Synchronizing pretrained kernel regressors with applications to American option pricing

Xuwei Yang; Anastasis Kratsios; Florian Krach; Matheus Grasselli; Aurelien Lucchi

doi:10.3934/fmf.2026001

Article Contents

March 2026, Volume 8, 23-77. Doi: 10.3934/fmf.2026001 open access

This volume Previous Article Physical returns in a pricing world: Towards forward-looking market risk measures Next Article

Synchronizing pretrained kernel regressors with applications to American option pricing

1.
Department of Mathematics and Statistics, McMaster University, Canada
2.
Department of Mathematics and Statistics, McMaster University, and Vector Institute, Canada
3.
Department of Mathematics, ETH Zürich, Switzerland
4.
Department of Mathematics and Computer Science, University of Basel, Switzerland

Received: January 24, 2024

Revised: November 17, 2025

Published online: January 28, 2026

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Abstract

American option pricing is a well-studied, computationally-heavy optimal stopping problem, which has motivated the need for light and accurate computational pipelines. Currently, the state-of-the-art light American option prices are built on randomly generated kernel regressors such as random neural networks; however, a single random (or pre-trained) kernel can yield unstable performance on new tasks. With this motivation, we study the problem of fine-tuning several pre-trained kernel regressors, trained on distinct datasets to a given novel dataset with a downstream view toward light American option pricing. This paper addresses the meta-optmization problem of the task of designing optimal transfer learning algorithms, which minimize a pathwise regret functional. Using techniques from optimal control, we construct the unique regret-optimal optimization algorithms for fine-tuning mixtures of pre-trained kernel regressors on $ \mathbb{R}^d $ sharing the same feature map. Our regret functional balances the objects of predictive power on the new dataset against transfer learning from other datasets under an algorithmic stability penalty. We show that an adversary which perturbs a proportion $ 0\leq q\leq 1 $ of training pairs by at most $ \varepsilon>0 $, across all training sets, cannot reduce the regret-optimal algorithm's regret by more than $ \mathcal{O}(\varepsilon (q \bar{N} )^{1/2} ) $, where $ \bar{N} $ is the aggregate number of training pairs. Further, we derive estimates of the computational complexity of our regret-optimal optimization algorithm. Our theoretical findings are used to improve state-of-the-art (SOTA) in computationally light American option pricing problems using random feature models, where we achieve SOTA performance compared to the benchmark light American option pricer of [32].

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Mathematics Subject Classification: 91-10, 49N90, 68T01.

Citation:

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Figure 1. Loss and energy $ {\mathcal L} $ for gradient descent (GD), the regret-optimal algorithm (RO), and the accelerated regret-optimal algorithm (ARO). Means and standard deviations over 10 runs are shown. Left: all iterations; middle: first 1000 iterations; right: iterations 500 to 1000 for our algorithms

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Figure 2. Loss and energy $ {\mathcal L} $ for the regret-optimal algorithm (RO) and the accelerated regret-optimal algorithm (ARO) with means and standard deviations over $ 10 $ runs. Left: $ \lambda = 10^{-4} $; right: $ \lambda = 2 $

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Table 1. Ablation Results: American Option Pricing. Relative performance (RP) and $ 95\% $-confidence-intervals for different optimization methods. We compare the local optimizers on the different datasets (LO-$ n $) with the mean local optimizer (MLO), the joint optimizer (JO), and our regret-optimal method (RO). The oracle local optimizer on the main dataset with additional training samples (standard: $ 100 $ samples per dataset) is included. See Table 2 for additional details

(A) Experiment $ 1 $ (compressed)			(b) Experiment 2
method	$ RP $	$ 95\% $-CI	method	$ RP $	$ 95\% $-CI
LO-$ 1 $	1.000	$ [0.991; 1.009] $	LO-$ 1 $	1.000	[0.993; 1.007]
LO-2, $ \dotsc $, LO-7	$ \geq 0.966 $	$ [\geq 0.957; \leq 0.994] $	LO-2	0.773	[0.765; 0.782]
LO-8, $ \dotsc $, LO-13	$ \leq 0.725 $	$ [\geq 0.696; \leq 0.736] $	LO-3	1.003	[0.994; 1.012]
MLO	0.823	$ [0.813; 0.833] $	MLO	0.932	[0.920; 0.944]
JO	0.886	$ [0.878; 0.894] $	JO	0.763	[0.754; 0.772]
JSO-$ \{1, \dotsc, 7\} $	1.062	$ [1.057; 1.067] $	RO ($ \eta=10 $)	0.993	[0.985; 1.000]
RO ($ \eta=100 $)	1.090	[1.086; 1.095]	RO ($ \eta=50 $)	1.040	[1.034; 1.046]
LO-1 (700 tr. samp.)	1.100	[1.097; 1.104]	RO ($ \eta=100$)	1.041	[1.036; 1.047]
LO-1 (50K tr. samp.)	1.194	[1.192; 1.195]	RO ($ \eta=100$)	1.041	[1.036; 1.047]

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Table 2. Relative performance (RP) and $ 95\% $-confidence-intervals for different optimization methods in Experiment 4.1. We compare the local optimizers on the different datasets (LO-$ n $) with the mean local optimizer (MLO), the joint optimizer (JO), and our regret-optimal method (RO). The oracle local optimizer on the main dataset with additional training samples (standard: $ 100 $ samples per dataset) is included. When the information sharing parameter $ \eta $ is set to $ 100 $, the regret-optimal (RO) algorithm outperforms all "non-oracle" baselines

method	$ RP $	$ 95\% $-CI
LO-1	1	$[0.991; 1.009]$
LO-2	0.985	$[0.977; 0.994]$
LO-3	0.966	$[0.958; 0.973]$
LO-4	0.975	$[0.966; 0.984]$
LO-5	0.966	$[0.957; 0.975]$
LO-6	0.979	$[0.971; 0.988]$
LO-7	0.978	$[0.971; 0.986]$
LO-8	0.721	$[0.711; 0.730]$
LO-9	0.72	$[0.711; 0.729]$
LO-10	0.706	$[0.696; 0.716]$
LO-11	0.725	$[0.715; 0.736]$
LO-12	0.707	$[0.697; 0.716]$
LO-13	0.718	$[0.708; 0.727]$
MLO	0.823	$[0.813; 0.833]$
JO	0.886	$[0.878; 0.894]$
JO (datasets 1-7)	1.062	$[1.057; 1.067]$
RO ($\eta=10$)	1	$[0.990; 1.010]$
RO ($\eta=100$)	1.091	[1.087; 1.095]
RO ($\eta=500$)	1.05	$[1.043; 1.057]$
LO-1 (700 tr. samp.)	1.1	$[1.097; 1.104]$
LO-1 (50K tr. samp.)	1.194	$[1.192; 1.195]$

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