[Paper Review] The Black-Scholes Equation and Certain Quantum Hamiltonians

Juan M. Romero, O. González-Gaxiola+2 more|arXiv (Cornell University)|Feb 8, 2010

Quantum Mechanics and Applications3 citations

TL;DR

This paper constructs a non-Hermitian quantum mechanics framework using a modified momentum operator, leading to two Hermitian and two non-Hermitian Hamiltonians. It establishes a generalized supersymmetric quantum mechanics with a dual structure and demonstrates that non-Hermitian Hamiltonians from this framework correspond exactly to key Hamiltonians in quantum finance, including the Black-Scholes, generalized Black-Scholes, and barrier option models.

ABSTRACT

In this paper a quantum mechanics is built by means of a non-Hermitian momentum operator. We have shown that it is possible to construct two Hermitian and two non-Hermitian type of Hamiltonians using this momentum operator. We can construct a generalized supersymmetric quantum mechanics that has a dual based on these Hamiltonians. In addition, it is shown that the non-Hermitian Hamiltonians of this theory can be related to Hamiltonians that naturally arise in the so-called quantum finance.

Motivation & Objective

• To develop a quantum mechanical framework based on a non-Hermitian momentum operator derived from a gauge-like transformation of standard momentum.
• To classify and construct four types of Hamiltonians—two Hermitian and two non-Hermitian—using this modified momentum operator.
• To extend supersymmetric quantum mechanics by constructing a generalized version with dual structures based on the new Hamiltonians.
• To establish a direct correspondence between the non-Hermitian Hamiltonians in this quantum framework and fundamental Hamiltonians in quantum finance, particularly the Black-Scholes model.

Proposed method

• Define a non-Hermitian momentum operator $ P_{(f)j} = e^f P_j e^{-f} $, where $ f $ is an arbitrary function, leading to a gauge-transformed momentum that breaks Hermiticity.
• Construct four Hamiltonians: $ H_1 $ and $ H_2 $ as Hermitian combinations $ \alpha^2 \vec{P}_{(f)}^\dagger \cdot \vec{P}_{(f)} $ and $ \alpha^2 \vec{P}_{(f)} \cdot \vec{P}_{(f)}^\dagger $, and $ H_3 $, $ H_4 $ as non-Hermitian forms $ \beta^2 \vec{P}_{(f)}^\dagger \cdot \vec{P}_{(f)}^\dagger $ and $ \beta^2 \vec{P}_{(f)} \cdot \vec{P}_{(f)} $.
• Formulate a generalized supersymmetric quantum mechanics by defining supercharges $ Q_1, Q_2 $ and $ Q_3, Q_4 $ such that $ Q^2 = 0 $, leading to a super-Hamiltonian $ H = \{Q_1, Q_2\} $ with a dual structure.
• Identify the one-dimensional form of $ H_{II} = \beta^2 \vec{P}_{(f)}^\dagger \cdot \vec{P}_{(f)}^{\dagger} + V_2(x) $, and show it matches the Black-Scholes Hamiltonian $ H_{BS} $ under specific parameter identifications: $ \beta^2 = \sigma^2/2 $, $ f(x) = \frac{1}{\sigma^2}(\sigma^2/2 - r)x $, and $ V_2(x) = r $.
• Extend the correspondence to generalized and barrier option models by adjusting $ f(x) $ and $ V_2(x) $ to match $ H_{BSG} $ and $ H_{BSB} $, respectively.

Experimental results

Research questions

• RQ1 Can a non-Hermitian momentum operator generate both Hermitian and non-Hermitian Hamiltonians while preserving physical consistency?
• RQ2 How can generalized supersymmetric quantum mechanics be constructed from such a non-Hermitian framework, and what is its dual structure?
• RQ3 What is the precise mathematical correspondence between the non-Hermitian Hamiltonians derived from the modified momentum operator and the Black-Scholes Hamiltonian in quantum finance?
• RQ4 Can the generalized Black-Scholes and barrier option Hamiltonians be recovered as special cases of this quantum mechanical model?

Key findings

• The non-Hermitian momentum operator $ P_{(f)j} = e^f P_j e^{-f} $ generates two Hermitian Hamiltonians $ H_1 $ and $ H_2 $, and two non-Hermitian Hamiltonians $ H_3 $ and $ H_4 $, all derived from the same underlying operator structure.
• A generalized supersymmetric quantum mechanics is constructed using supercharges $ Q_1 $ and $ Q_2 $, satisfying $ Q^2 = 0 $ and $ \{Q_1, Q_2\} = H $, with a dual formulation possible via $ Q_3 $ and $ Q_4 $, demonstrating a richer algebraic structure than standard supersymmetry.
• The non-Hermitian Hamiltonian $ H_{II} = \beta^2 \vec{P}_{(f)}^\dagger \cdot \vec{P}_{(f)}^{\dagger} + V_2(x) $ exactly reproduces the Black-Scholes Hamiltonian $ H_{BS} $ when $ \beta^2 = \sigma^2/2 $, $ f(x) = \frac{1}{\sigma^2}(\sigma^2/2 - r)x $, and $ V_2(x) = r $.
• The generalized Black-Scholes Hamiltonian $ H_{BSG} $ is recovered by allowing $ f(x) $ to depend on a time-varying drift function $ V(x) $, with $ f(x) = \int_0^x du \, \frac{1}{\sigma^2}(\sigma^2/2 - V(u)) $, showing the model's flexibility.
• The barrier option Hamiltonian $ H_{BSB} $ is also matched by setting $ V_2(x) = V(x) $, confirming the framework's applicability to path-dependent financial derivatives.
• The ground state of the Hamiltonians is explicitly found, and the model demonstrates that non-Hermitian operators with real spectra can yield physically meaningful quantum systems, especially in financial applications.