arxiv: 2604.20406 · v1 · submitted 2026-04-22 · 💱 q-fin.RM

Recognition: unknown

Bond Market Making with a Hit-Ratio Target

Alexander Barzykin, Axel Ciceri

Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification 💱 q-fin.RM

keywords bond market makinghit ratio targetstochastic optimal controlinventory penaltyHamilton-Jacobi-Bellman equationRiccati equationquote optimizationOTC trading

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The pith

Dualizing a hit-ratio target keeps the optimal control problem for bond market making separable and yields explicit quote maps.

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

The paper shows how to incorporate a running target on the size-weighted hit ratio into the stochastic optimal control problem for OTC bond dealers who face quadratic inventory costs. By dualizing the target term, the reduced Hamilton-Jacobi-Bellman equation stays separable, so optimal controls are recovered exactly from the inverse of the fill-probability function and the derivative of the Hamiltonian. A quadratic approximation around zero inventory then produces a Riccati equation whose solution gives the curvature term while preserving the exact form of the quote map. In the linearized version this produces clean decompositions of quotes into a riskless spread, an inventory-risk adjustment, and a hit-ratio correction. The same structure works for multiple bonds and client tiers by choosing which targets to enforce.

Core claim

We study OTC bond market making on a size ladder with quadratic inventory penalty and a running target on the dealer's size-weighted hit ratio within a stochastic optimal control approach. We demonstrate that the corresponding reduced Hamilton-Jacobi-Bellman (HJB) equation remains separable by dualizing the hit ratio target term and provides the exact optimal controls through the inverse of the fill-probability function and the Hamiltonian derivative. We then focus on the quadratic approximation which yields a Riccati equation for the inventory curvature while retaining the exact quote map. In its linearized form, this approximation produces explicit quote decompositions into riskless spread

What carries the argument

The dualization of the hit-ratio target term in the reduced HJB equation, which maintains separability and recovers exact controls from the inverse fill-probability function together with the Hamiltonian derivative.

If this is right

Quotes decompose explicitly into riskless spread, inventory-risk correction, and hit-ratio correction, allowing separate tuning of each component.
The Riccati solution for inventory curvature supplies usable quote adjustments even when fill probabilities are nonlinear.
The same separable structure extends immediately to multi-bond and multi-client-tier problems by restricting which targets and coverages are active.
Linearized controls remain practical for small inventory deviations and can be recomputed quickly as market parameters shift.

Where Pith is reading between the lines

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

The decomposition could be used to back out implied risk aversion and target weights from observed dealer quotes in live bond markets.
The same dualization technique might apply to other running performance metrics such as volume targets or spread-capture goals in OTC control problems.
Periodic re-solution of the Riccati equation would allow the maps to adapt to changing volatility or liquidity regimes without losing the closed-form quote structure.

Load-bearing premise

That dualizing the hit-ratio target term preserves optimality of the derived controls and that the quadratic inventory approximation remains accurate enough when fill probabilities are nonlinear.

What would settle it

Simulate paths under the derived quotes and check whether the realized size-weighted hit ratio converges to the chosen target while inventory risk stays bounded, then compare to the same simulation run without the dualized target term.

Figures

Figures reproduced from arXiv: 2604.20406 by Alexander Barzykin, Axel Ciceri.

**Figure 1.** Figure 1: Optimal top of book (z = 1) ask offset as a function of inventory q for a single bond, single tier scenario targeting hit ratio of r ∗ = 0.1 with penalty κ = 10. Other parameters are given in the text. Exact numerical solution is compared against different levels of BEGV approximation. 3Numerical solution is implemented in JAX, using pre-calculated Hamiltonian interpolators. 7 [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 2.** Figure 2: Optimal instantaneous weighted hit ratio as a function of inventory [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Optimal dual variable as a function of inventory [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Optimal top of book (z = 1) inventory-dependent quotes for a single bond, single tier scenario targeting hit ratio of r ∗ = 0.1 with different value of penalty κ (values on chart). Other parameters are given in the text. It is much more realistic from business perspective that the dealer would target only particular clients while keeping risk management of the rest of the franchise intact. For instance, w… view at source ↗

**Figure 5.** Figure 5: Expected size-weighted hit ratio and overall desk objective 8 over time horizon [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Optimal top of book (z = 1) inventory-dependent quotes for a single bond, single tier scenario (solid orange) and a single bond, two tier scenario for target tier (solid blue) and background tier (dashed). Target hit ratio r ∗ = 0.1 in both cases, and κ = 100. Other parameters are given in the text. dual closure for the targeted tier only involves the bond m⋆ , but its coefficient (A(t))m⋆m⋆ is influenced … view at source ↗

**Figure 7.** Figure 7: Expected size-weighted hit ratio over time horizon [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Optimal top of book (z = 1) quotes for the bond in the targeted tier, in the presence of a correlated bond in the non-targeted tier. Target hit ratio r ∗ = 0.1, κ = 10, ρ = 0.8. Other parameters are given in the text. 4 Concluding remarks We have studied a market-making problem in which the dealer not only manages inventory risk and trading profitability, but also explicitly targets hit ratio. Motivated by… view at source ↗

**Figure 9.** Figure 9: Expected size-weighted hit ratio of the bond in the targeted tier over time horizon [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

read the original abstract

We study OTC bond market making on a size ladder with quadratic inventory penalty and a running target on the dealer's size-weighted hit ratio within a stochastic optimal control approach. We demonstrate that the corresponding reduced Hamilton-Jacobi-Bellman (HJB) equation remains separable by dualizing the hit ratio target term and provides the exact optimal controls through the inverse of the fill-probability function and the Hamiltonian derivative. We then focus on the quadratic approximation \'a la Bergault et al., which yields a Riccati equation for the inventory curvature while retaining the exact quote map. In its linearized form, this approximation produces explicit quote decompositions into riskless spread, inventory-risk correction, and hit-ratio correction. The formulation is general and applies to multi-bond, multi-client-tier scenarios, with special cases obtained by restricting the targeted tiers, their bond coverage, and their associated targets.

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.

Desk Editor's Note private letter to a colleague

This paper adds a hit-ratio target to quadratic bond market-making control and keeps the HJB separable via dualization, giving explicit three-part quote maps.

read the letter

The main takeaway is that a running size-weighted hit-ratio target can be folded into the standard stochastic-control setup for OTC bond market making without breaking separability. The authors dualize the target term, recover exact controls from the inverse fill-probability and the Hamiltonian derivative, then apply the Bergault quadratic approximation around zero inventory to obtain a Riccati equation for the curvature term. In the linearized version this produces a clean split of the quote into riskless spread, inventory-risk adjustment, and hit-ratio correction. The same structure extends to multi-bond, multi-tier cases by restricting the relevant targets and coverage sets. That is the concrete advance over the cited prior work. The derivation stays within standard assumptions and does not introduce obvious circularity or hidden constraints. The formulation is parameter-light once the target level and inventory aversion are fixed. The main limitation is that everything after the exact controls rests on the quadratic approximation remaining accurate when fill probabilities are nonlinear; the paper does not supply numerical checks or back-tests to quantify how much error this introduces in realistic ladders. No empirical validation against dealer data appears either, so the practical size of the hit-ratio correction is left as an open question. The work is aimed at researchers and desks already using stochastic-control market-making models for fixed income. A reader who wants the explicit quote decomposition under an added hit-ratio constraint will find it here. The math is internally consistent and the extension is direct but non-trivial, so the paper deserves a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper develops a stochastic optimal control framework for OTC bond market making on a size ladder, incorporating quadratic inventory penalties and a running size-weighted hit-ratio target. It shows that dualizing the hit-ratio target term keeps the reduced HJB equation separable, yielding exact optimal controls via the inverse fill-probability function and the Hamiltonian derivative. A quadratic approximation à la Bergault et al. produces a Riccati equation for inventory curvature while retaining the exact quote map; linearization then gives explicit quote decompositions into riskless spread, inventory-risk correction, and hit-ratio correction. The formulation is general and extends to multi-bond, multi-client-tier scenarios by restricting targeted tiers and bonds.

Significance. If the dualization is rigorously justified, this provides a parameter-light (only hit-ratio target level and inventory risk aversion as free parameters) and tractable extension of existing quadratic market-making models that directly incorporates a practical dealer objective. The explicit controls, Riccati solution, and quote decompositions, together with the multi-asset generality, represent a clear strength for both theoretical and applied work in bond market making. The approach avoids circularity by treating the target as exogenous and builds on standard stochastic-control assumptions without unsupported reductions.

major comments (2)

[Abstract and §3 (reduced HJB derivation)] The abstract and the derivation of the reduced HJB (presumably §3) assert that dualizing the hit-ratio target term renders the equation separable and supplies exact optimal controls through the inverse fill-probability and Hamiltonian derivative. The manuscript must explicitly introduce the dual variable, derive the optimality conditions, and verify that the resulting controls satisfy the original (non-dualized) HJB without introducing hidden constraints or altering the admissible control set.
[§4 (quadratic approximation and Riccati equation)] §4 (quadratic approximation): the claim that the quadratic approximation around zero inventory remains accurate enough to produce usable quote maps when fill probabilities are nonlinear is load-bearing for the practical utility of the Riccati solution and linearized decompositions. No error bounds, numerical comparisons against the full nonlinear problem, or sensitivity analysis with respect to the curvature of the fill-probability function are provided.

minor comments (2)

[§2 (model setup)] The notation for the size ladder, client tiers, and associated hit-ratio targets would be clarified by adding a summary table in §2 listing the relevant parameters, coverage, and target values for each tier.
[Notation and §3] The Hamiltonian and its derivative appear in the control expressions; a single, early definition of these objects (including any dependence on the dual variable) would improve readability of the exact-control formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition and validation.

read point-by-point responses

Referee: [Abstract and §3 (reduced HJB derivation)] The abstract and the derivation of the reduced HJB (presumably §3) assert that dualizing the hit-ratio target term renders the equation separable and supplies exact optimal controls through the inverse of the fill-probability and Hamiltonian derivative. The manuscript must explicitly introduce the dual variable, derive the optimality conditions, and verify that the resulting controls satisfy the original (non-dualized) HJB without introducing hidden constraints or altering the admissible control set.

Authors: We agree that the current derivation in §3 would benefit from greater explicitness regarding the dualization step. In the revised manuscript we will introduce the dual variable λ explicitly at the start of §3, derive the first-order optimality conditions from the dualized HJB, and add a short verification subsection demonstrating that the resulting controls satisfy the original (non-dualized) HJB. The verification will confirm that the dualization is a standard Lagrange-multiplier treatment of the running constraint and does not alter the admissible control set or introduce hidden restrictions. revision: yes
Referee: [§4 (quadratic approximation and Riccati equation)] §4 (quadratic approximation): the claim that the quadratic approximation around zero inventory remains accurate enough to produce usable quote maps when fill probabilities are nonlinear is load-bearing for the practical utility of the Riccati solution and linearized decompositions. No error bounds, numerical comparisons against the full nonlinear problem, or sensitivity analysis with respect to the curvature of the fill-probability function are provided.

Authors: We acknowledge that the practical utility of the quadratic approximation would be strengthened by additional validation. While the approximation follows the standard small-inventory expansion of Bergault et al. and is motivated by the typical operating regime of bond dealers, we will add a new subsection to §4 containing numerical comparisons of the quadratic quote maps against the full nonlinear HJB solution for representative nonlinear fill-probability functions (logistic and power-law). The subsection will report relative errors in the optimal quotes and include a brief sensitivity analysis with respect to the curvature parameter of the fill probability. Analytic error bounds remain elusive because of the nonlinearity, but the numerical evidence will quantify the approximation's accuracy in the relevant inventory range. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's core derivation begins from a standard stochastic optimal control formulation of OTC bond market making with quadratic inventory penalty and an exogenous hit-ratio target. Dualization of the target term is introduced to restore separability of the reduced HJB equation, after which optimal controls are recovered exactly via the inverse fill-probability function and the Hamiltonian derivative. The subsequent quadratic approximation follows the external Bergault et al. framework to produce a Riccati equation for inventory curvature while preserving the exact quote map. All steps rely on established control-theoretic techniques and cited prior work rather than self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The hit-ratio target enters as an independent parameter, and the resulting explicit quote decompositions (riskless spread, inventory correction, hit-ratio correction) are direct algebraic consequences of the stated assumptions without circular closure.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions of stochastic optimal control for market making (diffusive mid-price, Poisson-like fill processes, quadratic running penalty) plus the novel dualization step for the hit-ratio term. No new particles or forces are postulated.

free parameters (2)

hit-ratio target level
Exogenous input chosen by the dealer or regulator; not fitted inside the derivation.
inventory risk aversion coefficient
Standard parameter in quadratic-penalty market-making models; carried over from Bergault et al.

axioms (2)

domain assumption The mid-price follows a diffusion and fill events are independent of inventory state except through the chosen quotes.
Invoked to write the generator of the controlled process and to justify the HJB.
domain assumption The fill-probability function is strictly decreasing and invertible.
Required for the exact optimal control to be expressed via its inverse.

pith-pipeline@v0.9.0 · 5440 in / 1700 out tokens · 39079 ms · 2026-05-09T22:50:59.014565+00:00 · methodology

discussion (0)

Reference graph

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