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Quantum-Theoretical Re-interpretation of Pricing Theory

Tian Xin

TL;DR

The paper reframes option pricing as a fully observable, quantum-inspired process on a price lattice, replacing hidden information with observable price-state transitions. It develops a dual-generator framework: a frequency-based generator $H_{\mathrm{freq}} = \hbar f(\hat{S})$ that fixes transition phases, and a translation-invariant jump generator $H_{\mathrm{conv}} = \hbar\sum_α K(α) T_α$ that governs propagation, with the two interacting under a Lindbladian, translation-covariant structure. The result is a nonlocal, risk-neutral pricing equation that converges to the Black–Scholes–Merton operator in the diffusive limit, while enabling heavy-tail and tail-wings through fractional-like limits; the framework also provides a practical FFT-based method for synthetic pricing and IV computation. The approach yields testable links between jump intensities and extreme-event probabilities, and lays out concrete paths for extensions to multi-asset lattices, nonlinear interactions, and empirical calibration of the jump spectrum. Overall, the work offers a principled, observable-first foundation for pricing theory with a natural path to empirical validation and broadening of the classical paradigm.

Abstract

Motivated by Heisenberg's observable-only stance, we replace latent "information" (filtrations, hidden diffusions, state variables) with observable transitions between price states. On a discrete price lattice with a Hilbert-space representation, shift operators and the spectral calculus of the price define observable frequency operators and a translation-invariant convolution generator. Combined with jump operators that encode transition intensities, this yields a completely positive, translation-covariant Lindblad semigroup. Under the risk-neutral condition the framework leads to a nonlocal pricing equation that is diagonal in Fourier space; in the small-mesh diffusive limit its generator converges to the classical Black-Scholes-Merton operator. We do not propose another parametric model. We propose a foundation for model construction that is observable, first-principles, and mathematically natural. Noncommutativity emerges from the observable shift algebra rather than being postulated. The jump-intensity ledger determines tail behavior and short-maturity smiles and implies testable links between extreme-event probabilities and implied-volatility wings. Future directions: (i) multi-asset systems on higher-dimensional lattices with vector shifts and block kernels; (ii) state- or flow-dependent kernels as "financial interactions" leading to nonlinear master equations while preserving linear risk-neutral pricing; (iii) empirical tests of the predicted scaling relations between jump intensities and market extremes.

Quantum-Theoretical Re-interpretation of Pricing Theory

TL;DR

The paper reframes option pricing as a fully observable, quantum-inspired process on a price lattice, replacing hidden information with observable price-state transitions. It develops a dual-generator framework: a frequency-based generator that fixes transition phases, and a translation-invariant jump generator that governs propagation, with the two interacting under a Lindbladian, translation-covariant structure. The result is a nonlocal, risk-neutral pricing equation that converges to the Black–Scholes–Merton operator in the diffusive limit, while enabling heavy-tail and tail-wings through fractional-like limits; the framework also provides a practical FFT-based method for synthetic pricing and IV computation. The approach yields testable links between jump intensities and extreme-event probabilities, and lays out concrete paths for extensions to multi-asset lattices, nonlinear interactions, and empirical calibration of the jump spectrum. Overall, the work offers a principled, observable-first foundation for pricing theory with a natural path to empirical validation and broadening of the classical paradigm.

Abstract

Motivated by Heisenberg's observable-only stance, we replace latent "information" (filtrations, hidden diffusions, state variables) with observable transitions between price states. On a discrete price lattice with a Hilbert-space representation, shift operators and the spectral calculus of the price define observable frequency operators and a translation-invariant convolution generator. Combined with jump operators that encode transition intensities, this yields a completely positive, translation-covariant Lindblad semigroup. Under the risk-neutral condition the framework leads to a nonlocal pricing equation that is diagonal in Fourier space; in the small-mesh diffusive limit its generator converges to the classical Black-Scholes-Merton operator. We do not propose another parametric model. We propose a foundation for model construction that is observable, first-principles, and mathematically natural. Noncommutativity emerges from the observable shift algebra rather than being postulated. The jump-intensity ledger determines tail behavior and short-maturity smiles and implies testable links between extreme-event probabilities and implied-volatility wings. Future directions: (i) multi-asset systems on higher-dimensional lattices with vector shifts and block kernels; (ii) state- or flow-dependent kernels as "financial interactions" leading to nonlinear master equations while preserving linear risk-neutral pricing; (iii) empirical tests of the predicted scaling relations between jump intensities and market extremes.

Paper Structure

This paper contains 55 sections, 29 theorems, 182 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Mathematical setting (price lattice and conjugacy). Grid, Hilbert space, and positive spectrum.
  2. Index separation and Fourier normalization.
  3. Transition Frequencies and Difference Structures:A Quantum Reinterpretation of Fourier Dynamics
  4. Setting and notation.
  5. Definition and Additivity of Transition Frequencies
  6. Difference Operator Representation of Transition Frequencies：A Quantum Reinterpretation of Fourier Dynamics
  7. Equivalent Paths and Transition Symmetry
  8. Transition Amplitude and Difference Structures: A Quantum Reinterpretation of Fourier Dynamics
  9. Transition Amplitudes and Propagator Structure (purely operator-theoretic)
  10. Basic facts on the basis $\{|n\rangle\}$.
  11. Short-time expansion.
  12. Translation-invariant generator
  13. Well-posedness in brief.
  14. Alternative Constructing Price-Type Observables from Frequencies and Amplitudes
  15. Translation-invariant amplitudes (Toeplitz class).
  16. ...and 40 more sections

Key Result

Lemma 1.1

Assume that for each fixed gap $\alpha\in\mathbb{Z}$ the transition frequency $\omega(n,n-\alpha)$ is independent of $n$. Then there exist constants $a,b\in\mathbb{R}$ such that

Figures (1)

  • Figure 1: Implied–volatility smiles from the spectral (FFT/DFT) solver for two kernel families: NearFieldExp_sym (symmetric near–field exponential) and HeavyTail_skew (skewed power–law tail). Parameters as in the text; each smile is obtained by a single FFT per maturity.

Theorems & Definitions (55)

  • Lemma 1.1: index homogeneity $\Rightarrow$ affine $f$
  • proof
  • Proposition 2.1: Column normalization and probability conservation
  • proof
  • Proposition 2.2: Composition law for amplitudes
  • proof
  • Remark 2.3: No probabilistic convolution at the level of $P_t$
  • Proposition 2.4: Convolution/self-adjoint generator on $\ell^2(\mathbb Z)$
  • proof
  • Corollary 2.5: Unitarity and translation invariance of the propagator
  • ...and 45 more