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arxiv: 2606.26024 · v1 · pith:OBRDZYWQnew · submitted 2026-06-24 · 💱 q-fin.PR

Matrix Approximation of Bachelier Option Prices and Greeks under Stochastic Volatility models

Elisa Al\`os , \`Oscar Bur\'es This is my paper

Pith reviewed 2026-06-25 19:32 UTC · model grok-4.3

classification 💱 q-fin.PR
keywords stochastic volatilityBachelier modeloption pricingGreeksSABR modelmatrix approximationrough Bergomi
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The pith

A linear algebra method computes Bachelier option prices and Greeks for infinitely many strikes from a fixed set of expectations.

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

The paper introduces a numerical technique for pricing options and their Greeks inside stochastic volatility models whose dynamics follow a Bachelier-type process. The technique rests on elementary linear algebra and separates the strike dependence from the underlying expectations, so that prices and sensitivities at every strike inside a convergence interval follow from the same finite collection of model expectations. For the SABR model the authors supply an explicit interval where the approximation is valid, and they demonstrate the procedure on both SABR and rough Bergomi dynamics. A reader would care because the computational effort stays constant no matter how many strikes or Greeks are required.

Core claim

Under stochastic volatility Bachelier-type models, the option price and its Greeks admit a matrix representation that isolates strike dependence; consequently a finite number of expectations suffices to recover the full price and Greek surfaces inside a stated interval of convergence.

What carries the argument

The matrix approximation of the Bachelier pricing operator obtained through elementary linear algebra, which encodes the model expectations once and then evaluates them at any strike by matrix-vector multiplication.

If this is right

  • Prices and Greeks for any number of strikes are obtained from the same fixed number of expectations.
  • An explicit convergence interval is supplied for the SABR model.
  • The same finite-expectation procedure applies to the rough Bergomi model.
  • Greeks are recovered at the same computational cost as the prices themselves.

Where Pith is reading between the lines

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

  • The fixed-cost structure could make repeated surface recalibrations practical in trading systems.
  • Analogous matrix factorizations might be tested on other convex payoffs beyond vanilla calls and puts.
  • Error bounds derived from the matrix norm could be used to choose the number of expectations in advance.

Load-bearing premise

The asset-price process must be a stochastic volatility Bachelier-type model for which the matrix approximation converges inside the interval of interest.

What would settle it

For the SABR model, evaluate the matrix approximation and the true prices at strikes lying outside the derived convergence range and check whether the difference exceeds ordinary numerical tolerance.

Figures

Figures reproduced from arXiv: 2606.26024 by Elisa Al\`os, \`Oscar Bur\'es.

Figure 5.1
Figure 5.1. Figure 5.1: Call prices and implied volatility smile in the SABR model with [PITH_FULL_IMAGE:figures/full_fig_p016_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Delta and Gamma of the options with Mmax = 4 and Nmax = 50. Example 5.2. We choose again the SABR model, but this time we choose as parameters σ0 = 25, ν = 0.4, ρ = −0.5 and 500 options with strikes k ∈ [40, 160] and time to maturity T = 5 years. We apply our method with Mmax = 10, Nmax = 50. In Figures 5.3 and 5.4 we plot the result of the computation of the option prices and the Greeks [PITH_FULL_IMAG… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Call prices and implied volatility smile in the SABR model with [PITH_FULL_IMAGE:figures/full_fig_p017_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Delta and Gamma of the options with Mmax = 10 and Nmax = 50. Example 5.3. We now consider the rough Bergomi model, that is, σ 2 t = σ 2 0 exp  ηWH t − η 2 t 2H 2  . In this case we haven’t derived a result regarding the region of convergence of our method, meaning that for the rough Bergomi model we do not know if our method provides convergent or asymptotic approximation formulas. However, we can stil… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Call prices and implied volatility smile in the rough Bergomi model with [PITH_FULL_IMAGE:figures/full_fig_p019_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Delta and Gamma of the options with Mmax = 6 and Nmax = 50. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_5_6.png] view at source ↗
read the original abstract

In this paper, we present a numerical method for option pricing and the computation of Greeks under stochastic volatility Bachelier-type models, based on elementary linear algebra. The method allows option prices and Greeks to be computed for infinitely many strikes (within a range of convergence) by evaluating only a finite number of expectations, independent of the number of strikes. For the SABR model, we derive an explicit range of convergence. Numerical examples are provided for both the SABR and the rough Bergomi models.

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

0 major / 2 minor

Summary. The manuscript presents a numerical method based on elementary linear algebra for computing Bachelier option prices and Greeks under stochastic volatility models. The central claim is that prices and Greeks for infinitely many strikes (within a derived range of convergence) can be obtained from only a finite number of expectations whose results are then combined via matrix operations, independent of strike count. An explicit convergence interval is supplied for the SABR model, and numerical illustrations are given for both SABR and rough Bergomi.

Significance. If the linear-algebra reduction and convergence statements hold, the method offers a practical efficiency gain for repeated pricing and Greek calculations across strike grids in models without closed forms. The explicit SABR convergence range is a concrete strength that supports falsifiable use of the technique.

minor comments (2)
  1. [Abstract / Introduction] The abstract states that an explicit range of convergence is derived for SABR; the introduction or §2 should restate this range with the precise conditions on model parameters so readers can immediately assess applicability.
  2. [Numerical examples] Numerical examples for SABR and rough Bergomi are mentioned; the corresponding section should include a table or figure comparing the matrix method against a benchmark (e.g., Monte Carlo) with reported error metrics and timings to quantify the claimed independence from strike count.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report correctly identifies the core contribution of the linear-algebra reduction that yields prices and Greeks for infinitely many strikes from a finite set of expectations, together with the explicit SABR convergence interval. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained linear algebra

full rationale

The paper frames its core contribution as a matrix-based numerical method grounded in elementary linear algebra properties, allowing reuse of a finite set of expectations across infinitely many strikes within a derived convergence range (explicitly supplied for SABR). No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the abstract and description present the approach as independent of the target quantities and externally verifiable via the stated convergence interval and numerical examples for SABR and rough Bergomi. This qualifies as a normal non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities; review is abstract-only.

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discussion (0)

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