the inability to predict them at a later time—and the quantum
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_10
145