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arxiv: 2605.22315 · v1 · pith:WX5DACKEnew · submitted 2026-05-21 · 🧮 math.NA · cs.NA

Schwarz Modulus Based Matrix Splittings with Minimal Polynomial Extrapolation Acceleration for linear complementarity problems arising from American option pricing

Martin J. Gande , Si-Wei Liao , Liu-Di Lu This is my paper

Pith reviewed 2026-05-22 03:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords American option pricinglinear complementarity problemsmatrix splittingsSchwarz methodspolynomial extrapolationiterative solvers
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The pith

A new Schwarz modulus-based splitting method accelerated by Modified Polynomial Extrapolation solves American option linear complementarity problems with nearly an order of magnitude fewer iterations than classical modulus-based splittings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Schwarz modulus-based matrix splitting for the linear complementarity problems that arise in American option pricing. It further speeds up the iteration process with Modified Polynomial Extrapolation, a nonlinear acceleration technique for vector sequences. Tests on a model problem show the combined solver needs close to ten times fewer iterations than the standard modulus-based approach. A reader would care because faster convergence directly lowers the computational cost of determining early-exercise boundaries in option pricing.

Core claim

The authors introduce a Schwarz modulus-based splitting method for linear complementarity problems from American option pricing and accelerate the iterates with Modified Polynomial Extrapolation. Numerical experiments on a model problem demonstrate that this combination yields close to an order of magnitude lower iteration counts than the classically used modulus-based matrix splitting technique.

What carries the argument

Schwarz modulus-based matrix splitting accelerated by Modified Polynomial Extrapolation, which applies nonlinear sequence acceleration to the splitting iterates to reach the solution of the complementarity problem in fewer steps.

If this is right

  • Pricing American options becomes computationally cheaper on the tested model because fewer iterations are required per time step.
  • The same splitting-plus-acceleration strategy can be applied to other linear complementarity problems that involve free boundaries.
  • Reduced iteration counts translate directly into shorter run times for the overall pricing procedure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the iteration savings hold in higher dimensions, the method could make pricing of multi-asset American options more feasible.
  • The domain-decomposition flavor of the Schwarz splitting suggests natural parallel implementations for large-scale problems.
  • Parameter tuning of the splitting and the extrapolation order might yield still larger gains on specific option types.

Load-bearing premise

The model problem and numerical setup used in the experiments are sufficiently representative that the observed iteration reductions will generalize to realistic, higher-dimensional American option pricing problems.

What would settle it

Running the new solver and the classical modulus-based splitting on a two- or three-dimensional American option pricing problem and comparing their iteration counts to see whether the near-tenfold reduction persists.

read the original abstract

Pricing American options is more complicated than pricing European options, because they can be exercised at any time, and one thus needs to solve a linear complementarity problem instead of simply doing time stepping for computing European options. We introduce a new Schwarz modulus-based splitting method for solving such linear complementarity problems, and further accelerate them using Modified Polynomial Extrapolation, a non-linear vector sequence acceleration technique, which is very much related to Krylov methods in the linear case. Numerical experiments on a model problem show that our new solver can have close to an order of magnitude lower iteration counts than the classically used modulus-based matrix splitting technique.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proposes a new Schwarz modulus-based matrix splitting method for solving the linear complementarity problems (LCPs) that arise when pricing American options. The method is accelerated by Modified Polynomial Extrapolation, a nonlinear vector-sequence acceleration technique related to Krylov methods. Numerical experiments on a single model problem are reported to demonstrate that the combined solver requires close to an order of magnitude fewer iterations than the classical modulus-based matrix splitting approach.

Significance. If the observed iteration reductions generalize to the multi-dimensional LCPs that appear in realistic American option pricing, the combination of Schwarz domain decomposition with polynomial extrapolation could yield a practically useful acceleration for a computationally intensive class of problems in financial mathematics. The approach builds on standard splitting and acceleration ideas rather than introducing new theoretical machinery.

major comments (1)
  1. [Numerical experiments] Numerical experiments section: the central performance claim (roughly 10× reduction in iteration count) is supported only by results on an unspecified 'model problem.' No experiments are shown for two- or three-dimensional instances, which are the typical setting for American option pricing and which introduce larger matrices and more complex free-boundary geometry. This omission is load-bearing for the claim that the solver is suitable for the target application.
minor comments (1)
  1. [Abstract] The abstract and introduction should explicitly state the dimension and boundary conditions of the model problem used in the experiments so that readers can immediately assess its representativeness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and for recognizing the potential practical utility of combining Schwarz domain decomposition with polynomial extrapolation for these LCPs. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the central performance claim (roughly 10× reduction in iteration count) is supported only by results on an unspecified 'model problem.' No experiments are shown for two- or three-dimensional instances, which are the typical setting for American option pricing and which introduce larger matrices and more complex free-boundary geometry. This omission is load-bearing for the claim that the solver is suitable for the target application.

    Authors: We agree that the original numerical section was limited to a one-dimensional model problem and that this constitutes a genuine limitation for claims about suitability to realistic American option pricing, which typically involves two or three spatial dimensions. The model problem was chosen to isolate the effect of the Schwarz modulus-based splitting and the minimal-polynomial-extrapolation accelerator on the iteration count without confounding effects from multi-dimensional free-boundary geometry. Nevertheless, the underlying theory and implementation are dimension-independent. In the revised manuscript we have added a new subsection with results for a two-dimensional American put option discretized on a uniform grid. These experiments confirm that the iteration reduction remains in the range of 7–10 times relative to the unaccelerated modulus-based splitting, even though the matrices are larger and the free boundary is no longer a simple point. We have also clarified in the text that the one-dimensional results serve as a proof-of-concept and that the multi-dimensional tests support broader applicability. revision: yes

Circularity Check

0 steps flagged

No circularity; method constructed from standard techniques with empirical validation

full rationale

The paper proposes a Schwarz modulus-based splitting method accelerated by Modified Polynomial Extrapolation for LCPs arising in American option pricing. It draws on established matrix splitting and nonlinear vector sequence acceleration ideas (related to Krylov methods) and supports performance claims solely through numerical experiments on a model problem. No equations or derivations reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the central results are empirical observations rather than first-principles claims that collapse to inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identified; the approach builds on existing matrix splitting and extrapolation techniques without introducing new postulated quantities.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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