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Selective Forgetting in Option Calibration: An Operator-Theoretic Gauss-Newton Framework

Ahmet Umur Özsoy

TL;DR

This work addresses the problem of removing the influence of selected market data from an already calibrated option-pricing model without full retraining. It introduces an operator-theoretic framework for selective forgetting that leverages the additive Gauss--Newton structure, enabling exact forgetting under a fixed linearization via two operators: sharded recompute and fast refactorization. The authors provide theoretical guarantees (local exactness, stability) and demonstrate, on synthetic Heston-calibrated data, that fast refactorization reproduces full retraining to machine precision while offering substantial computational speedups, with sharded recompute providing locality-based efficiency. The approach supports regulatory, data-quality, and auditing needs by enabling data deletion without reprocessing the entire dataset, and offers a foundation for extending unlearning to online and influence-diagnostic calibration workflows, all within a coherent operator-theoretic framework.

Abstract

Calibration of option pricing models is routinely repeated as markets evolve, yet modern systems lack an operator for removing data from a calibrated model without full retraining. When quotes become stale, corrupted, or subject to deletion requirements, existing calibration pipelines must rebuild the entire nonlinear least-squares problem, even if only a small subset of data must be excluded. In this work, we introduce a principled framework for selective forgetting (machine unlearning) in parametric option calibration. We provide stability guarantees, perturbation bounds, and show that the proposed operators satisfy local exactness under standard regularity assumptions.

Selective Forgetting in Option Calibration: An Operator-Theoretic Gauss-Newton Framework

TL;DR

This work addresses the problem of removing the influence of selected market data from an already calibrated option-pricing model without full retraining. It introduces an operator-theoretic framework for selective forgetting that leverages the additive Gauss--Newton structure, enabling exact forgetting under a fixed linearization via two operators: sharded recompute and fast refactorization. The authors provide theoretical guarantees (local exactness, stability) and demonstrate, on synthetic Heston-calibrated data, that fast refactorization reproduces full retraining to machine precision while offering substantial computational speedups, with sharded recompute providing locality-based efficiency. The approach supports regulatory, data-quality, and auditing needs by enabling data deletion without reprocessing the entire dataset, and offers a foundation for extending unlearning to online and influence-diagnostic calibration workflows, all within a coherent operator-theoretic framework.

Abstract

Calibration of option pricing models is routinely repeated as markets evolve, yet modern systems lack an operator for removing data from a calibrated model without full retraining. When quotes become stale, corrupted, or subject to deletion requirements, existing calibration pipelines must rebuild the entire nonlinear least-squares problem, even if only a small subset of data must be excluded. In this work, we introduce a principled framework for selective forgetting (machine unlearning) in parametric option calibration. We provide stability guarantees, perturbation bounds, and show that the proposed operators satisfy local exactness under standard regularity assumptions.

Paper Structure

This paper contains 7 sections, 7 theorems, 40 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulation of the unlearning problem
  3. Preliminaries
  4. Sharded recompute operator
  5. Fast refactor operator
  6. Illustrations
  7. Conclusion

Key Result

Proposition 1

(Shard-level exactness at a fixed linearization) Fix the reference $\theta^{\text{ref}}$. Consider the Gauss--Newton normal equations at that reference. If we recompute exactly $G'_k$, $H'_k$ for all affected shards on $D_k \setminus F$ and keep $(G_k, H_k)$ for unaffected shards, then the global sy is identical to the Gauss--Newton system built by running over the full retained set $D \setminus F

Figures (5)

  • Figure 1: An exemplary comparison of equivalence of retraining and fast factorization with a smaller sample
  • Figure 2: An exemplary comparison of equivalence of retraining and fast factorization with a larger sample
  • Figure 3: Example on the importance of the forgetting set for the sharded recomputation
  • Figure 4: Exemplary demonstration of the sharding positions in sharded recomputation
  • Figure 5: Exemplary comparison of relative parameter error and runtime across forgetting fractions

Theorems & Definitions (18)

  • Proposition 1
  • Proposition 2: Accuracy after one relinearization on affected shards
  • Remark 1: Quadratic accuracy under small residuals
  • proof : Proof of Proposition \ref{['thm:relinearization']}
  • Proposition 3: Stability of the unlearning update
  • proof
  • Remark 2: Robust loss control
  • Remark 3: Conditioning links: eigenvalues and Neumann bound
  • Proposition 4: Exactness under fixed linearization
  • proof
  • ...and 8 more