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arxiv: 2606.31387 · v1 · pith:4ATXSTHXnew · submitted 2026-06-30 · 💱 q-fin.TR

Signature-Based Optimal Execution for Statistical Arbitrage with Path-Dependent Trading Signals

Gianmarco Morbelli , Sven Karbach , Mike Derksen This is my paper

Pith reviewed 2026-07-01 02:58 UTC · model grok-4.3

classification 💱 q-fin.TR
keywords signature methodsoptimal executionstatistical arbitragepath-dependent signalsquadratic programmingtrading signalspairs trading
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The pith

Signature-linear trading speeds turn path-dependent execution into a finite-dimensional concave quadratic program.

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

The paper places both the alpha process and trading speed as linear functionals of the truncated signature of a time-augmented market path. This common basis lets the execution rule respond to the realised signal history while incorporating temporary impact, inventory risk, terminal liquidation and approximate dollar neutrality. The central result is a quadratic reduction theorem showing that any execution problem restricted to this policy class collapses to a concave quadratic program over the signature coefficients. Synthetic tests on mean-reverting log-spread dynamics and a historical equity pairs backtest both show the fitted policy delivers higher return on turnover than a classical z-score threshold rule.

Core claim

Within the class of signature-linear trading speeds, the restricted path-dependent execution problem becomes a finite-dimensional concave quadratic programme in the policy coefficients.

What carries the argument

The quadratic reduction theorem that converts the signature-linear policy class into a concave quadratic program.

If this is right

  • The execution problem becomes solvable with standard quadratic programming solvers.
  • The same workflow applies directly to mean-reverting spread models and to historical equity pairs trading.
  • Fitted policies achieve higher return on turnover than z-score benchmarks while enforcing dollar neutrality.
  • Signal generation and execution share the same truncated signature basis.

Where Pith is reading between the lines

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

  • The truncation level of the signature trades off expressivity against the dimension of the resulting quadratic program.
  • Real-time implementation would require streaming signature computations that keep pace with market data.
  • The reduction may extend to multi-asset portfolios by enlarging the signature feature space.
  • Signature-linear policies could serve as a common language for comparing execution rules across different predictive signals.

Load-bearing premise

Both the alpha process and the trading speed can be adequately represented as linear functionals of the truncated signature of the time-augmented market path.

What would settle it

A simulation or backtest in which the optimal signature-linear policy fails to outperform the z-score benchmark or the quadratic program does not recover the claimed optimum.

Figures

Figures reproduced from arXiv: 2606.31387 by Gianmarco Morbelli, Mike Derksen, Sven Karbach.

Figure 1
Figure 1. Figure 1: Synthetic benchmark against a classical z-score pairs-trading rule. The two assets follow the common-trend log-spread model of Section 3.1. We display the log spread, the z-score signal, price-scaled trading speed and inventory paths, and cumulative PnL for the signature strategy and the benchmark of Section 3.2. The plotted values are averages over 5000 test paths after a 600-step warm-up window used to e… view at source ↗
Figure 2
Figure 2. Figure 2: The historical Shell–BP deployment backtest on a single out-of-sample test window. The panels show the log spread, the z-score signal, price-scaled trading and inventory exposure in the two legs, and cumulative PnL of the signature strategy against the z-score benchmark of Section 3.2. The signature policy is calibrated on past fixed-length trading windows and then applied on the test window. The figure il… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of the empirically estimated Gram matrices Gψ and Gr to their closed-form OU moment targets, evaluated on the simulator time grid so that the comparison isolates Monte Carlo sampling error. The error decays at the O(M−1/2 ) Monte Carlo reference rate (grey), consistent with consistent estimation of the OU-projected blocks; imposing the spread model removes this sampling noise for those blocks. … view at source ↗
Figure 7
Figure 7. Figure 7: Effect of signature truncation order N. Out-of￾sample per-path distributions (boxes: 25–75%, whiskers: 5– 95%, diamonds: means) of the reduced objective, terminal wealth, terminal inventory norm, and turnover at N = 1 versus N = 2. Adding the second-order block (time ordering, L´evy area) shifts the reduced-objective and terminal-wealth accounting distributions upward, while the turnover and inventory dist… view at source ↗
Figure 5
Figure 5. Figure 5: Quadratic-reduction check. The matrix-form value θ ⊤Aθ + b ⊤θ + c versus the directly path-simulated objective Jsim(θ), for the optimiser θ ∗ (star) and 48 random coefficient vectors. All points lie on the identity line (maximum relative error 1.5 × 10−7 ), checking the matrix implementation of Theorem 2.9 numerically [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: sorted signed eigenvalues of −Aˆ (positive semidefinite under concavity) on a signed-log scale, with zero and the ridge level λridge marked. Plotting signed eigenvalues keeps any concavity violation visible. Right: heatmap of Aˆ showing its Kronecker block structure from signature Gram matrices and execution penalties. therefore makes the linear solve well posed precisely in those near-null signature… view at source ↗
Figure 8
Figure 8. Figure 8: reports the resulting mean accounting re￾turn on turnover under the canonical synthetic workflow. The purpose of this diagnostic is not to optimise these parameters, but to identify which modelling parameters materially affect out-of-sample accounting metrics. The model is especially sensitive to the ridge parameter, which controls the use of near-null directions in the empirical curvature matrix: too litt… view at source ↗
read the original abstract

We develop a signature-based framework for optimal execution in statistical arbitrage strategies with path-dependent predictive signals. Both the alpha process and the trading speed are modelled as linear functionals of the truncated signature of a time-augmented market path, placing signal generation and execution on the same truncated signature basis. This allows the trading rule to react to the realised history of the signal while accounting for temporary impact, inventory exposure, terminal liquidation, and approximate dollar neutrality The main contribution is a quadratic reduction theorem: within the class of signature-linear trading speeds, the restricted path-dependent execution problem becomes a finite-dimensional concave quadratic programme in the policy coefficients. After running synthetic experiments under a mean-reverting log-spread model, we find that the fitted policy achieves a higher return on turnover than a z-score classical threshold benchmark. We shows how the same workflow can be deployed on a historical equity pairs-trading backtest, where the fitted signature policy again outperforms the benchmark in accounting terms.

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

3 major / 2 minor

Summary. The manuscript develops a signature-based framework for optimal execution in statistical arbitrage with path-dependent predictive signals. Both the alpha process and trading speeds are represented as linear functionals of the truncated signature of a time-augmented market path. The central result is a quadratic reduction theorem showing that, within this signature-linear class, the execution problem (incorporating temporary impact, inventory, terminal liquidation, and approximate dollar neutrality) reduces to a finite-dimensional concave quadratic program over the policy coefficients. Synthetic experiments under a mean-reverting log-spread model and one equity pairs-trading backtest report that the fitted signature policy outperforms a classical z-score threshold benchmark in return-on-turnover and accounting metrics.

Significance. If the quadratic reduction theorem is correct and the signature-linear class is sufficiently expressive, the approach supplies a computationally tractable route to optimize genuinely path-dependent execution rules while remaining inside a concave quadratic program. This could usefully connect rough-path signature methods to practical stat-arb execution, especially when signals exhibit memory. The reported outperformance in the mean-reverting synthetic setting is consistent with low-order signatures capturing the dynamics, but broader applicability hinges on the class being dense enough for the true optimum.

major comments (3)
  1. [Section stating the quadratic reduction theorem] The quadratic reduction theorem is the load-bearing mathematical claim. The manuscript must supply a self-contained derivation (or at least the key steps establishing concavity and finite-dimensionality) rather than merely asserting the result; without it, the reduction from the infinite-dimensional path-dependent problem to the QP cannot be verified.
  2. [Synthetic experiments and backtest sections] The empirical claim that the signature-linear class captures relevant dynamics rests on outperformance versus a z-score benchmark in a mean-reverting log-spread model and a single equity backtest. No truncation-level sensitivity, no comparison against richer (non-linear or non-signature) policy classes, and no argument that the linear functional form can represent the optimal feedback law under impact and inventory penalties are provided; these omissions make the practical relevance of the restricted class untested.
  3. [Fitting and evaluation procedure] Policy coefficients are fitted directly to the same objective function later used to evaluate performance. This introduces dependence between the reported gains and the in-sample optimization procedure; out-of-sample or cross-validated evaluation would be required to substantiate generalization.
minor comments (2)
  1. [Abstract] Abstract contains the grammatical error 'We shows'.
  2. [Model setup] Notation for the time-augmented path and the precise truncation level should be stated once at the beginning and used consistently.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Section stating the quadratic reduction theorem] The quadratic reduction theorem is the load-bearing mathematical claim. The manuscript must supply a self-contained derivation (or at least the key steps establishing concavity and finite-dimensionality) rather than merely asserting the result; without it, the reduction from the infinite-dimensional path-dependent problem to the QP cannot be verified.

    Authors: We agree that a self-contained derivation is necessary for verifiability. In the revised manuscript we will insert a dedicated subsection containing the full proof of the quadratic reduction theorem. The proof will explicitly derive the finite-dimensional quadratic objective in the signature coefficients, confirm its concavity under the stated impact and penalty terms, and detail the steps that map the path-dependent execution problem onto the finite-dimensional QP. revision: yes

  2. Referee: [Synthetic experiments and backtest sections] The empirical claim that the signature-linear class captures relevant dynamics rests on outperformance versus a z-score benchmark in a mean-reverting log-spread model and a single equity backtest. No truncation-level sensitivity, no comparison against richer (non-linear or non-signature) policy classes, and no argument that the linear functional form can represent the optimal feedback law under impact and inventory penalties are provided; these omissions make the practical relevance of the restricted class untested.

    Authors: The experiments illustrate the workflow and the practical advantage of the reduction theorem rather than claim that the signature-linear class is universally optimal. We will add truncation-level sensitivity plots for both the synthetic and backtest settings. Direct comparisons with non-linear policy classes lie outside the paper’s scope, which focuses on tractable optimization inside the signature-linear family; we will clarify this motivation and note the universal-approximation property of signatures as the theoretical justification for the functional form. We acknowledge that an explicit argument showing the linear class can recover the true optimum under impact and inventory penalties is absent and will discuss this limitation in the revised text. revision: partial

  3. Referee: [Fitting and evaluation procedure] Policy coefficients are fitted directly to the same objective function later used to evaluate performance. This introduces dependence between the reported gains and the in-sample optimization procedure; out-of-sample or cross-validated evaluation would be required to substantiate generalization.

    Authors: We accept that the current evaluation is performed in-sample. For the synthetic experiments the data-generating process is known, so the optimization recovers the best policy inside the class for that exact model; we will state this explicitly. For the equity backtest we will implement a rolling-window or cross-validated evaluation procedure in the revision to provide evidence of out-of-sample performance. revision: yes

Circularity Check

0 steps flagged

No circularity: mathematical reduction is self-contained within explicitly restricted policy class; empirical results are direct comparisons to external benchmark.

full rationale

The core contribution is a quadratic reduction theorem that converts the execution problem into a concave QP strictly inside the signature-linear policy class; this is a finite-dimensional algebraic identity given the modeling ansatz and does not rely on data fitting or self-citation. The paper states the modeling choice (alpha and speed as linear signature functionals) upfront rather than smuggling it via prior work. Empirical sections report the performance of the QP solution versus a z-score threshold benchmark on both synthetic mean-reverting paths and a historical pairs backtest; these are standard out-of-class comparisons, not predictions of quantities already used as fitting targets. No load-bearing self-citation, no uniqueness theorem imported from the same authors, and no renaming of known results appear in the provided text. The derivation chain therefore remains independent of its own fitted outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework rests on the assumption that path-dependent functionals can be approximated by linear maps on truncated signatures; no new physical entities are introduced, but the truncation level and policy coefficients are chosen or fitted.

free parameters (2)
  • signature truncation level
    Determines the finite dimension of the basis used for both signal and policy; chosen to balance expressivity and computation.
  • policy coefficients
    Optimized within the quadratic program to maximize the objective that includes return on turnover.
axioms (1)
  • standard math The signature transform provides a faithful linear representation of path-dependent functionals up to the chosen truncation.
    Invoked to place both alpha and trading speed on the same basis.

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discussion (0)

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Reference graph

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