Option pricing: A K–theoretic perspective on diffusion models

Ioannis P. ZOIS; Ioannis P. ZOIS

doi:10.3934/QFE.2026017

Quantitative Finance and Economics

2026, Volume 10, Issue 3: 405-437. doi: 10.3934/QFE.2026017

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Research article

Option pricing: A K–theoretic perspective on diffusion models

Ioannis P. ZOIS ^{1,2
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1.
Exeter College, Turl Street, OX1 3DP, Oxford, UK
2.
Public Power Corporation (PPC) S.A., 30 Chalkokondyli Street, GR 104 32, Athens, Greece

Received: 22 December 2025 Revised: 03 April 2026 Accepted: 14 May 2026 Published: 14 July 2026
JEL Codes: G13, C02, C65

Background
Continuous–time option pricing under diffusion models leads, after the usual risk–neutral reduction and standard changes of variables, to parabolic partial differential equations generated by second–order elliptic operators with Gaussian heat kernels. This suggests a geometric reformulation of diffusion pricing in terms of Dirac–type operators and index theory. The purpose of this article was to develop such a structural viewpoint and to identify the invariant information it could reveal about payoff spaces.
Methods
We studied stylised Bachelier and Black–Scholes–Merton diffusion generators on compactified state spaces and realised them as Laplace–type operators of the form $ -D^2 $ for suitable Dirac–type operators. On the two–torus equipped with a $ U(1) $ line bundle of degree $ k $, we analysed the associated twisted Dirac operator using heat–kernel methods, the Atiyah–Singer index theorem, and an explicit theta–function construction of zero modes. We then related the resulting invariant sector to a finite–market diagnostic based on payoff vectors and Gram matrices.
Results
We obtained a Dirac realisation of stylised pricing generators on $ S^1 $ and $ T^2 $, proved that the twisted Dirac operator on $ T^2 $ has index $ k $, and gave an explicit Jacobi theta–function basis for its zero modes. These zero modes spanned a finite–dimensional diffusion–invariant subspace of payoff space in the compactified model. In addition, the finite–market example showed how the same viewpoint led to a rank/kernel diagnostic for payoff redundancy and constrained market dimension.
Conclusions
Index–theoretic methods provide a coherent structural language for diffusion–based option pricing. Although the compactified constructions are stylised rather than literal market models, they isolate stable geometric features of pricing generators and motivate finite–dimensional diagnostics that can complement standard quantitative–finance workflows.
- option pricing,
- Bachelier model,
- Black–Scholes–Merton model,
- Dirac operators,
- Atiyah–Singer index theorem,
- spectral triples,
- K–theory,
- theta functions
Citation: Ioannis P. ZOIS. Option pricing: A K–theoretic perspective on diffusion models[J]. Quantitative Finance and Economics, 2026, 10(3): 405-437. doi: 10.3934/QFE.2026017

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Abstract

Background

Continuous–time option pricing under diffusion models leads, after the usual risk–neutral reduction and standard changes of variables, to parabolic partial differential equations generated by second–order elliptic operators with Gaussian heat kernels. This suggests a geometric reformulation of diffusion pricing in terms of Dirac–type operators and index theory. The purpose of this article was to develop such a structural viewpoint and to identify the invariant information it could reveal about payoff spaces.

Methods

We studied stylised Bachelier and Black–Scholes–Merton diffusion generators on compactified state spaces and realised them as Laplace–type operators of the form $ -D^2 $ for suitable Dirac–type operators. On the two–torus equipped with a $ U(1) $ line bundle of degree $ k $, we analysed the associated twisted Dirac operator using heat–kernel methods, the Atiyah–Singer index theorem, and an explicit theta–function construction of zero modes. We then related the resulting invariant sector to a finite–market diagnostic based on payoff vectors and Gram matrices.

Results

We obtained a Dirac realisation of stylised pricing generators on $ S^1 $ and $ T^2 $, proved that the twisted Dirac operator on $ T^2 $ has index $ k $, and gave an explicit Jacobi theta–function basis for its zero modes. These zero modes spanned a finite–dimensional diffusion–invariant subspace of payoff space in the compactified model. In addition, the finite–market example showed how the same viewpoint led to a rank/kernel diagnostic for payoff redundancy and constrained market dimension.

Conclusions

Index–theoretic methods provide a coherent structural language for diffusion–based option pricing. Although the compactified constructions are stylised rather than literal market models, they isolate stable geometric features of pricing generators and motivate finite–dimensional diagnostics that can complement standard quantitative–finance workflows.

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