Optimal Adaptive Market Making: A Theoretical Framework for High-Yield Liquidity Provision in Perpetual Futures Markets

# A Theoretical Framework for High-Yield Liquidity Provision in Perpetual Futures Markets
Source: [https://arxiv.org/html/2607.11888](https://arxiv.org/html/2607.11888)
## Optimal Adaptive Market Making: A Theoretical Framework for High\-Yield Liquidity Provision in Perpetual Futures Markets

###### Abstract

We develop a rigorous theoretical framework for optimal market making in perpetual futures markets with zero maker fees, addressing the question:*under what conditions can a market maker achieve annualized returns exceeding 50%–200% on deployed capital?*We model the market maker’s problem as a stochastic optimal control problem on a filtered probability space, where the controls are adaptive bid–ask spreads and inventory hedging decisions across two exchanges\. Our first contribution is a complete*PnL decomposition theorem*that separates market\-making revenue into spread income, adverse selection loss, inventory carrying cost, hedging friction, and funding rate exposure, each with explicit dependence on market microstructure parameters\. Second, we derive the Hamilton–Jacobi–Bellman equation for the joint spread–inventory–hedging control problem under CARA utility and obtain a verification theorem guaranteeing optimality of the candidate solution\. Third, we establish*High\-APY Regime Theorems*that characterize, in terms of five dimensionless parameters, the precise regions where annualized Sharpe ratios exceed given thresholds, culminating in a*Master APY Formula*that unifies all cost channels into a single closed\-form expression\. Fourth, we analyze the*zero\-fee economics*unique to decentralized perpetual exchanges, proving that the elimination of maker fees expands the profitable parameter space and constitutes an economic moat, and derive optimal entry–exit thresholds with hysteresis for non\-stationary markets\. Fifth, we derive optimal*cross\-exchange hedging*policies incorporating funding rate dynamics, basis risk with imperfect correlation, and a hedge regime classification that provides a principled trichotomy for operational decision\-making\. Sixth, we introduce a*robustness margin*that quantifies how many standard deviations of parameter uncertainty a strategy can absorb while remaining profitable—a critical metric for practitioners facing non\-stationary microstructure\. Seventh, we establish an*exponential drawdown probability bound*linking maximum drawdown to the Sharpe ratio and prove that the product APY×\\timesVaR is a universal constant independent of strategy parameters, revealing the fundamental risk–return identity for market making\. Eighth, we prove that the*ergodic inventory distribution*under optimal control is Gaussian with varianceσq2=λ¯∗/\(2​η​k\)\\sigma\_\{q\}^\{2\}=\\bar\{\\lambda\}^\{\*\}/\(2\\eta k\), yielding closed\-form expressions for the optimal inventory penalization and long\-run cost rate\. We also establish a*Bayesian sequential estimation*framework for the informed trading fractionπ\\piwithO​\(n−1/2\)O\(n^\{\-1/2\}\)convergence guarantees, enabling fully adaptive, self\-tuning market\-making strategies\. Ninth, we extend the single\-pair framework to*multi\-pair portfolio allocation*, deriving the optimal Sharpe\-maximizing capital allocation acrossNNcorrelated trading pairs and proving that the diversification benefit saturates at1/ρ¯1/\\sqrt\{\\bar\{\\rho\}\}for equi\-correlated pairs\. Numerical analysis with twenty\-two figures reveals phase transitions between profitable and unprofitable regimes, with sharp boundaries governed by the ratio of adverse selection to spread capture\. Our framework unifies and extends the classical Avellaneda–Stoikov model, the Guéant–Lehalle–Fernandez\-Tapia inventory penalization approach, and the Glosten–Milgrom adverse selection paradigm into a single coherent theory with 65 references applicable to modern decentralized venue microstructure\.

Keywords:Market making, stochastic optimal control, adverse selection, perpetual futures, zero\-fee regime, HJB equation, high\-frequency trading, funding rate, drawdown risk\.

JEL Classification:G12, G13, C61, D82\.

## 1Introduction

Market making—the provision of continuous two\-sided liquidity through limit orders—is among the oldest and most studied problems in financial economics\. The classical theory, initiated byHo and Stoll \[[22](https://arxiv.org/html/2607.11888#bib.bib22)\]and formalized in the continuous\-time framework ofAvellaneda and Stoikov \[[4](https://arxiv.org/html/2607.11888#bib.bib4)\], characterizes the market maker \(MM\) as an agent who quotes bid and ask prices to maximize expected utility of terminal wealth while managing inventory risk\[[18](https://arxiv.org/html/2607.11888#bib.bib18)\]\. The past decade has witnessed a fundamental shift in market structure with the emergence of decentralized exchanges \(DEXs\) that operate on blockchain infrastructure, introducing novel fee structures, latency environments, and adverse selection dynamics that challenge classical assumptions\.

### 1\.1Motivation

Perpetual futures markets on decentralized platforms have grown to represent a significant fraction of total crypto derivatives volume, with daily volumes exceeding $10 billion across major venues as of 2025\[[21](https://arxiv.org/html/2607.11888#bib.bib21)\]\. A distinguishing feature of several decentralized perpetual exchanges is the*zero maker fee*regime: limit orders that add liquidity incur no execution fee \(ϕm=0\\phi\_\{m\}=0\), while taker orders pay a positive fee \(ϕt\>0\\phi\_\{t\}\>0\)\. This stands in stark contrast to centralized exchanges \(CEXs\), where maker fees typically range from 1\.5 to 5 basis points\.

The zero\-fee regime creates a fundamentally different economic landscape for market makers\. On a CEX, the minimum viable spread must exceed2​ϕm2\\phi\_\{m\}merely to cover fee costs, independent of adverse selection\. Under zero fees, this floor vanishes, and the sole binding constraint becomes adverse selection—the expected loss from trading against informed or faster counterparties\. This observation motivates our central question:

> *Under what microstructure conditions can a market maker in a zero\-fee perpetual futures market achieve sustained annualized returns \(APY\) of 50% or more on deployed capital, and what is the optimal adaptive strategy that attains these returns?*

### 1\.2Related Literature

Our work connects to several strands of literature\.

##### Classical market making theory\.

The foundational framework ofHo and Stoll \[[22](https://arxiv.org/html/2607.11888#bib.bib22)\]models a risk\-averse dealer who optimizes bid and ask prices under inventory uncertainty\.Avellaneda and Stoikov \[[4](https://arxiv.org/html/2607.11888#bib.bib4)\]extend this to continuous time with Poisson fill arrivals and CARA utility, deriving closed\-form optimal spreads as functions of inventory, volatility, and time horizon\.Guéant et al\. \[[20](https://arxiv.org/html/2607.11888#bib.bib20)\]introduce an inventory penalization approach that avoids the finite\-horizon constraint, yielding stationary optimal policies\.Cartea et al\. \[[9](https://arxiv.org/html/2607.11888#bib.bib9)\]provide a comprehensive treatment of algorithmic market making within the framework of stochastic optimal control\.

##### Adverse selection in market making\.

Glosten and Milgrom \[[17](https://arxiv.org/html/2607.11888#bib.bib17)\]andCopeland and Galai \[[10](https://arxiv.org/html/2607.11888#bib.bib10)\]establish the fundamental connection between adverse selection and bid–ask spreads: a market maker facing informed traders must widen spreads to compensate for expected losses on adversely selected fills\.Easley and O’Hara \[[14](https://arxiv.org/html/2607.11888#bib.bib14)\]introduce the PIN \(probability of informed trading\) model\.Foucault et al\. \[[15](https://arxiv.org/html/2607.11888#bib.bib15)\]analyze “toxic arbitrage” in fragmented markets, where speed advantages create latency\-based adverse selection\.Glosten and Harris \[[16](https://arxiv.org/html/2607.11888#bib.bib16)\]andHuang and Stoll \[[23](https://arxiv.org/html/2607.11888#bib.bib23)\]develop econometric decompositions of the bid–ask spread into adverse selection, inventory, and order\-processing components\.

##### DEX and DeFi market making\.

The DeFi literature has primarily focused on automated market makers \(AMMs\) such as Uniswap\[[1](https://arxiv.org/html/2607.11888#bib.bib1)\], analyzing impermanent loss\[[29](https://arxiv.org/html/2607.11888#bib.bib29)\], MEV extraction\[[11](https://arxiv.org/html/2607.11888#bib.bib11)\], and AMM design\[[2](https://arxiv.org/html/2607.11888#bib.bib2),[30](https://arxiv.org/html/2607.11888#bib.bib30)\]\. However, order\-book\-based DEXs—which support traditional limit order placement—have received comparatively less theoretical attention\. Recent work byMa et al\. \[[26](https://arxiv.org/html/2607.11888#bib.bib26)\]examines trader behavior on decentralized perpetual exchanges, andLalor and Swishchuk \[[24](https://arxiv.org/html/2607.11888#bib.bib24)\]simulate adverse selection in order\-book settings\. Our paper fills a gap by providing a rigorous optimal control theory for market making on order\-book DEXs with zero maker fees\.

##### Cross\-exchange market making\.

The problem of market making across multiple venues has been studied byBergault et al\. \[[5](https://arxiv.org/html/2607.11888#bib.bib5)\]\(multi\-asset\) andGuéant \[[19](https://arxiv.org/html/2607.11888#bib.bib19)\]\(general optimal MM\)\.Budish et al\. \[[7](https://arxiv.org/html/2607.11888#bib.bib7)\]andAquilina et al\. \[[3](https://arxiv.org/html/2607.11888#bib.bib3)\]analyze latency arbitrage in fragmented markets, which is directly relevant to cross\-exchange inventory hedging\. Our contribution is to embed the cross\-exchange hedging decision within the stochastic control problem, deriving joint optimal spread\-hedge policies\.

### 1\.3Contributions

This paper makes six principal contributions:

1. \(i\)PnL Decomposition Theorem\(Section[3](https://arxiv.org/html/2607.11888#S3)\): We provide a rigorous decomposition of market\-making PnL into four components—spread income, latency adverse selection, informed adverse selection, and inventory cost—each expressed as explicit functionals of market microstructure parameters\.
2. \(ii\)Optimal Spread–Inventory–Hedge Control\(Section[4](https://arxiv.org/html/2607.11888#S4)\): We formulate the MM’s problem as a three\-dimensional stochastic control problem \(spread, inventory gating, hedge timing\) and derive the HJB equation\. We prove a verification theorem showing that the candidate value function satisfies the HJB equation and the associated optimal policy is indeed optimal\.
3. \(iii\)High\-APY Regime Characterization\(Section[5](https://arxiv.org/html/2607.11888#S5)\): We derive necessary and sufficient conditions on five dimensionless parameters for the MM to achieve a target APY\. This yields a “phase diagram” of the parameter space separating profitable from unprofitable regimes\.
4. \(iv\)Zero\-Fee Economics\(Section[6](https://arxiv.org/html/2607.11888#S6)\): We analyze how the zero\-fee regime expands the profitable parameter space relative to standard CEX fee structures, quantifying the “fee advantage” in terms of additional tradeable markets and higher equilibrium fill rates\.
5. \(v\)Cross\-Exchange Optimal Hedging\(Section[7](https://arxiv.org/html/2607.11888#S7)\): We derive the optimal dynamic hedging policy when the MM can offset inventory on a second exchange at a known transaction cost, including the optimal hedge threshold, hedge ratio, dynamic hedge timing, and the impact on overall APY\.
6. \(vi\)Regime Robustness Under Parameter Uncertainty\(Section[5](https://arxiv.org/html/2607.11888#S5)\): We introduce a*robustness margin*ℛ\\mathcal\{R\}that quantifies the number of standard deviations of parameter uncertainty a strategy can absorb before becoming unprofitable\. This worst\-case APY analysis, based on ellipsoidal uncertainty sets and the Cauchy–Schwarz inequality, provides practitioners with a practical tool for stress\-testing market\-making strategies against non\-stationary microstructure conditions\.

### 1\.4Paper Organization

Section[2](https://arxiv.org/html/2607.11888#S2)develops the market model, including price dynamics, fill processes, and fee structures\. Section[3](https://arxiv.org/html/2607.11888#S3)presents the adverse selection theory and PnL decomposition\. Section[4](https://arxiv.org/html/2607.11888#S4)formulates and solves the stochastic optimal control problem\. Section[5](https://arxiv.org/html/2607.11888#S5)characterizes the high\-APY regime\. Section[6](https://arxiv.org/html/2607.11888#S6)analyzes zero\-fee economics\. Section[7](https://arxiv.org/html/2607.11888#S7)treats cross\-exchange hedging\. Section[8](https://arxiv.org/html/2607.11888#S8)presents numerical analysis of the parameter space\. Section[9](https://arxiv.org/html/2607.11888#S9)concludes\. All proofs are in the Appendix\.

## 2Market Model

We develop a continuous\-time model of market making on a decentralized perpetual futures exchange \(DEX\-A\) with a centralized exchange \(CEX\-B\) serving as the reference venue\. The model is defined on a filtered probability space\(Ω,ℱ,\{ℱt\}t≥0,ℙ\)\(\\Omega,\\mathcal\{F\},\\\{\\mathcal\{F\}\_\{t\}\\\}\_\{t\\geq 0\},\\mathbb\{P\}\)satisfying the usual conditions\.

### 2\.1Price Dynamics

###### Assumption 2\.1\(Reference Price\)\.

The CEX\-B reference mid\-priceStS\_\{t\}follows an arithmetic Brownian motion:

d​St=μ​d​t\+σ​d​Wt,\\mathrm\{d\}S\_\{t\}=\\mu\\,\\mathrm\{d\}t\+\\sigma\\,\\mathrm\{d\}W\_\{t\},\(1\)whereμ∈ℝ\\mu\\in\\mathbb\{R\}is the drift \(set to zero at high\-frequency time scales by efficient market arguments\),σ\>0\\sigma\>0is the instantaneous volatility, and\(Wt\)t≥0\(W\_\{t\}\)\_\{t\\geq 0\}is a standard Brownian motion\.

###### Assumption 2\.3\(DEX–CEX Price Link\)\.

The DEX\-A mid\-priceS~t\\tilde\{S\}\_\{t\}is linked to the CEX\-B reference price through a structural premium:

S~t=St\+βt,\\tilde\{S\}\_\{t\}=S\_\{t\}\+\\beta\_\{t\},\(2\)where the premium processβt\\beta\_\{t\}satisfies the Ornstein–Uhlenbeck dynamics:

d​βt=−κ​\(βt−β¯\)​d​t\+σβ​d​Wtβ,\\mathrm\{d\}\\beta\_\{t\}=\-\\kappa\(\\beta\_\{t\}\-\\bar\{\\beta\}\)\\,\\mathrm\{d\}t\+\\sigma\_\{\\beta\}\\,\\mathrm\{d\}W\_\{t\}^\{\\beta\},\(3\)with long\-run meanβ¯∈ℝ\\bar\{\\beta\}\\in\\mathbb\{R\}, mean\-reversion speedκ\>0\\kappa\>0, premium volatilityσβ\>0\\sigma\_\{\\beta\}\>0, andWtβW\_\{t\}^\{\\beta\}independent ofWtW\_\{t\}\.

The OU premium captures the empirical observation that DEX prices systematically deviate from CEX prices due to differences in liquidity, funding rates, and participant composition, but these deviations are transient and mean\-reverting\.

### 2\.2Fee Structure

###### Definition 2\.4\(Fee Regime\)\.

A*fee regime*is characterized by a pair\(ϕm,ϕt\)\(\\phi\_\{m\},\\phi\_\{t\}\)where:

- •ϕm≥0\\phi\_\{m\}\\geq 0is the maker fee \(charged on limit order fills\),
- •ϕt\>0\\phi\_\{t\}\>0is the taker fee \(charged on market order fills\)\.

We define three canonical regimes:

1. 1\.Zero\-fee regime:ϕmDEX=0\\phi\_\{m\}^\{\\mathrm\{DEX\}\}=0,ϕtDEX\>0\\phi\_\{t\}^\{\\mathrm\{DEX\}\}\>0\(DEX\-A\)\.
2. 2\.Standard CEX regime:ϕmCEX\>0\\phi\_\{m\}^\{\\mathrm\{CEX\}\}\>0,ϕtCEX\>ϕmCEX\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\>\\phi\_\{m\}^\{\\mathrm\{CEX\}\}\(CEX\-B\)\.
3. 3\.Maker rebate regime:ϕm<0\\phi\_\{m\}<0\(rebate\),ϕt\>0\\phi\_\{t\}\>0\(some CEXs\)\.

Table 1:Representative fee structures across venue types \(basis points\)
### 2\.3Market Maker’s Actions

The market maker continuously chooses:

1. 1\.Bid half\-spreadδtb≥0\\delta\_\{t\}^\{b\}\\geq 0: bid pricePtb=S~t−δtbP\_\{t\}^\{b\}=\\tilde\{S\}\_\{t\}\-\\delta\_\{t\}^\{b\}\.
2. 2\.Ask half\-spreadδta≥0\\delta\_\{t\}^\{a\}\\geq 0: ask pricePta=S~t\+δtaP\_\{t\}^\{a\}=\\tilde\{S\}\_\{t\}\+\\delta\_\{t\}^\{a\}\.
3. 3\.Hedge decisionht∈\{0,±1\}h\_\{t\}\\in\\\{0,\\pm 1\\\}: whether to execute a hedging trade on CEX\-B\.

The controls\(δb,δa,h\)\(\\delta^\{b\},\\delta^\{a\},h\)areℱt\\mathcal\{F\}\_\{t\}\-adapted processes\.

### 2\.4Fill Dynamics

###### Assumption 2\.5\(Poisson Fill Arrivals\)\.

Fill arrivals on the bid and ask sides are independent doubly stochastic Poisson processesNtbN\_\{t\}^\{b\}andNtaN\_\{t\}^\{a\}with stochastic intensities:

λb​\(δtb\)=Λ​exp⁡\(−k​δtb\),λa​\(δta\)=Λ​exp⁡\(−k​δta\),\\lambda^\{b\}\(\\delta\_\{t\}^\{b\}\)=\\Lambda\\exp\(\-k\\delta\_\{t\}^\{b\}\),\\qquad\\lambda^\{a\}\(\\delta\_\{t\}^\{a\}\)=\\Lambda\\exp\(\-k\\delta\_\{t\}^\{a\}\),\(4\)whereΛ\>0\\Lambda\>0is the baseline fill rate \(fills per unit time when the spread is zero\) andk\>0k\>0is the fill\-rate sensitivity to spread width\.

###### Definition 2\.7\(Fill Rate and Fill Frequency\)\.

The*fill rate*λ¯\\bar\{\\lambda\}is the expected number of fills per unit time:

λ¯=λb​\(δb⁣∗\)\+λa​\(δa⁣∗\),\\bar\{\\lambda\}=\\lambda^\{b\}\(\\delta^\{b\*\}\)\+\\lambda^\{a\}\(\\delta^\{a\*\}\),\(5\)whereδb⁣∗,δa⁣∗\\delta^\{b\*\},\\delta^\{a\*\}are the optimal half\-spreads\. The*fill frequency*f=λ¯/Λf=\\bar\{\\lambda\}/\\Lambdais the fill rate normalized by the baseline intensity\.

### 2\.5State Dynamics

Letqt∈ℝq\_\{t\}\\in\\mathbb\{R\}denote the MM’s inventory on DEX\-A,XtX\_\{t\}the cash balance, andHtH\_\{t\}the cumulative hedge position on CEX\-B\. The state evolves as:

##### Inventory:

d​qt=Q​d​Ntb−Q​d​Nta−d​Ht,\\mathrm\{d\}q\_\{t\}=Q\\,\\mathrm\{d\}N\_\{t\}^\{b\}\-Q\\,\\mathrm\{d\}N\_\{t\}^\{a\}\-\\mathrm\{d\}H\_\{t\},\(6\)whereQ\>0Q\>0is the order size \(assumed constant for tractability\) andd​Ht\\mathrm\{d\}H\_\{t\}is the hedge trade increment\.

##### Cash:

d​Xt=Pta​Q​d​Nta−Ptb​Q​d​Ntb−ϕm​\|Pt​Q\|​\(d​Nta\+d​Ntb\)−ϕtCEX​\|St​d​Ht\|,\\mathrm\{d\}X\_\{t\}=P\_\{t\}^\{a\}Q\\,\\mathrm\{d\}N\_\{t\}^\{a\}\-P\_\{t\}^\{b\}Q\\,\\mathrm\{d\}N\_\{t\}^\{b\}\-\\phi\_\{m\}\|P\_\{t\}Q\|\(\\mathrm\{d\}N\_\{t\}^\{a\}\+\\mathrm\{d\}N\_\{t\}^\{b\}\)\-\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\|S\_\{t\}\\,\\mathrm\{d\}H\_\{t\}\|,\(7\)where the third term captures DEX\-A maker fees \(zero under the zero\-fee regime\) and the fourth term captures CEX\-B taker fees on hedging trades\.

Under the zero\-fee regime \(ϕm=0\\phi\_\{m\}=0\), the cash dynamics simplify to:

d​Xt=\(S~t\+δta\)​Q​d​Nta−\(S~t−δtb\)​Q​d​Ntb−ϕtCEX​\|St​d​Ht\|\.\\mathrm\{d\}X\_\{t\}=\(\\tilde\{S\}\_\{t\}\+\\delta\_\{t\}^\{a\}\)Q\\,\\mathrm\{d\}N\_\{t\}^\{a\}\-\(\\tilde\{S\}\_\{t\}\-\\delta\_\{t\}^\{b\}\)Q\\,\\mathrm\{d\}N\_\{t\}^\{b\}\-\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\|S\_\{t\}\\,\\mathrm\{d\}H\_\{t\}\|\.\(8\)

##### Wealth:

The total marked\-to\-market wealth is:

Wt=Xt\+qt​S~t\+Ht​St\.W\_\{t\}=X\_\{t\}\+q\_\{t\}\\tilde\{S\}\_\{t\}\+H\_\{t\}S\_\{t\}\.\(9\)

### 2\.6Information Structure and Adverse Selection

###### Assumption 2\.8\(Information Asymmetry\)\.

The taker order flow is a mixture of two types:

1. 1\.Uninformed flow\(fraction1−π1\-\\pi\): orders that are independent of future price movements, arriving at rate\(1−π\)​Λ​e−k​δ\(1\-\\pi\)\\Lambda e^\{\-k\\delta\}\.
2. 2\.Informed flow\(fractionπ\\pi\): orders correlated with imminent price movements, arriving at rateπ​Λ​e−k​δ\\pi\\Lambda e^\{\-k\\delta\}, where a fill signals an expected adverse price move of magnitudeαinfo\>0\\alpha\_\{\\mathrm\{info\}\}\>0\.

The informed fractionπ∈\[0,1\)\\pi\\in\[0,1\)is the probability of informed trading \(analogous to the PIN ofEasley and O’Hara \[[14](https://arxiv.org/html/2607.11888#bib.bib14)\]\)\.

###### Definition 2\.9\(Adverse Selection Cost\)\.

The*adverse selection cost*per fill is the expected loss from the price movement conditional on being filled:

α=π⋅αinfo\+\(1−π\)⋅αlatency,\\alpha=\\pi\\cdot\\alpha\_\{\\mathrm\{info\}\}\+\(1\-\\pi\)\\cdot\\alpha\_\{\\mathrm\{latency\}\},\(10\)whereαinfo\\alpha\_\{\\mathrm\{info\}\}is the adverse selection from informed traders andαlatency\\alpha\_\{\\mathrm\{latency\}\}is the adverse selection from stale quotes \(latency arbitrage\)\.

The stale\-quote component arises because the MM cannot instantly cancel orders when the reference price moves\. If the cancel latency isΔ​t\>0\\Delta t\>0\(the time between a CEX\-B price move and the cancellation taking effect on DEX\-A\), then:

αlatency=cℓ⋅σ​Δ​t,\\alpha\_\{\\mathrm\{latency\}\}=c\_\{\\ell\}\\cdot\\sigma\\sqrt\{\\Delta t\},\(11\)wherecℓ\>0c\_\{\\ell\}\>0is a constant depending on the intensity of latency arbitrageurs and the queue position of the MM’s orders\.

### 2\.7Funding Rate Mechanism

Perpetual futures contracts do not expire; instead, they use a*funding rate*mechanism to anchor the contract price to the spot/reference price\.

###### Assumption 2\.10\(Funding Rate\)\.

The funding raterf​\(t\)r\_\{f\}\(t\)is settled at discrete intervals of lengthΔf\>0\\Delta\_\{f\}\>0\(typically 8 hours\)\. Between settlements,rf​\(t\)r\_\{f\}\(t\)follows an Ornstein–Uhlenbeck process:

d​rf​\(t\)=−κf​\(rf​\(t\)−r¯f\)​d​t\+σf​d​Wtf,\\mathrm\{d\}r\_\{f\}\(t\)=\-\\kappa\_\{f\}\(r\_\{f\}\(t\)\-\\bar\{r\}\_\{f\}\)\\,\\mathrm\{d\}t\+\\sigma\_\{f\}\\,\\mathrm\{d\}W\_\{t\}^\{f\},\(12\)wherer¯f≥0\\bar\{r\}\_\{f\}\\geq 0is the long\-run mean funding rate,κf\>0\\kappa\_\{f\}\>0is the mean\-reversion speed,σf\>0\\sigma\_\{f\}\>0is the funding rate volatility, andWtfW\_\{t\}^\{f\}is a Brownian motion independent of\(Wt,Wtβ\)\(W\_\{t\},W\_\{t\}^\{\\beta\}\)\.

At each settlement timetk=k​Δft\_\{k\}=k\\Delta\_\{f\}, a position of sizeqqincurs a funding payment:

Fk=rf​\(tk\)⋅qtk⋅Q⋅Stk\.F\_\{k\}=r\_\{f\}\(t\_\{k\}\)\\cdot q\_\{t\_\{k\}\}\\cdot Q\\cdot S\_\{t\_\{k\}\}\.\(13\)Long positions pay whenrf\>0r\_\{f\}\>0and receive whenrf<0r\_\{f\}<0\.

The funding rate adds a carry component to the MM’s PnL\. For a market maker with average inventoryq¯\\bar\{q\}, the annualized funding cost rate is:

Π˙funding=−r¯f⋅q¯⋅Q⋅S¯Δf\.\\dot\{\\Pi\}\_\{\\mathrm\{funding\}\}=\-\\frac\{\\bar\{r\}\_\{f\}\\cdot\\bar\{q\}\\cdot Q\\cdot\\bar\{S\}\}\{\\Delta\_\{f\}\}\.\(14\)This term enters the complete PnL decomposition of Section[3](https://arxiv.org/html/2607.11888#S3)and the hedging analysis of Section[7](https://arxiv.org/html/2607.11888#S7)\.

### 2\.8Parameter Summary

Table[2](https://arxiv.org/html/2607.11888#S2.T2)summarizes the model parameters and their economic interpretations\.

Table 2:Model parameters

## 3Adverse Selection Theory and PnL Decomposition

This section develops a precise mathematical framework for understanding the components of market\-making profitability\. We establish a fundamental PnL decomposition theorem that separates revenue into distinct, measurable components, each governed by identifiable microstructure parameters\.

### 3\.1Per\-Fill Economics

###### Definition 3\.1\(Fill\-Level Edge\)\.

For theii\-th fill occurring at timeτi\\tau\_\{i\}at pricePiP\_\{i\}on sidesi∈\{b,a\}s\_\{i\}\\in\\\{b,a\\\}\(bid/ask\), the*edge*is:

ei=\{Sτi−Piif​si=b​\(MM buys\),Pi−Sτiif​si=a​\(MM sells\),e\_\{i\}=\\begin\{cases\}S\_\{\\tau\_\{i\}\}\-P\_\{i\}&\\text\{if \}s\_\{i\}=b\\text\{ \(MM buys\)\},\\\\ P\_\{i\}\-S\_\{\\tau\_\{i\}\}&\\text\{if \}s\_\{i\}=a\\text\{ \(MM sells\)\},\\end\{cases\}\(15\)measured relative to the CEX\-B reference priceSτiS\_\{\\tau\_\{i\}\}at fill time\. A positive edge indicates a favorable fill; a negative edge indicates adverse selection\.

###### Definition 3\.2\(Realized Adverse Selection\)\.

For filliiat timeτi\\tau\_\{i\}, the*realized adverse selection*over horizonh\>0h\>0is:

αi​\(h\)=\{Sτi−Sτi\+hif​si=b,Sτi\+h−Sτiif​si=a,\\alpha\_\{i\}\(h\)=\\begin\{cases\}S\_\{\\tau\_\{i\}\}\-S\_\{\\tau\_\{i\}\+h\}&\\text\{if \}s\_\{i\}=b,\\\\ S\_\{\\tau\_\{i\}\+h\}\-S\_\{\\tau\_\{i\}\}&\\text\{if \}s\_\{i\}=a,\\end\{cases\}\(16\)measuring the adverse price movement over the interval\[τi,τi\+h\]\[\\tau\_\{i\},\\tau\_\{i\}\+h\]\. The expected adverse selection isα​\(h\)=𝔼​\[αi​\(h\)∣fill at​τi\]\\alpha\(h\)=\\mathbb\{E\}\[\\alpha\_\{i\}\(h\)\\mid\\text\{fill at \}\\tau\_\{i\}\]\.

###### Lemma 3\.3\(Edge–Spread–Adverse Selection Identity\)\.

The edge decomposes as:

ei=δi⏟half\-spread\+βτisi⏟premium−αi​\(h\)⏟adverse selection \(horizonh\)\+εi​\(h\)⏟residual,e\_\{i\}=\\underbrace\{\\delta\_\{i\}\}\_\{\\text\{half\-spread\}\}\+\\underbrace\{\\beta\_\{\\tau\_\{i\}\}^\{s\_\{i\}\}\}\_\{\\text\{premium\}\}\-\\underbrace\{\\alpha\_\{i\}\(h\)\}\_\{\\text\{adverse selection \(horizon $h$\)\}\}\+\\underbrace\{\\varepsilon\_\{i\}\(h\)\}\_\{\\text\{residual\}\},\(17\)whereδi\\delta\_\{i\}is the half\-spread at fill time,βτisi\\beta\_\{\\tau\_\{i\}\}^\{s\_\{i\}\}is the signed DEX–CEX premium contribution, andεi​\(h\)\\varepsilon\_\{i\}\(h\)is a mean\-zero residual capturing price movements beyond the adverse selection horizon\.

###### Proof\.

By definition of the bid/ask pricesPi=S~τi∓δi=Sτi\+βτi∓δiP\_\{i\}=\\tilde\{S\}\_\{\\tau\_\{i\}\}\\mp\\delta\_\{i\}=S\_\{\\tau\_\{i\}\}\+\\beta\_\{\\tau\_\{i\}\}\\mp\\delta\_\{i\}, we have:

ei\\displaystyle e\_\{i\}=±\(Sτi−Pi\)=±\(Sτi−Sτi−βτi±δi\)=δi∓βτi\.\\displaystyle=\\pm\(S\_\{\\tau\_\{i\}\}\-P\_\{i\}\)=\\pm\(S\_\{\\tau\_\{i\}\}\-S\_\{\\tau\_\{i\}\}\-\\beta\_\{\\tau\_\{i\}\}\\pm\\delta\_\{i\}\)=\\delta\_\{i\}\\mp\\beta\_\{\\tau\_\{i\}\}\.Adding and subtractingSτi\+hS\_\{\\tau\_\{i\}\+h\}:

ei\\displaystyle e\_\{i\}=δi∓βτi−αi​\(h\)\+\[Sτi\+h−Sτi\+αi​\(h\)∓βτi\]\.\\displaystyle=\\delta\_\{i\}\\mp\\beta\_\{\\tau\_\{i\}\}\-\\alpha\_\{i\}\(h\)\+\[S\_\{\\tau\_\{i\}\+h\}\-S\_\{\\tau\_\{i\}\}\+\\alpha\_\{i\}\(h\)\\mp\\beta\_\{\\tau\_\{i\}\}\]\.Reorganizing withβτisi\\beta\_\{\\tau\_\{i\}\}^\{s\_\{i\}\}absorbing the signed premium contribution yields \([17](https://arxiv.org/html/2607.11888#S3.E17)\)\. ∎

### 3\.2The PnL Decomposition Theorem

We now state the main decomposition result\.

###### Theorem 3\.4\(PnL Decomposition\)\.

Consider a market maker operating over the interval\[0,T\]\[0,T\]withnntotal fills\. The total marked\-to\-market PnL decomposes as:

ΠT=ΠTspread⏟spread income−ΠTAS⏟adverse selection loss−ΠTinv⏟inventory cost−ΠThedge⏟hedging friction−ΠTfee⏟fee cost,\\Pi\_\{T\}=\\underbrace\{\\Pi\_\{T\}^\{\\mathrm\{spread\}\}\}\_\{\\text\{spread income\}\}\-\\underbrace\{\\Pi\_\{T\}^\{\\mathrm\{AS\}\}\}\_\{\\text\{adverse selection loss\}\}\-\\underbrace\{\\Pi\_\{T\}^\{\\mathrm\{inv\}\}\}\_\{\\text\{inventory cost\}\}\-\\underbrace\{\\Pi\_\{T\}^\{\\mathrm\{hedge\}\}\}\_\{\\text\{hedging friction\}\}\-\\underbrace\{\\Pi\_\{T\}^\{\\mathrm\{fee\}\}\}\_\{\\text\{fee cost\}\},\(18\)where each component is defined as follows\.

\(i\) Spread income:

ΠTspread=Q​∑i=1nδi,\\Pi\_\{T\}^\{\\mathrm\{spread\}\}=Q\\sum\_\{i=1\}^\{n\}\\delta\_\{i\},\(19\)the cumulative half\-spread capture across all fills\.

\(ii\) Adverse selection loss:

ΠTAS=Q​∑i=1nαi,\\Pi\_\{T\}^\{\\mathrm\{AS\}\}=Q\\sum\_\{i=1\}^\{n\}\\alpha\_\{i\},\(20\)whereαi\\alpha\_\{i\}is the realized adverse price movement for fillii\.

\(iii\) Inventory carrying cost:

ΠTinv=−∫0Tqt​dSt=−∫0Tqt​σ​dWt,\\Pi\_\{T\}^\{\\mathrm\{inv\}\}=\-\\int\_\{0\}^\{T\}q\_\{t\}\\,\\mathrm\{d\}S\_\{t\}=\-\\int\_\{0\}^\{T\}q\_\{t\}\\sigma\\,\\mathrm\{d\}W\_\{t\},\(21\)the stochastic integral representing mark\-to\-market losses from holding inventory during price fluctuations\.

\(iv\) Hedging friction:

ΠThedge=ϕtCEX​∑j=1m\|Sτjh⋅Δ​Hj\|,\\Pi\_\{T\}^\{\\mathrm\{hedge\}\}=\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\sum\_\{j=1\}^\{m\}\|S\_\{\\tau\_\{j\}^\{h\}\}\\cdot\\Delta H\_\{j\}\|,\(22\)the cumulative taker fee paid onmmhedge trades\.

\(v\) Fee cost:

ΠTfee=ϕm⋅Q​∑i=1nSτi≈n⋅Q⋅ϕm⋅S¯,\\Pi\_\{T\}^\{\\mathrm\{fee\}\}=\\phi\_\{m\}\\cdot Q\\sum\_\{i=1\}^\{n\}S\_\{\\tau\_\{i\}\}\\approx n\\cdot Q\\cdot\\phi\_\{m\}\\cdot\\bar\{S\},\(23\)whereS¯\\bar\{S\}is the average fill price\. Under the zero\-fee regime,ΠTfee=0\\Pi\_\{T\}^\{\\mathrm\{fee\}\}=0\.

###### Proof\.

The proof follows from the cash dynamics \([8](https://arxiv.org/html/2607.11888#S2.E8)\) and the marked\-to\-market wealth definition \([9](https://arxiv.org/html/2607.11888#S2.E9)\)\. We decompose the total wealth changeΔ​WT=WT−W0\\Delta W\_\{T\}=W\_\{T\}\-W\_\{0\}as follows\.

Each bid fill at timeτi\\tau\_\{i\}contributes−Pib​Q\-P\_\{i\}^\{b\}Qto cash and\+Q\+Qto inventory\. The instantaneous PnL from the fill is:

Q​\(Sτi−Pib\)−ϕm​Q​Sτi=Q​\(δib\+βτi\)−ϕm​Q​Sτi\.Q\(S\_\{\\tau\_\{i\}\}\-P\_\{i\}^\{b\}\)\-\\phi\_\{m\}QS\_\{\\tau\_\{i\}\}=Q\(\\delta\_\{i\}^\{b\}\+\\beta\_\{\\tau\_\{i\}\}\)\-\\phi\_\{m\}QS\_\{\\tau\_\{i\}\}\.Summing over all fills and including the mark\-to\-market change from inventory and hedge positions yields the decomposition\. The inventory carrying cost arises from applying Itô’s lemma toqt​Stq\_\{t\}S\_\{t\}and separating the drift from the stochastic component\. Full details are in Appendix[A](https://arxiv.org/html/2607.11888#A1)\. ∎

### 3\.3Expected PnL Rate

Taking expectations and normalizing by time:

###### Corollary 3\.5\(Expected PnL Rate\)\.

Under stationary conditions \(constant optimal spreadsδ∗\\delta^\{\*\}, constant adverse selectionα\\alpha\), the expected PnL rate is:

Π˙≡𝔼​\[ΠT\]T=λ¯​Q​\(δ¯−α−ϕm​S¯\)−C˙inv−C˙hedge,\\dot\{\\Pi\}\\equiv\\frac\{\\mathbb\{E\}\[\\Pi\_\{T\}\]\}\{T\}=\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\-\\phi\_\{m\}\\bar\{S\}\)\-\\dot\{C\}\_\{\\mathrm\{inv\}\}\-\\dot\{C\}\_\{\\mathrm\{hedge\}\},\(24\)where:

- •λ¯=λb​\(δb⁣∗\)\+λa​\(δa⁣∗\)\\bar\{\\lambda\}=\\lambda^\{b\}\(\\delta^\{b\*\}\)\+\\lambda^\{a\}\(\\delta^\{a\*\}\)is the total fill rate,
- •δ¯=\(δb⁣∗\+δa⁣∗\)/2\\bar\{\\delta\}=\(\\delta^\{b\*\}\+\\delta^\{a\*\}\)/2is the average half\-spread,
- •C˙inv=12​γ​σ2​𝔼​\[qt2\]​Q2\\dot\{C\}\_\{\\mathrm\{inv\}\}=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}is the expected inventory cost rate,
- •C˙hedge=λ¯h⋅ϕtCEX⋅Q​S¯\\dot\{C\}\_\{\\mathrm\{hedge\}\}=\\bar\{\\lambda\}\_\{h\}\\cdot\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\cdot Q\\bar\{S\}is the expected hedging cost rate withλ¯h\\bar\{\\lambda\}\_\{h\}the hedge trade rate\.

### 3\.4Adverse Selection Decomposition: Latency vs\. Informed

We now formalize the two sources of adverse selection\.

###### Theorem 3\.6\(Two\-Source Adverse Selection Model\)\.

Under Assumptions[2\.5](https://arxiv.org/html/2607.11888#S2.Thmtheorem5)and[2\.8](https://arxiv.org/html/2607.11888#S2.Thmtheorem8), the expected adverse selection per fill decomposes as:

α=π⋅αinfo⏟informed component\+\(1−π\)⋅cℓ​σ​Δ​t⏟latency component,\\alpha=\\underbrace\{\\pi\\cdot\\alpha\_\{\\mathrm\{info\}\}\}\_\{\\text\{informed component\}\}\+\\underbrace\{\(1\-\\pi\)\\cdot c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}\}\_\{\\text\{latency component\}\},\(25\)where:

- •π\\piis the probability that a fill is from an informed trader,
- •αinfo\\alpha\_\{\\mathrm\{info\}\}is the expected price impact conditional on an informed fill,
- •cℓ\>0c\_\{\\ell\}\>0is the latency arbitrageur intensity,
- •σ\\sigmais the reference price volatility,
- •Δ​t\\Delta tis the cancel latency\.

Moreover, the total adverse selection is bounded:

cℓ​\(1−π\)​σ​Δ​t≤α≤π​αinfo\+cℓ​σ​Δ​t\.c\_\{\\ell\}\(1\-\\pi\)\\sigma\\sqrt\{\\Delta t\}\\leq\\alpha\\leq\\pi\\alpha\_\{\\mathrm\{info\}\}\+c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}\.\(26\)

###### Proof\.

By the law of total expectation, conditioning on fill type:

α\\displaystyle\\alpha=𝔼​\[αi\]=𝔼​\[αi∣informed\]​Pr⁡\(informed\)\+𝔼​\[αi∣uninformed\]​Pr⁡\(uninformed\)\\displaystyle=\\mathbb\{E\}\[\\alpha\_\{i\}\]=\\mathbb\{E\}\[\\alpha\_\{i\}\\mid\\text\{informed\}\]\\Pr\(\\text\{informed\}\)\+\\mathbb\{E\}\[\\alpha\_\{i\}\\mid\\text\{uninformed\}\]\\Pr\(\\text\{uninformed\}\)=π⋅αinfo\+\(1−π\)⋅𝔼​\[αi∣uninformed\]\.\\displaystyle=\\pi\\cdot\\alpha\_\{\\mathrm\{info\}\}\+\(1\-\\pi\)\\cdot\\mathbb\{E\}\[\\alpha\_\{i\}\\mid\\text\{uninformed\}\]\.For uninformed fills, adverse selection arises solely from the stale\-quote mechanism\. The expected adverse price movement during the cancel latency windowΔ​t\\Delta tis:

𝔼​\[\|Sτi\+Δ​t−Sτi\|∣fill at​τi\]=cℓ​σ​Δ​t,\\mathbb\{E\}\[\|S\_\{\\tau\_\{i\}\+\\Delta t\}\-S\_\{\\tau\_\{i\}\}\|\\mid\\text\{fill at \}\\tau\_\{i\}\]=c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\},wherecℓc\_\{\\ell\}depends on the conditional distribution of fills \(fills are more likely when the price has moved against the MM, introducing a selection bias that increasescℓc\_\{\\ell\}above the unconditional value of2/π\\sqrt\{2/\\pi\}\)\. The bounds follow from0≤αlatencyuninf≤cℓ​σ​Δ​t0\\leq\\alpha\_\{\\mathrm\{latency\}\}^\{\\mathrm\{uninf\}\}\\leq c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}\. ∎

###### Proposition 3\.7\(Optimal Cancel Latency\)\.

Given the adverse selection model \([25](https://arxiv.org/html/2607.11888#S3.E25)\) and a cancel\-on\-move mechanism with thresholdθ\>0\\theta\>0\(in price units\), the relationship between the cancel threshold and effective latency is:

Δ​teff​\(θ\)=θ2σ2,\\Delta t\_\{\\mathrm\{eff\}\}\(\\theta\)=\\frac\{\\theta^\{2\}\}\{\\sigma^\{2\}\},\(27\)and the expected fill rate as a function ofθ\\thetais:

λ​\(θ\)=Λ0​\(1−e−ρ​θ\),\\lambda\(\\theta\)=\\Lambda\_\{0\}\(1\-e^\{\-\\rho\\theta\}\),\(28\)whereρ\>0\\rho\>0governs the sensitivity of fill rate to the threshold\. The optimal thresholdθ∗\\theta^\{\*\}maximizes the PnL rate:

θ∗=arg​maxθ\>0⁡Λ0​\(1−e−ρ​θ\)​\(δ¯−π​αinfo−\(1−π\)​cℓ​θ\),\\theta^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{\\theta\>0\}\\Lambda\_\{0\}\(1\-e^\{\-\\rho\\theta\}\)\\left\(\\bar\{\\delta\}\-\\pi\\alpha\_\{\\mathrm\{info\}\}\-\(1\-\\pi\)c\_\{\\ell\}\\theta\\right\),\(29\)yielding the first\-order condition:

ρ​e−ρ​θ∗​\(δ¯−π​αinfo−\(1−π\)​cℓ​θ∗\)=\(1−π\)​cℓ​\(1−e−ρ​θ∗\)\.\\rho e^\{\-\\rho\\theta^\{\*\}\}\\bigl\(\\bar\{\\delta\}\-\\pi\\alpha\_\{\\mathrm\{info\}\}\-\(1\-\\pi\)c\_\{\\ell\}\\theta^\{\*\}\\bigr\)=\(1\-\\pi\)c\_\{\\ell\}\(1\-e^\{\-\\rho\\theta^\{\*\}\}\)\.\(30\)

###### Proof\.

The effective latency follows from the time for a Brownian motion to first reach thresholdθ\\theta:𝔼​\[inf\{t:\|Wt\|≥θ/σ\}\]=θ2/σ2\\mathbb\{E\}\[\\inf\\\{t:\|W\_\{t\}\|\\geq\\theta/\\sigma\\\}\]=\\theta^\{2\}/\\sigma^\{2\}\(first passage time of standard BM to levelθ/σ\\theta/\\sigma\)\. The optimalθ\\thetais obtained by differentiating \([29](https://arxiv.org/html/2607.11888#S3.E29)\) and setting the derivative to zero\. ∎

### 3\.5Optimal Latency Investment

The cancel thresholdθ∗\\theta^\{\*\}from Proposition[3\.7](https://arxiv.org/html/2607.11888#S3.Thmtheorem7)determines the effective adverse selection, but in practice the MM must also decide how much infrastructure budget to allocate toward latency reduction versus other objectives \(e\.g\., co\-location, monitoring bandwidth\)\. We formalize this as an optimal investment problem\.

###### Definition 3\.8\(Latency Cost Function\)\.

LetΔ​t​\(ℓ\)\\Delta t\(\\ell\)denote the achievable cancel latency as a function of infrastructure investmentℓ≥0\\ell\\geq 0\(in $/year\)\. We assumeΔ​t​\(ℓ\)\\Delta t\(\\ell\)is convex, decreasing, and satisfies:

Δ​t​\(ℓ\)=Δ​t01\+ℓ/ℓ0,\\Delta t\(\\ell\)=\\frac\{\\Delta t\_\{0\}\}\{1\+\\ell/\\ell\_\{0\}\},\(31\)whereΔ​t0\\Delta t\_\{0\}is the baseline latency with zero investment andℓ0\\ell\_\{0\}is the characteristic investment scale\.

###### Theorem 3\.9\(Optimal Latency Investment\)\.

Given the PnL rate from Corollary[3\.5](https://arxiv.org/html/2607.11888#S3.Thmtheorem5)with adverse selectionα​\(Δ​t\)=π​αinfo\+\(1−π\)​cℓ​σ​Δ​t\\alpha\(\\Delta t\)=\\pi\\alpha\_\{\\mathrm\{info\}\}\+\(1\-\\pi\)c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}, the optimal infrastructure investmentℓ∗\\ell^\{\*\}maximizes:

ℓ∗=arg​maxℓ≥0⁡\[Π˙​\(Δ​t​\(ℓ\)\)−ℓTyear\]\.\\ell^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{\\ell\\geq 0\}\\left\[\\dot\{\\Pi\}\(\\Delta t\(\\ell\)\)\-\\frac\{\\ell\}\{T\_\{\\mathrm\{year\}\}\}\\right\]\.\(32\)The first\-order condition yields:

ℓ∗=ℓ0​\(\(1−π\)​cℓ​σ​λ¯​Q​Tyear​Δ​t02​ℓ0−1\)\+,\\ell^\{\*\}=\\ell\_\{0\}\\left\(\\sqrt\{\\frac\{\(1\-\\pi\)c\_\{\\ell\}\\sigma\\bar\{\\lambda\}QT\_\{\\mathrm\{year\}\}\\sqrt\{\\Delta t\_\{0\}\}\}\{2\\ell\_\{0\}\}\}\-1\\right\)^\{\+\},\(33\)where\(x\)\+=max⁡\(x,0\)\(x\)^\{\+\}=\\max\(x,0\)\.

The resulting optimal latency is:

Δ​t∗=\(2​ℓ0\(1−π\)​cℓ​σ​λ¯​Q​Tyear\)2/3⋅Δ​t01/3\.\\Delta t^\{\*\}=\\left\(\\frac\{2\\ell\_\{0\}\}\{\(1\-\\pi\)c\_\{\\ell\}\\sigma\\bar\{\\lambda\}QT\_\{\\mathrm\{year\}\}\}\\right\)^\{2/3\}\\cdot\\Delta t\_\{0\}^\{1/3\}\.\(34\)

###### Proof\.

The PnL sensitivity to latency is:

∂Π˙∂Δ​t=−λ¯​Q⋅\(1−π\)​cℓ​σ2​Δ​t\.\\frac\{\\partial\\dot\{\\Pi\}\}\{\\partial\\Delta t\}=\-\\bar\{\\lambda\}Q\\cdot\\frac\{\(1\-\\pi\)c\_\{\\ell\}\\sigma\}\{2\\sqrt\{\\Delta t\}\}\.Using the chain rule withΔ​t​\(ℓ\)=Δ​t0/\(1\+ℓ/ℓ0\)\\Delta t\(\\ell\)=\\Delta t\_\{0\}/\(1\+\\ell/\\ell\_\{0\}\):

d​Δ​td​ℓ=−Δ​t0/ℓ0\(1\+ℓ/ℓ0\)2=−Δ​t​\(ℓ\)2Δ​t0​ℓ0\.\\frac\{\\mathrm\{d\}\\Delta t\}\{\\mathrm\{d\}\\ell\}=\-\\frac\{\\Delta t\_\{0\}/\\ell\_\{0\}\}\{\(1\+\\ell/\\ell\_\{0\}\)^\{2\}\}=\-\\frac\{\\Delta t\(\\ell\)^\{2\}\}\{\\Delta t\_\{0\}\\ell\_\{0\}\}\.The first\-order conditiond​\[Π˙−ℓ/Tyear\]/d​ℓ=0\\mathrm\{d\}\[\\dot\{\\Pi\}\-\\ell/T\_\{\\mathrm\{year\}\}\]/\\mathrm\{d\}\\ell=0gives:

λ¯​Q⋅\(1−π\)​cℓ​σ2​Δ​t∗⋅\(Δ​t∗\)2Δ​t0​ℓ0=1Tyear\.\\bar\{\\lambda\}Q\\cdot\\frac\{\(1\-\\pi\)c\_\{\\ell\}\\sigma\}\{2\\sqrt\{\\Delta t^\{\*\}\}\}\\cdot\\frac\{\(\\Delta t^\{\*\}\)^\{2\}\}\{\\Delta t\_\{0\}\\ell\_\{0\}\}=\\frac\{1\}\{T\_\{\\mathrm\{year\}\}\}\.Solving forΔ​t∗\\Delta t^\{\*\}and thenℓ∗=ℓ0​\(Δ​t0/Δ​t∗−1\)\\ell^\{\*\}=\\ell\_\{0\}\(\\Delta t\_\{0\}/\\Delta t^\{\*\}\-1\)yields \([33](https://arxiv.org/html/2607.11888#S3.E33)\)–\([34](https://arxiv.org/html/2607.11888#S3.E34)\)\. ∎

###### Corollary 3\.10\(Returns to Latency Investment\)\.

The marginal return on latency investment at the optimum is exactly1/Tyear1/T\_\{\\mathrm\{year\}\}\(the annualization factor\), implying that at the optimum, one additional dollar of annual infrastructure spending yields exactly one dollar of additional annual PnL\. Beyondℓ∗\\ell^\{\*\}, the returns are diminishing, consistent with the concavity ofΠ˙​\(Δ​t​\(ℓ\)\)\\dot\{\\Pi\}\(\\Delta t\(\\ell\)\)inℓ\\ell\.

### 3\.6Bayesian Sequential Estimation of Informed Fraction

In practice, the informed trading probabilityπ\\piis unknown and must be estimated online from observed fill data\. We formalize this as a Bayesian filtering problem, establishing convergence guarantees for the sequential estimator\.

###### Definition 3\.11\(Fill Signal\)\.

For filliiat timeτi\\tau\_\{i\}, define the*fill signal*:

Zi=\|Sτi\+h−Sτi\|,Z\_\{i\}=\|S\_\{\\tau\_\{i\}\+h\}\-S\_\{\\tau\_\{i\}\}\|,\(35\)the absolute price displacement over the adverse selection horizonh\>0h\>0\. Under the two\-source model \(Theorem[3\.6](https://arxiv.org/html/2607.11888#S3.Thmtheorem6)\),ZiZ\_\{i\}follows the mixture distribution:

Zi∼π⋅Finfo\+\(1−π\)⋅Fnoise,Z\_\{i\}\\sim\\pi\\cdot F\_\{\\mathrm\{info\}\}\+\(1\-\\pi\)\\cdot F\_\{\\mathrm\{noise\}\},\(36\)whereFinfoF\_\{\\mathrm\{info\}\}is the signal distribution conditional on an informed fill \(e\.g\., folded normal with meanαinfo\\alpha\_\{\\mathrm\{info\}\}\) andFnoise=\|𝒩​\(0,σ2​h\)\|F\_\{\\mathrm\{noise\}\}=\|\\mathcal\{N\}\(0,\\sigma^\{2\}h\)\|\.

###### Theorem 3\.12\(Bayesian Convergence forπ\\pi\)\.

Letπ0∼Beta​\(a0,b0\)\\pi\_\{0\}\\sim\\mathrm\{Beta\}\(a\_\{0\},b\_\{0\}\)be the prior on the informed fraction, and define the posterior afternnfills via the update:

π^n=a0\+∑i=1n𝟏​\{Zi\>θc\}a0\+b0\+n,\\hat\{\\pi\}\_\{n\}=\\frac\{a\_\{0\}\+\\sum\_\{i=1\}^\{n\}\\mathbf\{1\}\\\{Z\_\{i\}\>\\theta\_\{c\}\\\}\}\{a\_\{0\}\+b\_\{0\}\+n\},\(37\)whereθc=\(αinfo\+cℓ​σ​Δ​t\)/2\\theta\_\{c\}=\(\\alpha\_\{\\mathrm\{info\}\}\+c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}\)/2is the classification threshold\. Then:

1. 1\.Consistency:π^n→a\.s\.π\\hat\{\\pi\}\_\{n\}\\xrightarrow\{a\.s\.\}\\piasn→∞n\\to\\infty\.
2. 2\.Rate:The posterior concentrates at rate: 𝔼​\[\(π^n−π\)2\]≤π​\(1−π\)n\+O​\(1n2\)\.\\mathbb\{E\}\\bigl\[\(\\hat\{\\pi\}\_\{n\}\-\\pi\)^\{2\}\\bigr\]\\leq\\frac\{\\pi\(1\-\\pi\)\}\{n\}\+O\\left\(\\frac\{1\}\{n^\{2\}\}\\right\)\.\(38\)
3. 3\.Credible interval width:The95%95\\%credible interval has width: w0\.95​\(n\)=2⋅1\.96​π^n​\(1−π^n\)a0\+b0\+n∼3\.92n​π​\(1−π\)\.w\_\{0\.95\}\(n\)=2\\cdot 1\.96\\sqrt\{\\frac\{\\hat\{\\pi\}\_\{n\}\(1\-\\hat\{\\pi\}\_\{n\}\)\}\{a\_\{0\}\+b\_\{0\}\+n\}\}\\sim\\frac\{3\.92\}\{\\sqrt\{n\}\}\\sqrt\{\\pi\(1\-\\pi\)\}\.\(39\)

###### Proof\.

\(i\) By the strong law of large numbers,n−1​∑i=1n𝟏​\{Zi\>θc\}→pc≡π​Finfoc​\(θc\)\+\(1−π\)​Fnoisec​\(θc\)n^\{\-1\}\\sum\_\{i=1\}^\{n\}\\mathbf\{1\}\\\{Z\_\{i\}\>\\theta\_\{c\}\\\}\\to p\_\{c\}\\equiv\\pi F\_\{\\mathrm\{info\}\}^\{c\}\(\\theta\_\{c\}\)\+\(1\-\\pi\)F\_\{\\mathrm\{noise\}\}^\{c\}\(\\theta\_\{c\}\)a\.s\., whereFc=1−FF^\{c\}=1\-Fdenotes the survival function\. With an appropriate thresholdθc\\theta\_\{c\},pcp\_\{c\}is a monotone function ofπ\\pi, ensuring identifiability\. The Bernstein–von Mises theorem then givesπ^n→π\\hat\{\\pi\}\_\{n\}\\to\\pia\.s\.

\(ii\) The MSE bound follows from the variance of the Beta posterior:Var​\(π∣Z1,…,Zn\)=π^n​\(1−π^n\)/\(a0\+b0\+n\+1\)≤π​\(1−π\)/n\\mathrm\{Var\}\(\\pi\\mid Z\_\{1\},\\ldots,Z\_\{n\}\)=\\hat\{\\pi\}\_\{n\}\(1\-\\hat\{\\pi\}\_\{n\}\)/\(a\_\{0\}\+b\_\{0\}\+n\+1\)\\leq\\pi\(1\-\\pi\)/nforn≥a0\+b0\+1n\\geq a\_\{0\}\+b\_\{0\}\+1, using the fact that the prior contribution isO​\(n−2\)O\(n^\{\-2\}\)\.

\(iii\) The credible interval width follows from the normal approximation to the Beta posterior for largenn\. ∎

###### Corollary 3\.13\(Adaptive Adverse Selection Estimate\)\.

Using the sequential estimateπ^n\\hat\{\\pi\}\_\{n\}, the adaptive adverse selection cost is:

α^n=π^n⋅αinfo\+\(1−π^n\)⋅cℓ​σ​Δ​t,\\hat\{\\alpha\}\_\{n\}=\\hat\{\\pi\}\_\{n\}\\cdot\\alpha\_\{\\mathrm\{info\}\}\+\(1\-\\hat\{\\pi\}\_\{n\}\)\\cdot c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\},\(40\)and the optimal spread formula \(Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)\) applies withα\\alphareplaced byα^n\\hat\{\\alpha\}\_\{n\}, yielding a self\-tuning market\-making strategy\. The regret of usingα^n\\hat\{\\alpha\}\_\{n\}instead of the trueα\\alphasatisfies:

Regretn≤Cαn,Cα=3\.92​π​\(1−π\)⋅\(αinfo−cℓ​σ​Δ​t\),\\mathrm\{Regret\}\_\{n\}\\leq\\frac\{C\_\{\\alpha\}\}\{\\sqrt\{n\}\},\\qquad C\_\{\\alpha\}=3\.92\\sqrt\{\\pi\(1\-\\pi\)\}\\cdot\(\\alpha\_\{\\mathrm\{info\}\}\-c\_\{\\ell\}\\sigma\\sqrt\{\\Delta t\}\),\(41\)so the cumulative regret from parameter uncertainty isO​\(N\)O\(\\sqrt\{N\}\)overNNfills\.

### 3\.7Adverse Selection Intensity as a Dimensionless Parameter

###### Definition 3\.14\(Adverse Selection Ratio\)\.

The*adverse selection ratio*is the dimensionless quantity:

ξ=αδ¯,\\xi=\\frac\{\\alpha\}\{\\bar\{\\delta\}\},\(42\)measuring the fraction of spread income consumed by adverse selection\. The MM is profitable per fill whenξ<1\\xi<1\(ignoring inventory and hedging costs\)\.

## 4Optimal Market Making Algorithm

We formulate the market maker’s problem as a stochastic optimal control problem and derive the optimal spread–inventory policy via the Hamilton–Jacobi–Bellman equation\.

### 4\.1Control Problem Formulation

The state vector is\(t,Xt,qt,St,βt\)\(t,X\_\{t\},q\_\{t\},S\_\{t\},\\beta\_\{t\}\), whereXtX\_\{t\}is cash,qtq\_\{t\}is inventory,StS\_\{t\}is the reference price, andβt\\beta\_\{t\}is the DEX–CEX premium\. The controls are the half\-spreads𝒖t=\(δtb,δta\)∈\[0,δ¯max\]2\\bm\{u\}\_\{t\}=\(\\delta\_\{t\}^\{b\},\\delta\_\{t\}^\{a\}\)\\in\[0,\\bar\{\\delta\}\_\{\\max\}\]^\{2\}and the hedge decisionhth\_\{t\}\.

###### Definition 4\.1\(Admissible Controls\)\.

A control𝒖=\(δb,δa,h\)\\bm\{u\}=\(\\delta^\{b\},\\delta^\{a\},h\)is*admissible*if:

1. 1\.δtb,δta≥0\\delta^\{b\}\_\{t\},\\delta^\{a\}\_\{t\}\\geq 0areℱt\\mathcal\{F\}\_\{t\}\-progressively measurable,
2. 2\.hth\_\{t\}isℱt\\mathcal\{F\}\_\{t\}\-adapted with\|qt\+Ht\|≤q¯\|q\_\{t\}\+H\_\{t\}\|\\leq\\bar\{q\}\(inventory constraint\),
3. 3\.The resulting wealth process is integrable:𝔼​\[supt≤T\|Wt\|2\]<∞\\mathbb\{E\}\[\\sup\_\{t\\leq T\}\|W\_\{t\}\|^\{2\}\]<\\infty\.

Denote the set of admissible controls by𝒰\\mathcal\{U\}\.

The MM maximizes expected CARA utility of terminal wealth:

V​\(t,x,q,S,β\)=sup𝒖∈𝒰𝔼t​\[−exp⁡\(−γ​WT\)\],V\(t,x,q,S,\\beta\)=\\sup\_\{\\bm\{u\}\\in\\mathcal\{U\}\}\\mathbb\{E\}\_\{t\}\\left\[\-\\exp\\left\(\-\\gamma W\_\{T\}\\right\)\\right\],\(43\)whereWT=XT\+qT​S~T\+HT​STW\_\{T\}=X\_\{T\}\+q\_\{T\}\\tilde\{S\}\_\{T\}\+H\_\{T\}S\_\{T\}is the terminal wealth\.

### 4\.2Hamilton–Jacobi–Bellman Equation

We conjecture a separable value function of the form:

V​\(t,x,q,S,β\)=−exp⁡\(−γ​\(x\+q​\(S\+β\)\+θ​\(t,q,β\)\)\),V\(t,x,q,S,\\beta\)=\-\\exp\\left\(\-\\gamma\\bigl\(x\+q\(S\+\\beta\)\+\\theta\(t,q,\\beta\)\\bigr\)\\right\),\(44\)whereθ​\(t,q,β\)\\theta\(t,q,\\beta\)encodes the inventory penalty and premium adjustment\.

###### Theorem 4\.2\(HJB Equation\)\.

Under the zero\-fee regime \(ϕm=0\\phi\_\{m\}=0\) and Assumptions[2\.1](https://arxiv.org/html/2607.11888#S2.Thmtheorem1)–[2\.8](https://arxiv.org/html/2607.11888#S2.Thmtheorem8), the functionθ​\(t,q,β\)\\theta\(t,q,\\beta\)satisfies the HJB equation:

0=θt\\displaystyle 0=\\theta\_\{t\}\+σ22​γ​q2\+σβ22​θβ​β−κ​\(β−β¯\)​θβ\\displaystyle\+\\frac\{\\sigma^\{2\}\}\{2\}\\gamma q^\{2\}\+\\frac\{\\sigma\_\{\\beta\}^\{2\}\}\{2\}\\theta\_\{\\beta\\beta\}\-\\kappa\(\\beta\-\\bar\{\\beta\}\)\\theta\_\{\\beta\}\+supδb≥0\{Λ​e−k​δb​\(e−γ​\(δb−α−Δ\+​θ\)−1\)\}\\displaystyle\+\\sup\_\{\\delta^\{b\}\\geq 0\}\\left\\\{\\Lambda e^\{\-k\\delta^\{b\}\}\\left\(e^\{\-\\gamma\(\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\)\}\-1\\right\)\\right\\\}\+supδa≥0\{Λ​e−k​δa​\(e−γ​\(δa−α−Δ−​θ\)−1\)\},\\displaystyle\+\\sup\_\{\\delta^\{a\}\\geq 0\}\\left\\\{\\Lambda e^\{\-k\\delta^\{a\}\}\\left\(e^\{\-\\gamma\(\\delta^\{a\}\-\\alpha\-\\Delta^\{\-\}\\theta\)\}\-1\\right\)\\right\\\},\(45\)with terminal conditionθ​\(T,q,β\)=0\\theta\(T,q,\\beta\)=0for allq,βq,\\beta, and where:

Δ\+​θ​\(t,q,β\)=θ​\(t,q\+1,β\)−θ​\(t,q,β\),Δ−​θ​\(t,q,β\)=θ​\(t,q,β\)−θ​\(t,q−1,β\)\.\\Delta^\{\+\}\\theta\(t,q,\\beta\)=\\theta\(t,q\+1,\\beta\)\-\\theta\(t,q,\\beta\),\\quad\\Delta^\{\-\}\\theta\(t,q,\\beta\)=\\theta\(t,q,\\beta\)\-\\theta\(t,q\-1,\\beta\)\.\(46\)

###### Proof\.

Applying the dynamic programming principle to the value function \([44](https://arxiv.org/html/2607.11888#S4.E44)\), using Itô’s lemma for the diffusive components \(StS\_\{t\}andβt\\beta\_\{t\}\) and the jump contributions from Poisson fills, we obtain the HJB equation\. The key steps are:

Step 1\.Compute derivatives ofVV:

Vt\\displaystyle V\_\{t\}=−γ​θt⋅V,Vx=−γ​V,VS=−γ​q​V,\\displaystyle=\-\\gamma\\theta\_\{t\}\\cdot V,\\quad V\_\{x\}=\-\\gamma V,\\quad V\_\{S\}=\-\\gamma qV,VS​S\\displaystyle V\_\{SS\}=γ2​q2​V,Vβ=−γ​\(q\+θβ\)​V\.\\displaystyle=\\gamma^\{2\}q^\{2\}V,\\quad V\_\{\\beta\}=\-\\gamma\(q\+\\theta\_\{\\beta\}\)V\.
Step 2\.The generator of the diffusion part is:

ℒ​V=Vt\+σ22​VS​S\+σβ22​Vβ​β−κ​\(β−β¯\)​Vβ\.\\mathcal\{L\}V=V\_\{t\}\+\\frac\{\\sigma^\{2\}\}\{2\}V\_\{SS\}\+\\frac\{\\sigma\_\{\\beta\}^\{2\}\}\{2\}V\_\{\\beta\\beta\}\-\\kappa\(\\beta\-\\bar\{\\beta\}\)V\_\{\\beta\}\.
Step 3\.For a bid fill \(Poisson jump at rateλb\\lambda^\{b\}\), the value changes fromV​\(t,x,q,S,β\)V\(t,x,q,S,\\beta\)toV​\(t,x−Pb​Q,q\+Q,S,β\)V\(t,x\-P^\{b\}Q,q\+Q,S,\\beta\)\. Under the exponential ansatz and unit order sizeQ=1Q=1:

Vpost\-fillVpre\-fill=exp⁡\(−γ​\(−δb−β\+α\+Δ\+​θ\)\)=exp⁡\(−γ​\(δb\+β−α−Δ\+​θ\)\)\.\\frac\{V^\{\\text\{post\-fill\}\}\}\{V^\{\\text\{pre\-fill\}\}\}=\\exp\\left\(\-\\gamma\(\-\\delta^\{b\}\-\\beta\+\\alpha\+\\Delta^\{\+\}\\theta\)\\right\)=\\exp\\left\(\-\\gamma\(\\delta^\{b\}\+\\beta\-\\alpha\-\\Delta^\{\+\}\\theta\)\\right\)\.
Step 4\.Combining and dividing by−γ​V\>0\-\\gamma V\>0yields \([45](https://arxiv.org/html/2607.11888#S4.E45)\)\. The detailed computation is in Appendix[B](https://arxiv.org/html/2607.11888#A2)\. ∎

### 4\.3Optimal Spread Policy

###### Theorem 4\.3\(Optimal Half\-Spreads\)\.

The optimal half\-spreads that solve the HJB equation \([45](https://arxiv.org/html/2607.11888#S4.E45)\) are, in the small\-risk\-aversion regime \(γ≪k\\gamma\\ll k\):

δb⁣∗​\(t,q,β\)\\displaystyle\\delta^\{b\*\}\(t,q,\\beta\)=1k\+α\+Δ\+​θ​\(t,q,β\),\\displaystyle=\\frac\{1\}\{k\}\+\\alpha\+\\Delta^\{\+\}\\theta\(t,q,\\beta\),\(47\)δa⁣∗​\(t,q,β\)\\displaystyle\\delta^\{a\*\}\(t,q,\\beta\)=1k\+α\+Δ−​θ​\(t,q,β\)\.\\displaystyle=\\frac\{1\}\{k\}\+\\alpha\+\\Delta^\{\-\}\\theta\(t,q,\\beta\)\.\(48\)
Under the quadratic approximationθ​\(t,q,β\)≈−12​γ​σ2​\(T−t\)​q2\+g​\(β,t\)\\theta\(t,q,\\beta\)\\approx\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(T\-t\)q^\{2\}\+g\(\\beta,t\), these reduce to the*explicit optimal spread formulas*:

δb⁣∗​\(t,q\)\\displaystyle\\delta^\{b\*\}\(t,q\)=1k\+γ​σ2​\(T−t\)2−γ​σ2​\(T−t\)​q\+α,\\displaystyle=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\(T\-t\)\}\{2\}\-\\gamma\\sigma^\{2\}\(T\-t\)q\+\\alpha,\(49\)δa⁣∗​\(t,q\)\\displaystyle\\delta^\{a\*\}\(t,q\)=1k\+γ​σ2​\(T−t\)2\+γ​σ2​\(T−t\)​q\+α\.\\displaystyle=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\(T\-t\)\}\{2\}\+\\gamma\\sigma^\{2\}\(T\-t\)q\+\\alpha\.\(50\)

###### Proof\.

Maximizing the bid fill contribution in \([45](https://arxiv.org/html/2607.11888#S4.E45)\) overδb\\delta^\{b\}:

maxδb≥0⁡Λ​e−k​δb​\(e−γ​\(δb−α−Δ\+​θ\)−1\)\.\\max\_\{\\delta^\{b\}\\geq 0\}\\Lambda e^\{\-k\\delta^\{b\}\}\\left\(e^\{\-\\gamma\(\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\)\}\-1\\right\)\.Letψ=δb−α−Δ\+​θ\\psi=\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\. In the regimeγ​ψ≪1\\gamma\\psi\\ll 1, expande−γ​ψ≈1−γ​ψ\+12​γ2​ψ2e^\{\-\\gamma\\psi\}\\approx 1\-\\gamma\\psi\+\\frac\{1\}\{2\}\\gamma^\{2\}\\psi^\{2\}and retain leading terms:

Λ​e−k​δb​\(−γ​ψ\+O​\(γ2​ψ2\)\)\.\\Lambda e^\{\-k\\delta^\{b\}\}\\left\(\-\\gamma\\psi\+O\(\\gamma^\{2\}\\psi^\{2\}\)\\right\)\.The first\-order condition at leading order is:

−k​\(−γ​ψ\)\+\(−γ\)=0⟹ψ=1k⟹δb⁣∗=1k\+α\+Δ\+​θ\.\-k\(\-\\gamma\\psi\)\+\(\-\\gamma\)=0\\implies\\psi=\\frac\{1\}\{k\}\\implies\\delta^\{b\*\}=\\frac\{1\}\{k\}\+\\alpha\+\\Delta^\{\+\}\\theta\.Substituting the quadratic approximation forθ\\thetagives \([49](https://arxiv.org/html/2607.11888#S4.E49)\)–\([50](https://arxiv.org/html/2607.11888#S4.E50)\)\. See Appendix[C](https://arxiv.org/html/2607.11888#A3)for the full derivation including higher\-order corrections\. ∎

###### Corollary 4\.4\(Optimal Total Spread\)\.

The optimal total spread \(bid–ask\) is:

s∗​\(t,q\)=δb⁣∗\+δa⁣∗=2k\+γ​σ2​\(T−t\)\+2​α\.s^\{\*\}\(t,q\)=\\delta^\{b\*\}\+\\delta^\{a\*\}=\\frac\{2\}\{k\}\+\\gamma\\sigma^\{2\}\(T\-t\)\+2\\alpha\.\(51\)This is*independent of inventory*qq: inventory affects the positioning of the spread \(bid vs\. ask\) but not its width\.

###### Corollary 4\.5\(Reservation Price\)\.

The MM’s reservation price \(indifference price\) is:

rt=S~t−γ​σ2​\(T−t\)​qt,r\_\{t\}=\\tilde\{S\}\_\{t\}\-\\gamma\\sigma^\{2\}\(T\-t\)q\_\{t\},\(52\)which is below \(above\) the DEX mid\-price when the MM is long \(short\), creating mean\-reverting inventory dynamics\.

### 4\.4Verification Theorem

###### Theorem 4\.6\(Verification\)\.

Letθ∗​\(t,q,β\)\\theta^\{\*\}\(t,q,\\beta\)be aC1,2C^\{1,2\}solution to the HJB equation \([45](https://arxiv.org/html/2607.11888#S4.E45)\) with terminal conditionθ∗​\(T,q,β\)=0\\theta^\{\*\}\(T,q,\\beta\)=0\. Define the value function candidate:

V^​\(t,x,q,S,β\)=−exp⁡\(−γ​\(x\+q​\(S\+β\)\+θ∗​\(t,q,β\)\)\)\.\\hat\{V\}\(t,x,q,S,\\beta\)=\-\\exp\\left\(\-\\gamma\(x\+q\(S\+\\beta\)\+\\theta^\{\*\}\(t,q,\\beta\)\)\\right\)\.Then:

1. 1\.V^≥V\\hat\{V\}\\geq Vfor all admissible controls \(supermartingale property\)\.
2. 2\.The policy\(δb⁣∗,δa⁣∗\)\(\\delta^\{b\*\},\\delta^\{a\*\}\)defined by \([47](https://arxiv.org/html/2607.11888#S4.E47)\)–\([48](https://arxiv.org/html/2607.11888#S4.E48)\) attains the supremum:V^=V\\hat\{V\}=V\(martingale property under optimal control\)\.

Consequently,V^\\hat\{V\}is the value function and\(δb⁣∗,δa⁣∗\)\(\\delta^\{b\*\},\\delta^\{a\*\}\)is optimal\.

###### Proof sketch\.

Part 1\.For any admissible control, defineMt=V^​\(t,Xt,qt,St,βt\)M\_\{t\}=\\hat\{V\}\(t,X\_\{t\},q\_\{t\},S\_\{t\},\\beta\_\{t\}\)\. By Itô’s formula with jumps:

d​Mt=\(generator terms\)​d​t\+\(martingale terms\)​d​Wt\+\(jump terms\)\.\\mathrm\{d\}M\_\{t\}=\(\\text\{generator terms\}\)\\mathrm\{d\}t\+\(\\text\{martingale terms\}\)\\mathrm\{d\}W\_\{t\}\+\(\\text\{jump terms\}\)\.Sinceθ∗\\theta^\{\*\}solves the HJB equation, the drift ofMtM\_\{t\}is non\-positive for any admissible control \(the supremum in \([45](https://arxiv.org/html/2607.11888#S4.E45)\) is attained by the optimal control, so suboptimal controls yield negative drift\)\. HenceMtM\_\{t\}is a supermartingale, implyingV^​\(0,⋅\)≥𝔼​\[MT\]=𝔼​\[−e−γ​WT\]\\hat\{V\}\(0,\\cdot\)\\geq\\mathbb\{E\}\[M\_\{T\}\]=\\mathbb\{E\}\[\-e^\{\-\\gamma W\_\{T\}\}\]\.

Part 2\.Under the optimal control\(δb⁣∗,δa⁣∗\)\(\\delta^\{b\*\},\\delta^\{a\*\}\), the drift vanishes andMtM\_\{t\}becomes a true martingale \(bounded exponential moments from the CARA structure ensure uniform integrability\)\. HenceV^​\(0,⋅\)=𝔼​\[MT\]\\hat\{V\}\(0,\\cdot\)=\\mathbb\{E\}\[M\_\{T\}\]\. Full details in Appendix[H](https://arxiv.org/html/2607.11888#A8)\. ∎

### 4\.5Stationary Policy \(Infinite Horizon\)

For practical market making \(T→∞T\\to\\infty\), the time\-dependent terms vanish\. FollowingGuéant et al\. \[[20](https://arxiv.org/html/2607.11888#bib.bib20)\], we replace the terminal inventory penalty with a running penaltyϕ​\(q\)\\phi\(q\):

###### Proposition 4\.7\(Stationary Optimal Spreads with Inventory Penalization\)\.

Under the inventory penalization approachϕ​\(q\)=12​η​q2\\phi\(q\)=\\frac\{1\}\{2\}\\eta q^\{2\}, the stationary optimal half\-spreads are:

δb⁣∗​\(q\)\\displaystyle\\delta^\{b\*\}\(q\)=1k\+α\+η​q\+η2,\\displaystyle=\\frac\{1\}\{k\}\+\\alpha\+\\eta q\+\\frac\{\\eta\}\{2\},\(53\)δa⁣∗​\(q\)\\displaystyle\\delta^\{a\*\}\(q\)=1k\+α−η​q\+η2\.\\displaystyle=\\frac\{1\}\{k\}\+\\alpha\-\\eta q\+\\frac\{\\eta\}\{2\}\.\(54\)The parameterη\>0\\eta\>0controls inventory aversion, analogous toγ​σ2​\(T−t\)\\gamma\\sigma^\{2\}\(T\-t\)in the finite\-horizon solution\.

### 4\.6Inventory Gating

In addition to spread skewing, practical algorithms employ hard inventory limits\.

###### Definition 4\.9\(Inventory\-Gated Policy\)\.

An*inventory\-gated*policy augments the optimal spreads with side suppression:

δgatedb​\(q\)=\{δb⁣∗​\(q\)if​q<q¯,\+∞if​q≥q¯,δgateda​\(q\)=\{δa⁣∗​\(q\)if​q\>−q¯,\+∞if​q≤−q¯,\\delta^\{b\}\_\{\\mathrm\{gated\}\}\(q\)=\\begin\{cases\}\\delta^\{b\*\}\(q\)&\\text\{if \}q<\\bar\{q\},\\\\ \+\\infty&\\text\{if \}q\\geq\\bar\{q\},\\end\{cases\}\\qquad\\delta^\{a\}\_\{\\mathrm\{gated\}\}\(q\)=\\begin\{cases\}\\delta^\{a\*\}\(q\)&\\text\{if \}q\>\-\\bar\{q\},\\\\ \+\\infty&\\text\{if \}q\\leq\-\\bar\{q\},\\end\{cases\}\(55\)whereq¯\\bar\{q\}is the maximum absolute inventory\.

###### Proposition 4\.10\(Value of Inventory Gating\)\.

Under the gated policy \([55](https://arxiv.org/html/2607.11888#S4.E55)\), the expected inventory variance satisfies:

𝔼​\[qt2\]≤λ¯2​η​k\+O​\(e−2​η​k​t\),\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\\leq\\frac\{\\bar\{\\lambda\}\}\{2\\eta k\}\+O\(e^\{\-2\\eta kt\}\),\(56\)and the maximum drawdown from inventory risk is bounded:

Pr⁡\(maxt≤T⁡\|qt⋅Δ​St\|\>L\)≤2​exp⁡\(−L22​q¯2​σ2​T\),\\Pr\\left\(\\max\_\{t\\leq T\}\|q\_\{t\}\\cdot\\Delta S\_\{t\}\|\>L\\right\)\\leq 2\\exp\\left\(\-\\frac\{L^\{2\}\}\{2\\bar\{q\}^\{2\}\\sigma^\{2\}T\}\\right\),\(57\)whereΔ​St=St−S0\\Delta S\_\{t\}=S\_\{t\}\-S\_\{0\}\.

### 4\.7Convergence of Inventory Penalization to Finite\-Horizon Solution

The stationary policy \(Proposition[4\.7](https://arxiv.org/html/2607.11888#S4.Thmtheorem7)\) uses the penalization parameterη\\etaas a proxy for the time\-dependent risk aversionγ​σ2​\(T−t\)\\gamma\\sigma^\{2\}\(T\-t\)in the finite\-horizon solution\. We establish the rate at which the two policies agree\.

###### Theorem 4\.11\(Convergence Rate\)\.

LetδFH⁣∗​\(t,q\)\\delta^\{\\mathrm\{FH\}\*\}\(t,q\)denote the finite\-horizon optimal half\-spread from Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)with horizonTT, and letδIP⁣∗​\(q\)\\delta^\{\\mathrm\{IP\}\*\}\(q\)denote the stationary inventory\-penalized half\-spread from Proposition[4\.7](https://arxiv.org/html/2607.11888#S4.Thmtheorem7)withη=γ​σ2​τη\\eta=\\gamma\\sigma^\{2\}\\tau\_\{\\eta\}for a chosen reference timeτη\>0\\tau\_\{\\eta\}\>0\. Then for allt≤T−τηt\\leq T\-\\tau\_\{\\eta\}:

\|δFH⁣∗​\(t,q\)−δIP⁣∗​\(q\)\|≤γ​σ2​\|T−t−τη\|⋅\(\|q\|\+12\)\.\\left\|\\delta^\{\\mathrm\{FH\}\*\}\(t,q\)\-\\delta^\{\\mathrm\{IP\}\*\}\(q\)\\right\|\\leq\\gamma\\sigma^\{2\}\|T\-t\-\\tau\_\{\\eta\}\|\\cdot\\left\(\|q\|\+\\frac\{1\}\{2\}\\right\)\.\(58\)In particular, att=T−τηt=T\-\\tau\_\{\\eta\}, the two policies coincide exactly\. The expected PnL difference over a horizon of lengthτη\\tau\_\{\\eta\}satisfies:

\|𝔼​\[ΠFH\]−𝔼​\[ΠIP\]\|≤Λ​γ2​σ4​τη36​k⋅𝔼​\[\(\|qt\|\+12\)2\]\.\\left\|\\mathbb\{E\}\[\\Pi^\{\\mathrm\{FH\}\}\]\-\\mathbb\{E\}\[\\Pi^\{\\mathrm\{IP\}\}\]\\right\|\\leq\\frac\{\\Lambda\\gamma^\{2\}\\sigma^\{4\}\\tau\_\{\\eta\}^\{3\}\}\{6k\}\\cdot\\mathbb\{E\}\\left\[\\left\(\|q\_\{t\}\|\+\\tfrac\{1\}\{2\}\\right\)^\{2\}\\right\]\.\(59\)

###### Proof\.

From \([49](https://arxiv.org/html/2607.11888#S4.E49)\), the finite\-horizon bid half\-spread at timettis:

δFH,b⁣∗​\(t,q\)=1k\+γ​σ2​\(T−t\)2−γ​σ2​\(T−t\)​q\+α\.\\delta^\{\\mathrm\{FH\},b\*\}\(t,q\)=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\(T\-t\)\}\{2\}\-\\gamma\\sigma^\{2\}\(T\-t\)q\+\\alpha\.The stationary penalized spread withη=γ​σ2​τη\\eta=\\gamma\\sigma^\{2\}\\tau\_\{\\eta\}is:

δIP,b⁣∗​\(q\)=1k\+γ​σ2​τη2−γ​σ2​τη​q\+α\.\\delta^\{\\mathrm\{IP\},b\*\}\(q\)=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\\tau\_\{\\eta\}\}\{2\}\-\\gamma\\sigma^\{2\}\\tau\_\{\\eta\}q\+\\alpha\.The difference is:

δFH,b⁣∗​\(t,q\)−δIP,b⁣∗​\(q\)=γ​σ2​\[\(T−t\)−τη\]​\(12−q\)\.\\delta^\{\\mathrm\{FH\},b\*\}\(t,q\)\-\\delta^\{\\mathrm\{IP\},b\*\}\(q\)=\\gamma\\sigma^\{2\}\[\(T\-t\)\-\\tau\_\{\\eta\}\]\\left\(\\frac\{1\}\{2\}\-q\\right\)\.Taking absolute values and noting\|1/2−q\|≤\|q\|\+1/2\|1/2\-q\|\\leq\|q\|\+1/2yields \([58](https://arxiv.org/html/2607.11888#S4.E58)\)\.

For the PnL bound, the fill rate under spreadδ\\deltaisΛ​e−k​δ\\Lambda e^\{\-k\\delta\}\. A spread perturbationε\\varepsilonchanges the fill rate by approximately−k​Λ​e−k​δ​ε\-k\\Lambda e^\{\-k\\delta\}\\varepsilonand the per\-fill edge by\+ε\+\\varepsilon, so the net PnL rate change isO​\(ε2\)O\(\\varepsilon^\{2\}\)at the optimum \(envelope theorem\)\. Integratingε​\(t\)=γ​σ2​\(T−t−τη\)​\(\|q\|\+1/2\)\\varepsilon\(t\)=\\gamma\\sigma^\{2\}\(T\-t\-\\tau\_\{\\eta\}\)\(\|q\|\+1/2\)overt∈\[T−τη,T\]t\\in\[T\-\\tau\_\{\\eta\},T\]:

∫0τηε​\(s\)2​ds=γ2​σ4​𝔼​\[\(\|q\|\+1/2\)2\]​∫0τηs2​ds=γ2​σ4​τη33​𝔼​\[\(\|q\|\+1/2\)2\]\.\\int\_\{0\}^\{\\tau\_\{\\eta\}\}\\varepsilon\(s\)^\{2\}\\mathrm\{d\}s=\\gamma^\{2\}\\sigma^\{4\}\\mathbb\{E\}\[\(\|q\|\+1/2\)^\{2\}\]\\int\_\{0\}^\{\\tau\_\{\\eta\}\}s^\{2\}\\mathrm\{d\}s=\\frac\{\\gamma^\{2\}\\sigma^\{4\}\\tau\_\{\\eta\}^\{3\}\}\{3\}\\mathbb\{E\}\[\(\|q\|\+1/2\)^\{2\}\]\.Multiplying byΛ/\(2​k\)\\Lambda/\(2k\)\(the second\-order PnL sensitivity at the optimum\) gives \([59](https://arxiv.org/html/2607.11888#S4.E59)\)\. ∎

### 4\.8Ergodic Inventory Distribution

Under the optimal stationary policy, the inventory processqtq\_\{t\}is mean\-reverting\. We characterize its long\-run distribution, which determines the expected inventory cost rate and informs risk management\.

###### Theorem 4\.13\(Ergodic Inventory Distribution\)\.

Under the stationary optimal spread policy \([53](https://arxiv.org/html/2607.11888#S4.E53)\)–\([54](https://arxiv.org/html/2607.11888#S4.E54)\) with inventory gating atq¯\\bar\{q\}, the inventory processqtq\_\{t\}admits a unique stationary distributionπ∞\\pi\_\{\\infty\}with the following properties:

1. 1\.Gaussian approximation:For largeq¯\\bar\{q\}\(i\.e\., the gating constraint is rarely binding\), the stationary distribution is approximately: q∞≈d𝒩​\(0,σq2\),σq2=λ¯∗2​η​k,q\_\{\\infty\}\\stackrel\{\{\\scriptstyle d\}\}\{\{\\approx\}\}\\mathcal\{N\}\\\!\\left\(0,\\sigma\_\{q\}^\{2\}\\right\),\\qquad\\sigma\_\{q\}^\{2\}=\\frac\{\\bar\{\\lambda\}^\{\*\}\}\{2\\eta k\},\(60\)whereλ¯∗=2​Λ​exp⁡\(−1−k​α−k​η/2\)\\bar\{\\lambda\}^\{\*\}=2\\Lambda\\exp\(\-1\-k\\alpha\-k\\eta/2\)is the total fill rate under the optimal policy atq=0q=0\.
2. 2\.Inventory half\-life:The expected time for inventory to decay fromq0q\_\{0\}toq0/2q\_\{0\}/2is: t1/2=ln⁡2η​k​λ¯∗⋅1σq2\.t\_\{1/2\}=\\frac\{\\ln 2\}\{\\eta k\\bar\{\\lambda\}^\{\*\}\}\\cdot\\frac\{1\}\{\\sigma\_\{q\}^\{2\}\}\.\(61\)
3. 3\.Expected inventory cost:The long\-run average inventory cost rate is: C˙inv∞=12​γ​σ2​Q2​𝔼​\[q∞2\]=γ​σ2​Q2​λ¯∗4​η​k\.\\dot\{C\}\_\{\\mathrm\{inv\}\}^\{\\infty\}=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\mathbb\{E\}\[q\_\{\\infty\}^\{2\}\]=\\frac\{\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}^\{\*\}\}\{4\\eta k\}\.\(62\)

###### Proof\.

Part 1\.Under the optimal policy, the inventory drift at levelqqis:

μq​\(q\)=λb​\(q\)−λa​\(q\)=Λ​e−k​δb⁣∗​\(q\)−Λ​e−k​δa⁣∗​\(q\)\.\\mu\_\{q\}\(q\)=\\lambda^\{b\}\(q\)\-\\lambda^\{a\}\(q\)=\\Lambda e^\{\-k\\delta^\{b\*\}\(q\)\}\-\\Lambda e^\{\-k\\delta^\{a\*\}\(q\)\}\.Substituting \([53](https://arxiv.org/html/2607.11888#S4.E53)\)–\([54](https://arxiv.org/html/2607.11888#S4.E54)\):

μq​\(q\)=λ¯∗​\(q\)⋅sinh⁡\(−η​k​q\)/cosh⁡\(η​k​q/2\),\\mu\_\{q\}\(q\)=\\bar\{\\lambda\}^\{\*\}\(q\)\\cdot\\sinh\(\-\\eta kq\)/\\cosh\(\\eta kq/2\),which forη​k​\|q\|≪1\\eta k\|q\|\\ll 1linearizes toμq​\(q\)≈−2​η​k​λ¯∗​q/2=−η​k​λ¯∗​q\\mu\_\{q\}\(q\)\\approx\-2\\eta k\\bar\{\\lambda\}^\{\*\}q/2=\-\\eta k\\bar\{\\lambda\}^\{\*\}q\. The total jump variance rate \(from both sides\) isνq​\(q\)≈λ¯∗\\nu\_\{q\}\(q\)\\approx\\bar\{\\lambda\}^\{\*\}for smallqq\. The resulting Ornstein–Uhlenbeck\-type jump process has stationary varianceσq2=νq/\(2​\|μq′​\(0\)\|\)=λ¯∗/\(2​η​k\)\\sigma\_\{q\}^\{2\}=\\nu\_\{q\}/\(2\|\\mu\_\{q\}^\{\\prime\}\(0\)\|\)=\\bar\{\\lambda\}^\{\*\}/\(2\\eta k\)\[cf\.[9](https://arxiv.org/html/2607.11888#bib.bib9), Proposition 4\.1\]\.

Part 2\.The inventory half\-life follows from the linearized decay𝔼​\[qt∣q0\]=q0​e−η​k​λ¯∗​t/\(2​σq2\)\\mathbb\{E\}\[q\_\{t\}\\mid q\_\{0\}\]=q\_\{0\}e^\{\-\\eta k\\bar\{\\lambda\}^\{\*\}t/\(2\\sigma\_\{q\}^\{2\}\)\}\. Settinge−r​t1/2=1/2e^\{\-rt\_\{1/2\}\}=1/2givest1/2=ln⁡2/rt\_\{1/2\}=\\ln 2/rwithr=η​k​λ¯∗/\(2​σq2\)r=\\eta k\\bar\{\\lambda\}^\{\*\}/\(2\\sigma\_\{q\}^\{2\}\)\. Substitutingσq2\\sigma\_\{q\}^\{2\}yields \([61](https://arxiv.org/html/2607.11888#S4.E61)\)\.

Part 3\.Direct substitution of𝔼​\[q∞2\]=σq2\\mathbb\{E\}\[q\_\{\\infty\}^\{2\}\]=\\sigma\_\{q\}^\{2\}into the inventory cost formula\. ∎

###### Corollary 4\.14\(Optimal Inventory Penalization\)\.

Minimizing the total cost \(inventory cost \+ missed spread from over\-wide spreads\) overη\\etayields the optimal inventory penalization:

η∗=\(γ​σ22​k​λ¯∗\)1/2,\\eta^\{\*\}=\\left\(\\frac\{\\gamma\\sigma^\{2\}\}\{2k\\bar\{\\lambda\}^\{\*\}\}\\right\)^\{1/2\},\(63\)which balances inventory risk against fill rate reduction\. The resulting minimal total cost rate is:

C˙∗=Q2​γ​σ2​λ¯∗2​k\.\\dot\{C\}^\{\*\}=Q^\{2\}\\sqrt\{\\frac\{\\gamma\\sigma^\{2\}\\bar\{\\lambda\}^\{\*\}\}\{2k\}\}\.\(64\)

## 5High\-APY Conditions

This section establishes the theoretical conditions under which a market maker can achieve high annualized percentage yields \(APY\) on deployed capital\. We derive explicit parameter boundaries and phase diagrams separating profitable from unprofitable regimes\.

### 5\.1APY Definition and Formula

###### Definition 5\.1\(Market\-Making APY\)\.

The*annualized percentage yield*of a market\-making strategy with expected PnL rateΠ˙\\dot\{\\Pi\}and deployed capitalKKis:

APY=Π˙⋅TyearK×100%,\\mathrm\{APY\}=\\frac\{\\dot\{\\Pi\}\\cdot T\_\{\\mathrm\{year\}\}\}\{K\}\\times 100\\%,\(65\)whereTyearT\_\{\\mathrm\{year\}\}is the number of trading seconds per year \(approximately365\.25×24×3600≈3\.156×107365\.25\\times 24\\times 3600\\approx 3\.156\\times 10^\{7\}s for 24/7 crypto markets\)\.

###### Definition 5\.2\(Capital Utilization\)\.

The deployed capital is:

K=Kmargin\+Kbuffer,K=K\_\{\\mathrm\{margin\}\}\+K\_\{\\mathrm\{buffer\}\},\(66\)whereKmargin=q¯⋅Q⋅S¯/ℓK\_\{\\mathrm\{margin\}\}=\\bar\{q\}\\cdot Q\\cdot\\bar\{S\}/\\ellis the margin requirement \(with leverageℓ\\ell\), andKbufferK\_\{\\mathrm\{buffer\}\}is the buffer for hedging and drawdowns\. For perpetual futures withℓ\\ell\-fold leverage:

K=q¯⋅Q⋅S¯ℓ​\(1\+ηbuf\),K=\\frac\{\\bar\{q\}\\cdot Q\\cdot\\bar\{S\}\}\{\\ell\}\(1\+\\eta\_\{\\mathrm\{buf\}\}\),\(67\)whereηbuf≥0\\eta\_\{\\mathrm\{buf\}\}\\geq 0is the buffer fraction\.

### 5\.2The APY Formula

Combining Definition[5\.1](https://arxiv.org/html/2607.11888#S5.Thmtheorem1)with Corollary[3\.5](https://arxiv.org/html/2607.11888#S3.Thmtheorem5):

###### Theorem 5\.3\(APY Formula\)\.

Under the optimal policy of Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)with zero maker fees, the expected APY is:

APY=λ¯​Q​\(δ¯−α\)⋅TyearK−\(C˙inv\+C˙hedge\)⋅TyearK\.\\mathrm\{APY\}=\\frac\{\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)\\cdot T\_\{\\mathrm\{year\}\}\}\{K\}\-\\frac\{\(\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\)\\cdot T\_\{\\mathrm\{year\}\}\}\{K\}\.\(68\)
Substituting the optimal spreadδ¯∗=1/k\+γ​σ2​\(T−t\)/2\+α\\bar\{\\delta\}^\{\*\}=1/k\+\\gamma\\sigma^\{2\}\(T\-t\)/2\+\\alphaand the fill rateλ¯=2​Λ​e−k​δ¯∗\\bar\{\\lambda\}=2\\Lambda e^\{\-k\\bar\{\\delta\}^\{\*\}\}:

APY=2​Λ​e−k​\(1/k\+γ​σ2​τ/2\+α\)⋅Q⋅\(1/k\+γ​σ2​τ/2\)⋅TyearK−\(C˙inv\+C˙hedge\)​TyearK,\\mathrm\{APY\}=\\frac\{2\\Lambda e^\{\-k\(1/k\+\\gamma\\sigma^\{2\}\\tau/2\+\\alpha\)\}\\cdot Q\\cdot\(1/k\+\\gamma\\sigma^\{2\}\\tau/2\)\\cdot T\_\{\\mathrm\{year\}\}\}\{K\}\-\\frac\{\(\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\)T\_\{\\mathrm\{year\}\}\}\{K\},\(69\)whereτ=T−t\\tau=T\-tis the residual time horizon\.

### 5\.3Dimensionless Parameters

To characterize the high\-APY regime, we introduce five dimensionless parameters:

###### Definition 5\.4\(Dimensionless Parameter Set\)\.

ξ\\displaystyle\\xi=αδ¯∈\[0,∞\)\\displaystyle=\\frac\{\\alpha\}\{\\bar\{\\delta\}\}\\in\[0,\\infty\)\(adverse selection ratio\),\\displaystyle\\text\{\(adverse selection ratio\)\},\(70\)ϕ\\displaystyle\\phi=ϕmδ¯∈\[0,1\)\\displaystyle=\\frac\{\\phi\_\{m\}\}\{\\bar\{\\delta\}\}\\in\[0,1\)\(fee ratio\),\\displaystyle\\text\{\(fee ratio\)\},\(71\)ν\\displaystyle\\nu=Λ/fref\\displaystyle=\\Lambda/f\_\{\\mathrm\{ref\}\}\(normalized fill rate\),\\displaystyle\\text\{\(normalized fill rate\)\},\(72\)ρinv\\displaystyle\\rho\_\{\\mathrm\{inv\}\}=C˙invλ¯​Q​δ¯\\displaystyle=\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\}\{\\bar\{\\lambda\}Q\\bar\{\\delta\}\}\(inventory cost ratio\),\\displaystyle\\text\{\(inventory cost ratio\)\},\(73\)ρhedge\\displaystyle\\rho\_\{\\mathrm\{hedge\}\}=C˙hedgeλ¯​Q​δ¯\\displaystyle=\\frac\{\\dot\{C\}\_\{\\mathrm\{hedge\}\}\}\{\\bar\{\\lambda\}Q\\bar\{\\delta\}\}\(hedging cost ratio\)\.\\displaystyle\\text\{\(hedging cost ratio\)\}\.\(74\)

In terms of these parameters, the APY simplifies to:

APY=APY0⋅\(1−ξ−ϕ−ρinv−ρhedge\),\\mathrm\{APY\}=\\mathrm\{APY\}\_\{0\}\\cdot\(1\-\\xi\-\\phi\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\),\(75\)whereAPY0=λ¯​Q​δ¯​Tyear/K\\mathrm\{APY\}\_\{0\}=\\bar\{\\lambda\}Q\\bar\{\\delta\}T\_\{\\mathrm\{year\}\}/Kis the*gross APY*\(APY with no costs\)\.

### 5\.4High\-APY Regime Theorems

###### Theorem 5\.5\(Necessary Condition for Positive APY\)\.

The market maker earns positive expected PnL if and only if:

ξ\+ϕ\+ρinv\+ρhedge<1\.\\xi\+\\phi\+\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}<1\.\(76\)Under zero fees \(ϕ=0\\phi=0\) and without hedging \(ρhedge=0\\rho\_\{\\mathrm\{hedge\}\}=0\), this reduces to:

ξ\+ρinv<1⇔α<δ¯−C˙invλ¯​Q\.\\xi\+\\rho\_\{\\mathrm\{inv\}\}<1\\iff\\alpha<\\bar\{\\delta\}\-\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\}\{\\bar\{\\lambda\}Q\}\.\(77\)

###### Theorem 5\.6\(High\-APY Regime\)\.

For a target APY ofA%A\\%, the required condition on the parameter space is:

λ¯​Q​\(δ¯−α−ϕm​S¯\)\>A⋅K100⋅Tyear\+C˙inv\+C˙hedge\.\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\-\\phi\_\{m\}\\bar\{S\}\)\>\\frac\{A\\cdot K\}\{100\\cdot T\_\{\\mathrm\{year\}\}\}\+\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\.\(78\)
In the zero\-fee regime with no hedging costs and optimal spreads, this yields the*critical adverse selection*:

α<α∗​\(A\)≡1k\+γ​σ2​τ2−A⋅K100⋅Tyear⋅λ¯​Q−C˙invλ¯​Q\.\\alpha<\\alpha^\{\*\}\(A\)\\equiv\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\\tau\}\{2\}\-\\frac\{A\\cdot K\}\{100\\cdot T\_\{\\mathrm\{year\}\}\\cdot\\bar\{\\lambda\}Q\}\-\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\}\{\\bar\{\\lambda\}Q\}\.\(79\)The MM achieves APY≥A%\\geq A\\%if and only ifα<α∗​\(A\)\\alpha<\\alpha^\{\*\}\(A\)\.

###### Proof\.

From the APY formula \([68](https://arxiv.org/html/2607.11888#S5.E68)\), settingAPY≥A\\mathrm\{APY\}\\geq Aand solving forα\\alpha:

λ¯​Q​\(δ¯−α\)​TyearK−\(C˙inv\+C˙hedge\)​TyearK\\displaystyle\\frac\{\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)T\_\{\\mathrm\{year\}\}\}\{K\}\-\\frac\{\(\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\)T\_\{\\mathrm\{year\}\}\}\{K\}≥A/100,\\displaystyle\\geq A/100,δ¯−α\\displaystyle\\bar\{\\delta\}\-\\alpha≥A​K100⋅Tyear​λ¯​Q\+C˙inv\+C˙hedgeλ¯​Q,\\displaystyle\\geq\\frac\{AK\}\{100\\cdot T\_\{\\mathrm\{year\}\}\\bar\{\\lambda\}Q\}\+\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\}\{\\bar\{\\lambda\}Q\},α\\displaystyle\\alpha≤δ¯−A​K100⋅Tyear​λ¯​Q−C˙inv\+C˙hedgeλ¯​Q\.\\displaystyle\\leq\\bar\{\\delta\}\-\\frac\{AK\}\{100\\cdot T\_\{\\mathrm\{year\}\}\\bar\{\\lambda\}Q\}\-\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\+\\dot\{C\}\_\{\\mathrm\{hedge\}\}\}\{\\bar\{\\lambda\}Q\}\.Substitutingδ¯=1/k\+γ​σ2​τ/2\+α\\bar\{\\delta\}=1/k\+\\gamma\\sigma^\{2\}\\tau/2\+\\alphafrom the optimal spread formula, theα\\alphaterms cancel, confirming self\-consistency\. The bound \([79](https://arxiv.org/html/2607.11888#S5.E79)\) uses the pre\-adverse\-selection component of the optimal spread\. ∎

### 5\.5Phase Diagram

The parameter space can be partitioned into regions based on achievable APY:

###### Corollary 5\.7\(APY Phase Boundaries\)\.

In the\(ξ,ν\)\(\\xi,\\nu\)\-plane \(adverse selection ratio vs\. normalized fill rate\), the boundary for APY=A%=A\\%is:

ν=A⋅K/\(100⋅Tyear⋅Q​δ¯\)1−ξ−ρinv−ρhedge,\\nu=\\frac\{A\\cdot K/\(100\\cdot T\_\{\\mathrm\{year\}\}\\cdot Q\\bar\{\\delta\}\)\}\{1\-\\xi\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\},\(80\)which is a hyperbola in\(ξ,ν\)\(\\xi,\\nu\)\-space with vertical asymptote atξ=1−ρinv−ρhedge\\xi=1\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\.

### 5\.6Sharpe Ratio Analysis

###### Definition 5\.9\(Market\-Making Sharpe Ratio\)\.

The annualized Sharpe ratio of the MM strategy is:

SR=𝔼​\[ΠT/T\]Var​\[ΠT/T\]⋅Tyear\.\\mathrm\{SR\}=\\frac\{\\mathbb\{E\}\[\\Pi\_\{T\}/T\]\}\{\\sqrt\{\\mathrm\{Var\}\[\\Pi\_\{T\}/T\]\}\}\\cdot\\sqrt\{T\_\{\\mathrm\{year\}\}\}\.\(81\)

###### Theorem 5\.10\(Sharpe Ratio Under Optimal Policy\)\.

Under the optimal policy with stationary parameters, the Sharpe ratio is:

SR=λ¯​Q​\(δ¯−α\)−C˙inv−C˙hedgeλ¯​Q2​Var​\[ei\]\+σ2​𝔼​\[qt2\]​Q2⋅Tyear,\\mathrm\{SR\}=\\frac\{\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)\-\\dot\{C\}\_\{\\mathrm\{inv\}\}\-\\dot\{C\}\_\{\\mathrm\{hedge\}\}\}\{\\sqrt\{\\bar\{\\lambda\}Q^\{2\}\\mathrm\{Var\}\[e\_\{i\}\]\+\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}\}\}\\cdot\\sqrt\{T\_\{\\mathrm\{year\}\}\},\(82\)whereVar​\[ei\]\\mathrm\{Var\}\[e\_\{i\}\]is the per\-fill edge variance and𝔼​\[qt2\]\\mathbb\{E\}\[q\_\{t\}^\{2\}\]is the stationary inventory variance\.

For a market maker with negligible inventory risk \(e\.g\., through aggressive hedging\), the Sharpe ratio simplifies to:

SR≈\(δ¯−α\)​λ¯Var​\[ei\]⋅Tyear\.\\mathrm\{SR\}\\approx\\frac\{\(\\bar\{\\delta\}\-\\alpha\)\\sqrt\{\\bar\{\\lambda\}\}\}\{\\sqrt\{\\mathrm\{Var\}\[e\_\{i\}\]\}\}\\cdot\\sqrt\{T\_\{\\mathrm\{year\}\}\}\.\(83\)

### 5\.7Regime Robustness Under Parameter Uncertainty

In practice, the dimensionless parameters\(ξ,ϕ,ν,ρinv,ρhedge\)\(\\xi,\\phi,\\nu,\\rho\_\{\\mathrm\{inv\}\},\\rho\_\{\\mathrm\{hedge\}\}\)are not known precisely\. We analyze the robustness of the high\-APY regime to parameter perturbations\.

###### Definition 5\.12\(Parameter Uncertainty Set\)\.

Let𝜽0=\(ξ0,ϕ0,ν0,ρinv,0,ρhedge,0\)\\bm\{\\theta\}\_\{0\}=\(\\xi\_\{0\},\\phi\_\{0\},\\nu\_\{0\},\\rho\_\{\\mathrm\{inv\},0\},\\rho\_\{\\mathrm\{hedge\},0\}\)denote the nominal parameter vector\. The*uncertainty set*at confidence levelϵ\>0\\epsilon\>0is:

Θϵ=\{𝜽:∑j=15\(θj−θj,0σj\)2≤χ52​\(1−ϵ\)\},\\Theta\_\{\\epsilon\}=\\left\\\{\\bm\{\\theta\}:\\sum\_\{j=1\}^\{5\}\\left\(\\frac\{\\theta\_\{j\}\-\\theta\_\{j,0\}\}\{\\sigma\_\{j\}\}\\right\)^\{2\}\\leq\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)\\right\\\},\(84\)whereσj\\sigma\_\{j\}is the estimation uncertainty for parameterjjandχ52​\(1−ϵ\)\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)is the chi\-squared critical value\.

###### Theorem 5\.13\(Worst\-Case APY Under Parameter Uncertainty\)\.

LetAPY​\(𝛉\)\\mathrm\{APY\}\(\\bm\{\\theta\}\)be given by \([75](https://arxiv.org/html/2607.11888#S5.E75)\)\. The worst\-case APY over the uncertainty setΘϵ\\Theta\_\{\\epsilon\}is:

APYworst=APY0​\(1−ξ0−ϕ0−ρinv,0−ρhedge,0−χ52​\(1−ϵ\)⋅‖𝒘‖2\),\\mathrm\{APY\}\_\{\\mathrm\{worst\}\}=\\mathrm\{APY\}\_\{0\}\\left\(1\-\\xi\_\{0\}\-\\phi\_\{0\}\-\\rho\_\{\\mathrm\{inv\},0\}\-\\rho\_\{\\mathrm\{hedge\},0\}\-\\sqrt\{\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)\}\\cdot\\\|\\bm\{w\}\\\|\_\{2\}\\right\),\(85\)where𝐰=\(σξ,σϕ,0,σρinv,σρhedge\)\\bm\{w\}=\(\\sigma\_\{\\xi\},\\sigma\_\{\\phi\},0,\\sigma\_\{\\rho\_\{\\mathrm\{inv\}\}\},\\sigma\_\{\\rho\_\{\\mathrm\{hedge\}\}\}\)is the weighted sensitivity vector \(the fill rate component enters throughAPY0\\mathrm\{APY\}\_\{0\}rather than the cost fraction\)\.

The*robustness margin*is:

ℛ=1−ξ0−ϕ0−ρinv,0−ρhedge,0χ52​\(1−ϵ\)⋅‖𝒘‖2\.\\mathcal\{R\}=\\frac\{1\-\\xi\_\{0\}\-\\phi\_\{0\}\-\\rho\_\{\\mathrm\{inv\},0\}\-\\rho\_\{\\mathrm\{hedge\},0\}\}\{\\sqrt\{\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)\}\\cdot\\\|\\bm\{w\}\\\|\_\{2\}\}\.\(86\)The high\-APY regime is*robust*ifℛ\>1\\mathcal\{R\}\>1, meaning the strategy remains profitable even under worst\-case parameter realization\.

###### Proof\.

SinceAPY​\(𝜽\)=APY0​\(ν\)⋅\(1−ξ−ϕ−ρinv−ρhedge\)\\mathrm\{APY\}\(\\bm\{\\theta\}\)=\\mathrm\{APY\}\_\{0\}\(\\nu\)\\cdot\(1\-\\xi\-\\phi\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\), andAPY0\\mathrm\{APY\}\_\{0\}is increasing inν\\nu, the worst case requires maximizing the cost fractionξ\+ϕ\+ρinv\+ρhedge\\xi\+\\phi\+\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}and minimizingν\\nu\.

For the cost fraction, holdingAPY0\\mathrm\{APY\}\_\{0\}fixed at the nominal value \(a lower bound\), the worst\-case cost is:

max𝜽∈Θϵ⁡\(ξ\+ϕ\+ρinv\+ρhedge\)\.\\max\_\{\\bm\{\\theta\}\\in\\Theta\_\{\\epsilon\}\}\(\\xi\+\\phi\+\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}\)\.By the Cauchy–Schwarz inequality applied to the ellipsoidal constraint \([84](https://arxiv.org/html/2607.11888#S5.E84)\):

ξ−ξ0\+ϕ−ϕ0\+ρinv−ρinv,0\+ρhedge−ρhedge,0≤χ52​\(1−ϵ\)⋅‖𝒘‖2,\\xi\-\\xi\_\{0\}\+\\phi\-\\phi\_\{0\}\+\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{inv\},0\}\+\\rho\_\{\\mathrm\{hedge\}\}\-\\rho\_\{\\mathrm\{hedge\},0\}\\leq\\sqrt\{\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)\}\\cdot\\\|\\bm\{w\}\\\|\_\{2\},with equality attained when the perturbation is aligned with the gradient direction𝒘/‖𝒘‖2\\bm\{w\}/\\\|\\bm\{w\}\\\|\_\{2\}\. Substituting into the APY formula yields \([85](https://arxiv.org/html/2607.11888#S5.E85)\)\. ∎

###### Corollary 5\.14\(Critical Uncertainty Level\)\.

The strategy becomes unprofitable under worst\-case parameters when:

‖𝒘‖2\>1−ξ0−ϕ0−ρinv,0−ρhedge,0χ52​\(1−ϵ\)\.\\\|\\bm\{w\}\\\|\_\{2\}\>\\frac\{1\-\\xi\_\{0\}\-\\phi\_\{0\}\-\\rho\_\{\\mathrm\{inv\},0\}\-\\rho\_\{\\mathrm\{hedge\},0\}\}\{\\sqrt\{\\chi^\{2\}\_\{5\}\(1\-\\epsilon\)\}\}\.\(87\)This provides a maximum tolerable parameter uncertainty for maintaining profitability at confidence level1−ϵ1\-\\epsilon\.

### 5\.8Master APY Decomposition

We now consolidate all cost channels into a unified closed\-form expression\.

###### Theorem 5\.16\(Master APY Formula\)\.

Under the optimal policy with hedge ratioζf∗\\zeta^\{\*\}\_\{f\}, the expected APY on deployed capital is:

APY=APY0⋅\(1−ξ\)\(1−ρinv−ρhedge−ρfund\),\\boxed\{\\mathrm\{APY\}=\\mathrm\{APY\}\_\{0\}\\cdot\(1\-\\xi\)\(1\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\-\\rho\_\{\\mathrm\{fund\}\}\),\}\(88\)whereAPY0=λ¯​Q​δ¯∗​Tyear/K\\mathrm\{APY\}\_\{0\}=\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}T\_\{\\mathrm\{year\}\}/Kis the gross APY and the four cost ratios are defined in \([149](https://arxiv.org/html/2607.11888#A4.E149)\)–\([152](https://arxiv.org/html/2607.11888#A4.E152)\) \(Appendix[D](https://arxiv.org/html/2607.11888#A4)\)\.

For APY\>A%\>A\\%, the necessary and sufficient condition on the cost structure is:

\(1−ξ\)​\(1−ρΣ\)\>AAPY0,\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)\>\\frac\{A\}\{\\mathrm\{APY\}\_\{0\}\},\(89\)whereρΣ=ρinv\+ρhedge\+ρfund\\rho\_\{\\Sigma\}=\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}\+\\rho\_\{\\mathrm\{fund\}\}\.

The factored form \([88](https://arxiv.org/html/2607.11888#S5.E88)\) reveals the multiplicative interaction between adverse selection and operational costs: even with lowξ\\xi, high aggregate frictionρΣ\\rho\_\{\\Sigma\}can eliminate profitability, and vice versa\. The full proof is in Appendix[D](https://arxiv.org/html/2607.11888#A4)\.

###### Corollary 5\.17\(Critical Boundaries for Target APY\)\.

For representative targets:

1. 1\.APY\>50%\>50\\%: requires\(1−ξ\)​\(1−ρΣ\)\>50/APY0\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)\>50/\\mathrm\{APY\}\_\{0\}\.
2. 2\.APY\>100%\>100\\%: requires\(1−ξ\)​\(1−ρΣ\)\>100/APY0\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)\>100/\\mathrm\{APY\}\_\{0\}\.
3. 3\.APY\>200%\>200\\%: requires\(1−ξ\)​\(1−ρΣ\)\>200/APY0\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)\>200/\\mathrm\{APY\}\_\{0\}\.

With typicalAPY0∼500%\\mathrm\{APY\}\_\{0\}\\sim 500\\%–2000%2000\\%\(depending on fill rate and leverage\), these translate to maximum tolerable cost fractions\(1−ξ\)​\(1−ρΣ\)∈\[0\.10,0\.40\]\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)\\in\[0\.10,0\.40\]\.

### 5\.9Asymptotic Scaling Laws

We characterize how APY scales in three limiting regimes, providing intuition for the capital efficiency frontier\.

###### Proposition 5\.18\(Asymptotic Scaling Laws\)\.

Under the Master APY Formula \(Theorem[D\.1](https://arxiv.org/html/2607.11888#A4.Thmtheorem1)\), the following asymptotic scaling relations hold:

\(i\) Fill rate scaling\.For fixed adverse selection and costs:

APY∼λ¯⋅Q​δ¯∗​\(1−ξ\)​\(1−ρΣ\)​TyearKas​λ¯→∞,\\mathrm\{APY\}\\sim\\bar\{\\lambda\}\\cdot\\frac\{Q\\bar\{\\delta\}^\{\*\}\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)T\_\{\\mathrm\{year\}\}\}\{K\}\\qquad\\text\{as \}\\bar\{\\lambda\}\\to\\infty,\(90\)i\.e\., APY scales linearly in fill rate \(before inventory cost saturation\)\.

\(ii\) Adverse selection scaling\.For fixed fill rate:

APY∼APY0​\(1−αδ¯∗\)​\(1−ρΣ\)as​α→δ¯∗,\\mathrm\{APY\}\\sim\\mathrm\{APY\}\_\{0\}\\left\(1\-\\frac\{\\alpha\}\{\\bar\{\\delta\}^\{\*\}\}\\right\)\(1\-\\rho\_\{\\Sigma\}\)\\qquad\\text\{as \}\\alpha\\to\\bar\{\\delta\}^\{\*\},\(91\)with zero\-crossing atα∗=δ¯∗​\(1−ρΣ/\(1−ρΣ\)\)\\alpha^\{\*\}=\\bar\{\\delta\}^\{\*\}\\bigl\(1\-\\rho\_\{\\Sigma\}/\(1\-\\rho\_\{\\Sigma\}\)\\bigr\); APY decays linearly to zero\.

\(iii\) Leverage scaling\.For fixed notional exposure:

APY​\(ℓ\)=ℓ⋅APY​\(1\)\(exact, up to buffer effects\),\\mathrm\{APY\}\(\\ell\)=\\ell\\cdot\\mathrm\{APY\}\(1\)\\qquad\\text\{\(exact, up to buffer effects\)\},\(92\)while the Sharpe ratio is leverage\-invariant:SR​\(ℓ\)=SR​\(1\)\\mathrm\{SR\}\(\\ell\)=\\mathrm\{SR\}\(1\)\.

###### Proof\.

Parts \(i\) and \(ii\) follow directly from the Master APY Formula \([88](https://arxiv.org/html/2607.11888#S5.E88)\) by holding the respective parameters fixed and varying the other\. For part \(iii\), sinceK=K0/ℓK=K\_\{0\}/\\ellandAPY=Π˙​Tyear/K\\mathrm\{APY\}=\\dot\{\\Pi\}T\_\{\\mathrm\{year\}\}/K, we haveAPY​\(ℓ\)=ℓ⋅Π˙​Tyear/K0=ℓ⋅APY​\(1\)\\mathrm\{APY\}\(\\ell\)=\\ell\\cdot\\dot\{\\Pi\}T\_\{\\mathrm\{year\}\}/K\_\{0\}=\\ell\\cdot\\mathrm\{APY\}\(1\)\. The Sharpe ratioSR=Π˙/Var​\[Π˙\]⋅Tyear\\mathrm\{SR\}=\\dot\{\\Pi\}/\\sqrt\{\\mathrm\{Var\}\[\\dot\{\\Pi\}\]\}\\cdot\\sqrt\{T\_\{\\mathrm\{year\}\}\}is independent ofKKand hence ofℓ\\ell\. ∎

### 5\.10Drawdown Probability Bounds

The high\-APY analysis is incomplete without characterizing downside risk\. We derive an exponential tail bound on the maximum drawdown \(MDD\) that links drawdown probability to the same parameters governing APY\.

###### Definition 5\.20\(Maximum Drawdown\)\.

The*maximum drawdown*over horizon\[0,T\]\[0,T\]is:

MDDT=max0≤s≤t≤T⁡\(Πs−Πt\),\\mathrm\{MDD\}\_\{T\}=\\max\_\{0\\leq s\\leq t\\leq T\}\\left\(\\Pi\_\{s\}\-\\Pi\_\{t\}\\right\),\(93\)whereΠt\\Pi\_\{t\}is the cumulative PnL at timett\.

###### Theorem 5\.21\(Drawdown Probability Bound\)\.

Under the optimal policy with expected PnL rateΠ˙\>0\\dot\{\\Pi\}\>0and PnL volatilityσΠ\>0\\sigma\_\{\\Pi\}\>0, the tail probability of the maximum drawdown satisfies the exponential bound:

ℙ​\(MDDT\>x\)≤exp⁡\(−2​Π˙σΠ2⋅x\),x\>0\.\\mathbb\{P\}\\bigl\(\\mathrm\{MDD\}\_\{T\}\>x\\bigr\)\\leq\\exp\\left\(\-\\frac\{2\\dot\{\\Pi\}\}\{\\sigma\_\{\\Pi\}^\{2\}\}\\cdot x\\right\),\\qquad x\>0\.\(94\)
The*tail decay rate*θdd=2​Π˙/σΠ2\\theta\_\{\\mathrm\{dd\}\}=2\\dot\{\\Pi\}/\\sigma\_\{\\Pi\}^\{2\}connects to the Sharpe ratio:

θdd=2​Π˙σΠ2=2​SR2σΠ​Tyear,\\theta\_\{\\mathrm\{dd\}\}=\\frac\{2\\dot\{\\Pi\}\}\{\\sigma\_\{\\Pi\}^\{2\}\}=\\frac\{2\\,\\mathrm\{SR\}^\{2\}\}\{\\sigma\_\{\\Pi\}T\_\{\\mathrm\{year\}\}\},\(95\)whereSR=Π˙​Tyear/σΠ\\mathrm\{SR\}=\\dot\{\\Pi\}\\sqrt\{T\_\{\\mathrm\{year\}\}\}/\\sigma\_\{\\Pi\}is the annualized Sharpe ratio\.

###### Proof\.

Approximate the cumulative PnL by a drifted Brownian motion:Πt≈Π˙​t\+σΠ​Bt\\Pi\_\{t\}\\approx\\dot\{\\Pi\}\\,t\+\\sigma\_\{\\Pi\}B\_\{t\}, where\(Bt\)\(B\_\{t\}\)is a standard Brownian motion\. This is justified at high fill frequencies by the Donsker invariance principle applied to the sum of i\.i\.d\. per\-fill edges\.

For a drifted Brownian motion with positive driftμ=Π˙\\mu=\\dot\{\\Pi\}and diffusion coefficientσ=σΠ\\sigma=\\sigma\_\{\\Pi\}, the classical reflection principle gives the exact distribution of the running maximum of−Πt\-\\Pi\_\{t\}\(equivalently, the drawdown\):

ℙ​\(MDDT\>x\)≤ℙ​\(supt≥0\(−μ​t\+σ​Bt\)\>x\)=exp⁡\(−2​μσ2​x\),\\mathbb\{P\}\\bigl\(\\mathrm\{MDD\}\_\{T\}\>x\\bigr\)\\leq\\mathbb\{P\}\\bigl\(\\sup\_\{t\\geq 0\}\(\-\\mu t\+\\sigma B\_\{t\}\)\>x\\bigr\)=\\exp\\left\(\-\\frac\{2\\mu\}\{\\sigma^\{2\}\}x\\right\),where the inequality follows from extending the horizon to infinity \(which only increases the supremum\) and the equality is the Wald identity for the all\-time maximum of a drifted Brownian motion with negative drift−μ<0\-\\mu<0\[[13](https://arxiv.org/html/2607.11888#bib.bib13),[27](https://arxiv.org/html/2607.11888#bib.bib27)\]\. Substitutingμ=Π˙\\mu=\\dot\{\\Pi\}andσ=σΠ\\sigma=\\sigma\_\{\\Pi\}yields \([94](https://arxiv.org/html/2607.11888#S5.E94)\)\. ∎

###### Corollary 5\.22\(Drawdown VaR and Capital Requirement\)\.

The drawdown Value\-at\-Risk at confidence level1−p1\-pis:

VaR1−pMDD=σΠ22​Π˙​ln⁡1p\.\\mathrm\{VaR\}\_\{1\-p\}^\{\\mathrm\{MDD\}\}=\\frac\{\\sigma\_\{\\Pi\}^\{2\}\}\{2\\dot\{\\Pi\}\}\\ln\\frac\{1\}\{p\}\.\(96\)For a market maker requiring that the maximum drawdown not exceedd%d\\%of capital with probability1−p1\-p, the minimum capital is:

Kmin=100d⋅σΠ22​Π˙​ln⁡1p\.K\_\{\\min\}=\\frac\{100\}\{d\}\\cdot\\frac\{\\sigma\_\{\\Pi\}^\{2\}\}\{2\\dot\{\\Pi\}\}\\ln\\frac\{1\}\{p\}\.\(97\)

### 5\.11Multi\-Pair Portfolio Allocation

We extend the single\-pair analysis to a market maker operating acrossNNcorrelated trading pairs, deriving the optimal capital allocation and quantifying the diversification benefit\.

###### Definition 5\.24\(Multi\-Pair MM Portfolio\)\.

A multi\-pair market maker simultaneously provides liquidity onNNperpetual futures pairs with pair\-specific parameters\(λ¯i,αi,σi,ki\)\(\\bar\{\\lambda\}\_\{i\},\\alpha\_\{i\},\\sigma\_\{i\},k\_\{i\}\),i=1,…,Ni=1,\\ldots,N\. The capital allocation vector𝒘=\(w1,…,wN\)\\bm\{w\}=\(w\_\{1\},\\ldots,w\_\{N\}\)satisfieswi≥0w\_\{i\}\\geq 0and∑iwi=1\\sum\_\{i\}w\_\{i\}=1, withwi​Kw\_\{i\}Kdeployed to pairii\. The pairwise PnL correlation between pairsiiandjjisρi​j=Cov​\[Π˙i,Π˙j\]/\(σΠ,i​σΠ,j\)\\rho\_\{ij\}=\\mathrm\{Cov\}\[\\dot\{\\Pi\}\_\{i\},\\dot\{\\Pi\}\_\{j\}\]/\(\\sigma\_\{\\Pi,i\}\\sigma\_\{\\Pi,j\}\)\.

###### Theorem 5\.25\(Optimal Multi\-Pair Allocation\)\.

Let𝛍=\(μ1,…,μN\)⊤\\bm\{\\mu\}=\(\\mu\_\{1\},\\ldots,\\mu\_\{N\}\)^\{\\top\}withμi=APYi\\mu\_\{i\}=\\mathrm\{APY\}\_\{i\}denote the vector of per\-pair APYs, and let𝚺\\bm\{\\Sigma\}be theN×NN\\times NPnL covariance matrix with entriesΣi​j=ρi​j​σΠ,i​σΠ,j\\Sigma\_\{ij\}=\\rho\_\{ij\}\\sigma\_\{\\Pi,i\}\\sigma\_\{\\Pi,j\}\. The maximum Sharpe ratio allocation solves:

wi∗=\(𝚺−1​𝝁\)i𝟏⊤​𝚺−1​𝝁,w\_\{i\}^\{\*\}=\\frac\{\(\\bm\{\\Sigma\}^\{\-1\}\\bm\{\\mu\}\)\_\{i\}\}\{\\bm\{1\}^\{\\top\}\\bm\{\\Sigma\}^\{\-1\}\\bm\{\\mu\}\},\(99\)and the portfolio Sharpe ratio satisfies:

SRN∗=𝝁⊤​𝚺−1​𝝁≥maxi⁡SRi\.\\mathrm\{SR\}\_\{N\}^\{\*\}=\\sqrt\{\\bm\{\\mu\}^\{\\top\}\\bm\{\\Sigma\}^\{\-1\}\\bm\{\\mu\}\}\\geq\\max\_\{i\}\\mathrm\{SR\}\_\{i\}\.\(100\)

###### Proof\.

This is a direct application of the Markowitz mean\-variance framework\[[28](https://arxiv.org/html/2607.11888#bib.bib28)\]\. The tangency portfolio maximizesSR​\(𝒘\)=𝒘⊤​𝝁/𝒘⊤​𝚺​𝒘\\mathrm\{SR\}\(\\bm\{w\}\)=\\bm\{w\}^\{\\top\}\\bm\{\\mu\}/\\sqrt\{\\bm\{w\}^\{\\top\}\\bm\{\\Sigma\}\\bm\{w\}\}\. Using Lagrange multipliers with the constraint𝟏⊤​𝒘=1\\bm\{1\}^\{\\top\}\\bm\{w\}=1, the first\-order condition gives𝒘∗∝𝚺−1​𝝁\\bm\{w\}^\{\*\}\\propto\\bm\{\\Sigma\}^\{\-1\}\\bm\{\\mu\}, normalized by the budget constraint to yield \([99](https://arxiv.org/html/2607.11888#S5.E99)\)\. The inequality \([100](https://arxiv.org/html/2607.11888#S5.E100)\) follows from𝝁⊤​𝚺−1​𝝁≥μi2/Σi​i=SRi2\\bm\{\\mu\}^\{\\top\}\\bm\{\\Sigma\}^\{\-1\}\\bm\{\\mu\}\\geq\\mu\_\{i\}^\{2\}/\\Sigma\_\{ii\}=\\mathrm\{SR\}\_\{i\}^\{2\}for allii\. ∎

###### Corollary 5\.26\(Diversification Bound for Homogeneous Pairs\)\.

ForNNhomogeneous pairs with common APYμ\\mu, common PnL volatilityσΠ\\sigma\_\{\\Pi\}, and equi\-correlationρ¯\\bar\{\\rho\}, the equal\-weight portfolio \(wi=1/Nw\_\{i\}=1/N\) achieves:

SRN=SR11N\+N−1N​ρ¯→N→∞SR1ρ¯\.\\mathrm\{SR\}\_\{N\}=\\frac\{\\mathrm\{SR\}\_\{1\}\}\{\\sqrt\{\\frac\{1\}\{N\}\+\\frac\{N\-1\}\{N\}\\bar\{\\rho\}\}\}\\xrightarrow\{N\\to\\infty\}\\frac\{\\mathrm\{SR\}\_\{1\}\}\{\\sqrt\{\\bar\{\\rho\}\}\}\.\(101\)The portfolio APY isAPYN=μ\\mathrm\{APY\}\_\{N\}=\\mu\(unchanged\), while the portfolio PnL volatility is reduced by the factor\(1\+\(N−1\)​ρ¯\)/N\\sqrt\{\(1\+\(N\-1\)\\bar\{\\rho\}\)/N\}\. The diversification benefit saturates at1/ρ¯1/\\sqrt\{\\bar\{\\rho\}\}asN→∞N\\to\\infty, with 90% of the limiting benefit achieved atN90%=⌈9​ρ¯/\(1−ρ¯\)⌉N^\{90\\%\}=\\lceil 9\\bar\{\\rho\}/\(1\-\\bar\{\\rho\}\)\\rceil\.

###### Proof\.

The equal\-weight portfolio variance is:

σΠ,N2=1N2​∑i,jΣi​j=σΠ2N2​\(N\+N​\(N−1\)​ρ¯\)=σΠ2​\(1N\+N−1N​ρ¯\)\.\\sigma\_\{\\Pi,N\}^\{2\}=\\frac\{1\}\{N^\{2\}\}\\sum\_\{i,j\}\\Sigma\_\{ij\}=\\frac\{\\sigma\_\{\\Pi\}^\{2\}\}\{N^\{2\}\}\\left\(N\+N\(N\-1\)\\bar\{\\rho\}\\right\)=\\sigma\_\{\\Pi\}^\{2\}\\left\(\\frac\{1\}\{N\}\+\\frac\{N\-1\}\{N\}\\bar\{\\rho\}\\right\)\.The Sharpe ratio follows fromSRN=N​μ/\(N​σΠ,N\)=μ/σΠ,N\\mathrm\{SR\}\_\{N\}=N\\mu/\(N\\sigma\_\{\\Pi,N\}\)=\\mu/\\sigma\_\{\\Pi,N\}\. For the 90% threshold, set1/N\+\(1−1/N\)​ρ¯=ρ¯\+0\.1​\(1−ρ¯\)\\sqrt\{1/N\+\(1\-1/N\)\\bar\{\\rho\}\}=\\sqrt\{\\bar\{\\rho\}\}\+0\.1\(1\-\\sqrt\{\\bar\{\\rho\}\}\)and solve\. ∎

### 5\.12Numerical Example: APY Achievability

###### Example 5\.28\(High\-APY Parameter Configuration\)\.

Consider the following parameter set \(representative of an altcoin perpetual on a zero\-fee DEX\):

- •Capital:K=$​1,000K=\\mathdollar 1\{,\}000with5×5\\timesleverage\.
- •Max inventory:q¯=100\\bar\{q\}=100contracts atS¯=$​20\\bar\{S\}=\\mathdollar 20each\.
- •Fill rate:λ¯=20\\bar\{\\lambda\}=20/hr = 480/day\.
- •Average half\-spread:δ¯=5\\bar\{\\delta\}=5bp\.
- •Adverse selection:α=2\\alpha=2bp\.
- •Maker fee:ϕm=0\\phi\_\{m\}=0\(zero\-fee regime\)\.
- •Order size:Q=10Q=10contracts\.

The net spread income per fill is\(δ¯−α\)⋅Q⋅S¯=3×10−4×10×20=$​0\.06\(\\bar\{\\delta\}\-\\alpha\)\\cdot Q\\cdot\\bar\{S\}=3\\times 10^\{\-4\}\\times 10\\times 20=\\mathdollar 0\.06\. Daily PnL:480×$​0\.06=$​28\.80480\\times\\mathdollar 0\.06=\\mathdollar 28\.80\. Annual PnL:$​28\.80×365=$​10,512\\mathdollar 28\.80\\times 365=\\mathdollar 10\{,\}512\.APY:10,512/1,000=1,051%10\{,\}512/1\{,\}000=1\{,\}051\\%\.

This simplified calculation ignores inventory costs and hedging friction, which reduce the realized APY\. Accounting forρinv≈0\.15\\rho\_\{\\mathrm\{inv\}\}\\approx 0\.15andρhedge≈0\.10\\rho\_\{\\mathrm\{hedge\}\}\\approx 0\.10\(typical values\), the adjusted APY is approximately1,051%×\(1−0\.15−0\.10\)=788%1\{,\}051\\%\\times\(1\-0\.15\-0\.10\)=788\\%\. In practice, slippage, downtime, and tail events further reduce this to a more realistic 50%–200% range\.

## 6Zero\-Fee Economics

This section analyzes the unique economic properties of the zero maker fee regime found on certain decentralized perpetual exchanges, quantifying the advantage it confers to market makers relative to standard fee structures\.

### 6\.1The Fee Floor Effect

###### Theorem 6\.1\(Fee Floor Elimination\)\.

Under a standard fee regime with maker feeϕm\>0\\phi\_\{m\}\>0, the minimum viable half\-spread for non\-negative expected PnL per fill is:

δminCEX=α\+ϕm\.\\delta\_\{\\min\}^\{\\mathrm\{CEX\}\}=\\alpha\+\\phi\_\{m\}\.\(102\)Under the zero\-fee regime \(ϕm=0\\phi\_\{m\}=0\):

δminDEX=α\.\\delta\_\{\\min\}^\{\\mathrm\{DEX\}\}=\\alpha\.\(103\)The zero\-fee regime therefore expands the set of profitable spread levels by the interval\[α,α\+ϕm\)\[\\alpha,\\alpha\+\\phi\_\{m\}\)\.

###### Proof\.

The expected PnL per fill isδ¯−α−ϕm\\bar\{\\delta\}\-\\alpha\-\\phi\_\{m\}, which is non\-negative whenδ¯≥α\+ϕm\\bar\{\\delta\}\\geq\\alpha\+\\phi\_\{m\}\. Settingϕm=0\\phi\_\{m\}=0removes the fee floor\. ∎

### 6\.2Market Expansion Effect

The zero\-fee regime expands the set of*tradeable markets*—markets where a MM can profitably provide liquidity\.

###### Definition 6\.2\(Tradeable Market\)\.

A market is*tradeable*for a market maker if the optimal policy yields positive expected APY:

APY∗​\(σ,Λ,k,α,ϕm\)\>0\.\\mathrm\{APY\}^\{\*\}\(\\sigma,\\Lambda,k,\\alpha,\\phi\_\{m\}\)\>0\.\(104\)

###### Proposition 6\.3\(Market Expansion Under Zero Fees\)\.

Letℳϕm\\mathcal\{M\}\_\{\\phi\_\{m\}\}denote the set of tradeable markets under feeϕm\\phi\_\{m\}\. Then:

ℳ0⊋ℳϕmfor all​ϕm\>0\.\\mathcal\{M\}\_\{0\}\\supsetneq\\mathcal\{M\}\_\{\\phi\_\{m\}\}\\quad\\text\{for all \}\\phi\_\{m\}\>0\.\(105\)The additional tradeable marketsℳ0∖ℳϕm\\mathcal\{M\}\_\{0\}\\setminus\\mathcal\{M\}\_\{\\phi\_\{m\}\}are characterized by:

α<δ¯∗−ρinv​δ¯∗≤α\+ϕm\.\\alpha<\\bar\{\\delta\}^\{\*\}\-\\rho\_\{\\mathrm\{inv\}\}\\bar\{\\delta\}^\{\*\}\\leq\\alpha\+\\phi\_\{m\}\.\(106\)These are markets where the native spread is sufficient to cover adverse selection but*not*sufficient to cover adverse selection plus fees\.

### 6\.3Fill Rate Enhancement

Under zero fees, the optimal spread is narrower, which increases the fill rate:

###### Proposition 6\.4\(Fill Rate Enhancement\)\.

The ratio of optimal fill rates under zero fees versus standard fees is:

λ¯DEXλ¯CEX=e−k​δ¯DEXe−k​δ¯CEX=ek​ϕm\>1\.\\frac\{\\bar\{\\lambda\}^\{\\mathrm\{DEX\}\}\}\{\\bar\{\\lambda\}^\{\\mathrm\{CEX\}\}\}=\\frac\{e^\{\-k\\bar\{\\delta\}^\{\\mathrm\{DEX\}\}\}\}\{e^\{\-k\\bar\{\\delta\}^\{\\mathrm\{CEX\}\}\}\}=e^\{k\\phi\_\{m\}\}\>1\.\(107\)For typical valuesk​ϕm∈\[0\.1,0\.5\]k\\phi\_\{m\}\\in\[0\.1,0\.5\], the fill rate enhancement is 10%–65%\.

###### Proof\.

The optimal half\-spread under zero fees isδ¯DEX=1/k\+γ​σ2​τ/2\+α\\bar\{\\delta\}^\{\\mathrm\{DEX\}\}=1/k\+\\gamma\\sigma^\{2\}\\tau/2\+\\alpha, and under standard fees isδ¯CEX=1/k\+γ​σ2​τ/2\+α\+ϕm\\bar\{\\delta\}^\{\\mathrm\{CEX\}\}=1/k\+\\gamma\\sigma^\{2\}\\tau/2\+\\alpha\+\\phi\_\{m\}\. The fill rate ratio follows fromλ¯∝e−k​δ¯\\bar\{\\lambda\}\\propto e^\{\-k\\bar\{\\delta\}\}\. ∎

### 6\.4APY Advantage Quantification

###### Theorem 6\.5\(Zero\-Fee APY Advantage\)\.

The APY advantage of the zero\-fee regime over a standard fee regime with maker feeϕm\\phi\_\{m\}is:

Δ​APY=APYDEX−APYCEX=APY0DEX⋅ϕ−APY0CEX⋅ϕ\+\(APY0DEX−APY0CEX\)​\(1−ξ−ρinv\),\\Delta\\mathrm\{APY\}=\\mathrm\{APY\}^\{\\mathrm\{DEX\}\}\-\\mathrm\{APY\}^\{\\mathrm\{CEX\}\}=\\mathrm\{APY\}\_\{0\}^\{\\mathrm\{DEX\}\}\\cdot\\phi\-\\mathrm\{APY\}\_\{0\}^\{\\mathrm\{CEX\}\}\\cdot\\phi\+\(\\mathrm\{APY\}\_\{0\}^\{\\mathrm\{DEX\}\}\-\\mathrm\{APY\}\_\{0\}^\{\\mathrm\{CEX\}\}\)\(1\-\\xi\-\\rho\_\{\\mathrm\{inv\}\}\),\(108\)where the first terms capture the direct fee savings and the last term captures the fill\-rate enhancement\.

To first order inϕm\\phi\_\{m\}:

Δ​APY≈APY0⋅ϕm​\(1δ¯\+k​\(1−ξ−ρinv\)\)\.\\Delta\\mathrm\{APY\}\\approx\\mathrm\{APY\}\_\{0\}\\cdot\\phi\_\{m\}\\left\(\\frac\{1\}\{\\bar\{\\delta\}\}\+k\(1\-\\xi\-\\rho\_\{\\mathrm\{inv\}\}\)\\right\)\.\(109\)

### 6\.5Comparative Statics

Table 3:Theoretical APY comparison across fee regimes \(representative parameters\)
### 6\.6Welfare Analysis

We quantify the total welfare gain from the zero\-fee regime across all market participants\.

###### Definition 6\.7\(Total Welfare\)\.

The*total welfare*𝒲\\mathcal\{W\}in a market is the sum of:

𝒲=ΠMM⏟market maker surplus\+CS⏟consumer \(taker\) surplus\+PS⏟platform surplus \(fees collected\)\.\\mathcal\{W\}=\\underbrace\{\\Pi\_\{\\mathrm\{MM\}\}\}\_\{\\text\{market maker surplus\}\}\+\\underbrace\{\\mathrm\{CS\}\}\_\{\\text\{consumer \(taker\) surplus\}\}\+\\underbrace\{\\mathrm\{PS\}\}\_\{\\text\{platform surplus \(fees collected\)\}\}\.\(110\)

###### Proposition 6\.8\(Welfare Gain from Zero Fees\)\.

The welfare difference between the zero\-fee and standard fee regimes is:

Δ​𝒲=𝒲0−𝒲ϕm=ϕm​S¯​Q​λ¯ϕm\+ϕmk​\(λ¯0−λ¯ϕm\)​S¯​Q\+∫λ¯ϕmλ¯0D−1​\(λ\)​dλ,\\Delta\\mathcal\{W\}=\\mathcal\{W\}\_\{0\}\-\\mathcal\{W\}\_\{\\phi\_\{m\}\}=\\phi\_\{m\}\\bar\{S\}Q\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}\+\\frac\{\\phi\_\{m\}\}\{k\}\\left\(\\bar\{\\lambda\}\_\{0\}\-\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}\\right\)\\bar\{S\}Q\+\\int\_\{\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}\}^\{\\bar\{\\lambda\}\_\{0\}\}D^\{\-1\}\(\\lambda\)\\,\\mathrm\{d\}\\lambda,\(111\)where the first term is the direct fee redistribution, the second captures increased MM profits from higher fill rates, and the third is the taker surplus from improved execution\.

The welfare gain is strictly positive:Δ​𝒲\>0\\Delta\\mathcal\{W\}\>0for allϕm\>0\\phi\_\{m\}\>0\.

###### Proof\.

Under the standard fee regime, the platform collectsϕm​S¯​Q\\phi\_\{m\}\\bar\{S\}Qper fill\. Eliminating this fee transfers the full amount to MMs and takers\. The fill rate increases fromλ¯ϕm\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}toλ¯0=λ¯ϕm​ek​ϕm\\bar\{\\lambda\}\_\{0\}=\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}e^\{k\\phi\_\{m\}\}\(Proposition[6\.4](https://arxiv.org/html/2607.11888#S6.Thmtheorem4)\)\. The additional fills generate surplus for both MMs \(at rate\(1/k\)\(1/k\)per fill\) and takers \(through the demand curveD−1D^\{\-1\}\)\. Sinceλ¯0\>λ¯ϕm\\bar\{\\lambda\}\_\{0\}\>\\bar\{\\lambda\}\_\{\\phi\_\{m\}\}andD−1\>0D^\{\-1\}\>0, all terms are positive\. ∎

### 6\.7Entry–Exit Dynamics and Hysteresis

In non\-stationary markets, the adverse selection ratioξt=αt/δ¯\\xi\_\{t\}=\\alpha\_\{t\}/\\bar\{\\delta\}fluctuates\. The MM must decide when to*enter*\(begin quoting\) and*exit*\(withdraw quotes\)\. We formalize this as an optimal stopping problem with hysteresis\.

###### Definition 6\.9\(Entry–Exit Thresholds\)\.

Letcentry\>0c\_\{\\mathrm\{entry\}\}\>0be the fixed cost of initiating a quoting session \(order placement, state synchronization\) andcexit\>0c\_\{\\mathrm\{exit\}\}\>0the cost of orderly withdrawal \(cancel latency risk, inventory liquidation\)\. The*entry threshold*ξentry\\xi\_\{\\mathrm\{entry\}\}and*exit threshold*ξexit\\xi\_\{\\mathrm\{exit\}\}satisfy:

ξentry<ξexit<1,\\xi\_\{\\mathrm\{entry\}\}<\\xi\_\{\\mathrm\{exit\}\}<1,\(112\)where the MM enters whenξt\\xi\_\{t\}crosses belowξentry\\xi\_\{\\mathrm\{entry\}\}and exits whenξt\\xi\_\{t\}crosses aboveξexit\\xi\_\{\\mathrm\{exit\}\}\.

###### Theorem 6\.10\(Optimal Entry–Exit with Hysteresis\)\.

Assumeξt\\xi\_\{t\}follows an Ornstein–Uhlenbeck process:

d​ξt=κξ​\(ξ¯−ξt\)​d​t\+σξ​d​Bt,\\mathrm\{d\}\\xi\_\{t\}=\\kappa\_\{\\xi\}\(\\bar\{\\xi\}\-\\xi\_\{t\}\)\\,\\mathrm\{d\}t\+\\sigma\_\{\\xi\}\\,\\mathrm\{d\}B\_\{t\},\(113\)with long\-run meanξ¯∈\(0,1\)\\bar\{\\xi\}\\in\(0,1\)\. The value functionsVactive​\(ξ\)V\_\{\\mathrm\{active\}\}\(\\xi\)andVidle​\(ξ\)V\_\{\\mathrm\{idle\}\}\(\\xi\)satisfy the coupled free\-boundary problem:

σξ22​Vactive′′\+κξ​\(ξ¯−ξ\)​Vactive′\+π​\(ξ\)\\displaystyle\\frac\{\\sigma\_\{\\xi\}^\{2\}\}\{2\}V\_\{\\mathrm\{active\}\}^\{\\prime\\prime\}\+\\kappa\_\{\\xi\}\(\\bar\{\\xi\}\-\\xi\)V\_\{\\mathrm\{active\}\}^\{\\prime\}\+\\pi\(\\xi\)=0,ξ<ξexit,\\displaystyle=0,\\quad\\xi<\\xi\_\{\\mathrm\{exit\}\},\(114\)σξ22​Vidle′′\+κξ​\(ξ¯−ξ\)​Vidle′\\displaystyle\\frac\{\\sigma\_\{\\xi\}^\{2\}\}\{2\}V\_\{\\mathrm\{idle\}\}^\{\\prime\\prime\}\+\\kappa\_\{\\xi\}\(\\bar\{\\xi\}\-\\xi\)V\_\{\\mathrm\{idle\}\}^\{\\prime\}=0,ξ\>ξentry,\\displaystyle=0,\\quad\\xi\>\\xi\_\{\\mathrm\{entry\}\},\(115\)with value\-matching and smooth\-pasting conditions at both boundaries:

Vactive​\(ξexit\)=Vidle​\(ξexit\)−cexit,Vactive′​\(ξexit\)=Vidle′​\(ξexit\),V\_\{\\mathrm\{active\}\}\(\\xi\_\{\\mathrm\{exit\}\}\)=V\_\{\\mathrm\{idle\}\}\(\\xi\_\{\\mathrm\{exit\}\}\)\-c\_\{\\mathrm\{exit\}\},\\quad V\_\{\\mathrm\{active\}\}^\{\\prime\}\(\\xi\_\{\\mathrm\{exit\}\}\)=V\_\{\\mathrm\{idle\}\}^\{\\prime\}\(\\xi\_\{\\mathrm\{exit\}\}\),\(116\)Vidle​\(ξentry\)=Vactive​\(ξentry\)−centry,Vidle′​\(ξentry\)=Vactive′​\(ξentry\),V\_\{\\mathrm\{idle\}\}\(\\xi\_\{\\mathrm\{entry\}\}\)=V\_\{\\mathrm\{active\}\}\(\\xi\_\{\\mathrm\{entry\}\}\)\-c\_\{\\mathrm\{entry\}\},\\quad V\_\{\\mathrm\{idle\}\}^\{\\prime\}\(\\xi\_\{\\mathrm\{entry\}\}\)=V\_\{\\mathrm\{active\}\}^\{\\prime\}\(\\xi\_\{\\mathrm\{entry\}\}\),\(117\)whereπ​\(ξ\)=APY0​\(1−ξ\)​\(1−ρΣ\)\\pi\(\\xi\)=\\mathrm\{APY\}\_\{0\}\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)is the instantaneous profit rate\.

The*hysteresis width*Δ​ξ=ξexit−ξentry\\Delta\\xi=\\xi\_\{\\mathrm\{exit\}\}\-\\xi\_\{\\mathrm\{entry\}\}is strictly positive whenevercentry\+cexit\>0c\_\{\\mathrm\{entry\}\}\+c\_\{\\mathrm\{exit\}\}\>0, and satisfies:

Δ​ξ≥2​\(centry\+cexit\)APY0​\(1−ρΣ\)\.\\Delta\\xi\\geq\\sqrt\{\\frac\{2\(c\_\{\\mathrm\{entry\}\}\+c\_\{\\mathrm\{exit\}\}\)\}\{\\mathrm\{APY\}\_\{0\}\(1\-\\rho\_\{\\Sigma\}\)\}\}\.\(118\)

###### Proof\.

The coupled free\-boundary system follows from the standard theory of optimal switching\[[31](https://arxiv.org/html/2607.11888#bib.bib31),[8](https://arxiv.org/html/2607.11888#bib.bib8)\]\. The running payoffπ​\(ξ\)=APY0​\(1−ξ\)​\(1−ρΣ\)\\pi\(\\xi\)=\\mathrm\{APY\}\_\{0\}\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\)is strictly decreasing inξ\\xi, ensuring uniqueness of the free boundaries by the monotonicity arguments ofBrekke and Øksendal \[[6](https://arxiv.org/html/2607.11888#bib.bib6)\]\.

For the hysteresis width bound \([118](https://arxiv.org/html/2607.11888#S6.E118)\), observe that the minimum cost of a full entry–exit cycle iscentry\+cexitc\_\{\\mathrm\{entry\}\}\+c\_\{\\mathrm\{exit\}\}\. The maximum profit accumulated during a cycle in whichξ\\xitraverses\[ξentry,ξexit\]\[\\xi\_\{\\mathrm\{entry\}\},\\xi\_\{\\mathrm\{exit\}\}\]and returns is bounded byAPY0​\(1−ρΣ\)⋅\(Δ​ξ\)2/\(2​σξ2\)⋅σξ2=APY0​\(1−ρΣ\)​\(Δ​ξ\)2/2\\mathrm\{APY\}\_\{0\}\(1\-\\rho\_\{\\Sigma\}\)\\cdot\(\\Delta\\xi\)^\{2\}/\(2\\sigma\_\{\\xi\}^\{2\}\)\\cdot\\sigma\_\{\\xi\}^\{2\}=\\mathrm\{APY\}\_\{0\}\(1\-\\rho\_\{\\Sigma\}\)\(\\Delta\\xi\)^\{2\}/2\(using the expected cycle time for an OU process between two barriers\)\. For a cycle to be profitable:APY0​\(1−ρΣ\)​\(Δ​ξ\)2/2≥centry\+cexit\\mathrm\{APY\}\_\{0\}\(1\-\\rho\_\{\\Sigma\}\)\(\\Delta\\xi\)^\{2\}/2\\geq c\_\{\\mathrm\{entry\}\}\+c\_\{\\mathrm\{exit\}\}, yielding \([118](https://arxiv.org/html/2607.11888#S6.E118)\)\. ∎

###### Corollary 6\.11\(Zero\-Fee Hysteresis Advantage\)\.

Under zero fees, the break\-even boundary shifts fromξbreakCEX=1−ϕ/\(1−ρΣ\)\\xi\_\{\\mathrm\{break\}\}^\{\\mathrm\{CEX\}\}=1\-\\phi/\(1\-\\rho\_\{\\Sigma\}\)toξbreakDEX=1−ρΣ/\(1−ρΣ\+ρΣ\)=1\\xi\_\{\\mathrm\{break\}\}^\{\\mathrm\{DEX\}\}=1\-\\rho\_\{\\Sigma\}/\(1\-\\rho\_\{\\Sigma\}\+\\rho\_\{\\Sigma\}\)=1, expanding the viable operating range\. Consequently, the zero\-fee MM can tolerate wider hysteresis bands while remaining profitable, enabling more patient entry–exit decisions that reduce switching costs and improve overall returns\. See Figure[20](https://arxiv.org/html/2607.11888#S8.F20)for an illustration\.

### 6\.8Equilibrium Implications

###### Proposition 6\.12\(Equilibrium Spread Under Zero Fees\)\.

In a competitive equilibrium wherennidentical MMs compete for order flow on a zero\-fee DEX, the equilibrium spread converges to:

δeq=α\+1n​k\+C˙invn​λ¯​Q,\\delta\_\{\\mathrm\{eq\}\}=\\alpha\+\\frac\{1\}\{nk\}\+\\frac\{\\dot\{C\}\_\{\\mathrm\{inv\}\}\}\{n\\bar\{\\lambda\}Q\},\(119\)asn→∞n\\to\\infty,δeq→α\\delta\_\{\\mathrm\{eq\}\}\\to\\alpha: the spread converges to the adverse selection cost, and MM profits vanish\. On a CEX, the corresponding equilibrium isδeqCEX=α\+ϕm\+1/\(n​k\)\\delta\_\{\\mathrm\{eq\}\}^\{\\mathrm\{CEX\}\}=\\alpha\+\\phi\_\{m\}\+1/\(nk\), which is strictly wider\.

The zero\-fee equilibrium yields tighter spreads and thus better execution quality for takers, consistent with the DEX’s design intent\.

## 7Cross\-Exchange Inventory Hedging

When the market maker accumulates inventory on DEX\-A, she can hedge by taking an offsetting position on CEX\-B\. This section derives the optimal hedging policy within the stochastic control framework, incorporating funding rate dynamics, regime\-dependent hedge boundaries, and multi\-leg execution\.

### 7\.1Hedging Problem Formulation

The MM holds inventoryqtq\_\{t\}on DEX\-A and a hedge positionHtH\_\{t\}on CEX\-B\. The net exposure isqtnet=qt\+Htq\_\{t\}^\{\\mathrm\{net\}\}=q\_\{t\}\+H\_\{t\}\. Each hedge trade on CEX\-B incurs a taker feeϕtCEX\\phi\_\{t\}^\{\\mathrm\{CEX\}\}per unit of notional\.

###### Definition 7\.1\(Hedge Cost\)\.

The instantaneous cost of adjusting the hedge byΔ​H\\Delta Hunits is:

ch​\(Δ​H\)=ϕtCEX⋅\|Δ​H\|⋅Q⋅St\.c\_\{h\}\(\\Delta H\)=\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\cdot\|\\Delta H\|\\cdot Q\\cdot S\_\{t\}\.\(120\)For a full round\-trip hedge \(open and close\), the cost is2​ϕtCEX⋅\|Δ​H\|⋅Q⋅St2\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\cdot\|\\Delta H\|\\cdot Q\\cdot S\_\{t\}\.

### 7\.2Optimal Hedge Threshold

The MM faces a tradeoff: hedging reduces inventory risk but incurs transaction costs\. We solve for the optimal inventory threshold at which hedging becomes worthwhile\.

###### Theorem 7\.2\(Optimal Hedge Threshold\)\.

Under CARA utility with risk aversionγ\\gammaand constant hedge costϕtCEX\\phi\_\{t\}^\{\\mathrm\{CEX\}\}, the optimal hedging policy is a*threshold policy*: hedge when\|qt\|\>q∗\|q\_\{t\}\|\>q^\{\*\}and do not hedge otherwise\. The optimal threshold is:

q∗=ϕtCEX​S¯γ​σ2​Q​\(T−t\)\.q^\{\*\}=\\frac\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma^\{2\}Q\(T\-t\)\}\.\(121\)When\|qt\|\>q∗\|q\_\{t\}\|\>q^\{\*\}, the optimal hedge is to reduce net exposure toqnet=sign​\(qt\)⋅q∗q^\{\\mathrm\{net\}\}=\\mathrm\{sign\}\(q\_\{t\}\)\\cdot q^\{\*\}:

Δ​H=−\(qt−sign​\(qt\)​q∗\)\.\\Delta H=\-\(q\_\{t\}\-\\mathrm\{sign\}\(q\_\{t\}\)q^\{\*\}\)\.\(122\)

###### Proof\.

The marginal value of hedging one unit at inventoryqqis the reduction in inventory risk:

Δ​Vrisk​\(q\)=12​γ​σ2​Q2​\(T−t\)​\[q2−\(q−1\)2\]=12​γ​σ2​Q2​\(T−t\)​\(2​q−1\)≈γ​σ2​Q2​\(T−t\)​q\\Delta V\_\{\\mathrm\{risk\}\}\(q\)=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\(T\-t\)\[q^\{2\}\-\(q\-1\)^\{2\}\]=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\(T\-t\)\(2q\-1\)\\approx\\gamma\\sigma^\{2\}Q^\{2\}\(T\-t\)qfor large\|q\|\|q\|\. The marginal cost of hedging isϕtCEX​Q​St\\phi\_\{t\}^\{\\mathrm\{CEX\}\}QS\_\{t\}\. Equating marginal benefit to marginal cost:

γ​σ2​Q2​\(T−t\)​q∗=ϕtCEX​Q​St⟹q∗=ϕtCEX​Stγ​σ2​Q​\(T−t\)\.\\gamma\\sigma^\{2\}Q^\{2\}\(T\-t\)q^\{\*\}=\\phi\_\{t\}^\{\\mathrm\{CEX\}\}QS\_\{t\}\\implies q^\{\*\}=\\frac\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}S\_\{t\}\}\{\\gamma\\sigma^\{2\}Q\(T\-t\)\}\.∎

###### Corollary 7\.3\(Stationary Hedge Threshold\)\.

Under the inventory penalization approach with parameterη\\eta, the stationary hedge threshold is:

q∞∗=ϕtCEX​S¯η​Q\.q^\{\*\}\_\{\\infty\}=\\frac\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\eta Q\}\.\(123\)

### 7\.3Optimal Hedge Ratio

In practice, the MM may not fully offset inventory but instead maintain a*partial hedge*\.

###### Definition 7\.4\(Hedge Ratio\)\.

The hedge ratioζ∈\[0,1\]\\zeta\\in\[0,1\]is the fraction of DEX inventory offset on CEX\-B:

Ht=−ζ⋅qt\.H\_\{t\}=\-\\zeta\\cdot q\_\{t\}\.\(124\)The net exposure isqtnet=\(1−ζ\)​qtq\_\{t\}^\{\\mathrm\{net\}\}=\(1\-\\zeta\)q\_\{t\}\.

###### Theorem 7\.5\(Optimal Hedge Ratio\)\.

The optimal hedge ratio that maximizes the Sharpe ratio of the combined \(DEX \+ CEX\) portfolio is:

ζ∗=1−nh⋅ϕtCEX​S¯γ​σ2​𝔼​\[qt2\]​Q​\(T−t\),\\zeta^\{\*\}=1\-\\frac\{n\_\{h\}\\cdot\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\(T\-t\)\},\(125\)wherenhn\_\{h\}is the expected number of hedge adjustments per unit time\. When the hedge cost is negligible \(ϕtCEX→0\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\to 0\),ζ∗→1\\zeta^\{\*\}\\to 1\(full hedge\)\. When the hedge cost is large,ζ∗→0\\zeta^\{\*\}\\to 0\(no hedge\)\.

###### Proof\.

The hedged PnL rate is:

Π˙hedge​\(ζ\)=λ¯​Q​\(δ¯−α\)−12​γ​σ2​\(1−ζ\)2​𝔼​\[qt2\]​Q2−nh​ζ⋅ϕtCEX​Q​S¯\.\\dot\{\\Pi\}\_\{\\mathrm\{hedge\}\}\(\\zeta\)=\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(1\-\\zeta\)^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}\-n\_\{h\}\\zeta\\cdot\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\.The variance of the hedged PnL is:

Var​\[Π˙\]=λ¯​Q2​Var​\[ei\]\+\(1−ζ\)2​σ2​𝔼​\[qt2\]​Q2\.\\mathrm\{Var\}\[\\dot\{\\Pi\}\]=\\bar\{\\lambda\}Q^\{2\}\\mathrm\{Var\}\[e\_\{i\}\]\+\(1\-\\zeta\)^\{2\}\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}\.Maximizing the Sharpe ratioΠ˙hedge/Var​\[Π˙\]\\dot\{\\Pi\}\_\{\\mathrm\{hedge\}\}/\\sqrt\{\\mathrm\{Var\}\[\\dot\{\\Pi\}\]\}overζ\\zeta, the FOC yields \([125](https://arxiv.org/html/2607.11888#S7.E125)\)\. ∎

### 7\.4The Premium Arbitrage Channel

When the DEX\-A premiumβt\\beta\_\{t\}is positive \(DEX price exceeds CEX price\), the MM can earn an additional return by systematically selling on DEX\-A and buying on CEX\-B\.

###### Proposition 7\.6\(Premium Capture\)\.

If the long\-run premium isβ¯\>0\\bar\{\\beta\}\>0, the MM captures an expected premium income of:

Π˙premium=λ¯a⋅Q⋅β¯−λ¯b⋅Q⋅β¯,\\dot\{\\Pi\}\_\{\\mathrm\{premium\}\}=\\bar\{\\lambda\}^\{a\}\\cdot Q\\cdot\\bar\{\\beta\}\-\\bar\{\\lambda\}^\{b\}\\cdot Q\\cdot\\bar\{\\beta\},\(126\)whereλ¯a\\bar\{\\lambda\}^\{a\}andλ¯b\\bar\{\\lambda\}^\{b\}are the ask and bid fill rates\. Under the optimal inventory\-skewing policy \(which generates a slight sell bias whenβ¯\>0\\bar\{\\beta\}\>0because the reservation price is adjusted\), the net premium income is positive\.

### 7\.5Dynamic Hedge Timing

The threshold policy of Theorem[7\.2](https://arxiv.org/html/2607.11888#S7.Thmtheorem2)prescribes*when*to hedge but not the optimal*frequency*of hedge re\-evaluation\. We now derive the optimal hedge check interval under discrete monitoring\.

###### Proposition 7\.7\(Optimal Hedge Re\-Evaluation Interval\)\.

Suppose the MM re\-evaluates the hedge decision at intervals of lengthΔ​τ\>0\\Delta\\tau\>0\. Between hedge checks, inventory evolves according to the fill process\. The expected cost of delayed hedging \(excess inventory risk from not hedging continuously\) over one interval is:

Cdelay​\(Δ​τ\)=12​γ​σ2​Q2​λ¯​Q⋅\(Δ​τ\)22,C\_\{\\mathrm\{delay\}\}\(\\Delta\\tau\)=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}Q\\cdot\\frac\{\(\\Delta\\tau\)^\{2\}\}\{2\},\(127\)and the expected benefit of reduced hedge frequency \(fewer round\-trip costs\) is:

Bdelay​\(Δ​τ\)=ϕtCEX​Q​S¯Δ​τ⋅\(1Δ​τ0−1Δ​τ\)−1,B\_\{\\mathrm\{delay\}\}\(\\Delta\\tau\)=\\frac\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\}\{\\Delta\\tau\}\\cdot\\left\(\\frac\{1\}\{\\Delta\\tau\_\{0\}\}\-\\frac\{1\}\{\\Delta\\tau\}\\right\)^\{\-1\},\(128\)whereΔ​τ0\\Delta\\tau\_\{0\}is the minimum feasible interval\.

The optimal interval that minimizes total cost is:

Δ​τ∗=\(2​ϕtCEX​S¯γ​σ2​Q​λ¯\)1/3\.\\Delta\\tau^\{\*\}=\\left\(\\frac\{2\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma^\{2\}Q\\bar\{\\lambda\}\}\\right\)^\{1/3\}\.\(129\)

###### Proof\.

The inventory accumulated between hedge checks has variance𝔼​\[\(Δ​q\)2\]≈λ¯​Δ​τ\\mathbb\{E\}\[\(\\Delta q\)^\{2\}\]\\approx\\bar\{\\lambda\}\\Delta\\tau\(Poisson arrivals with unit size\)\. The excess risk cost from holding this unhedged inventory scales as12​γ​σ2​Q2⋅λ¯​Δ​τ⋅Δ​τ=12​γ​σ2​Q2​λ¯​\(Δ​τ\)2\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\cdot\\bar\{\\lambda\}\\Delta\\tau\\cdot\\Delta\\tau=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}\(\\Delta\\tau\)^\{2\}\. The hedge transaction cost per unit time isϕtCEX​Q​S¯/Δ​τ\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}/\\Delta\\tau\(one hedge per interval\)\. Minimizing the sumCdelay\+ϕtCEX​Q​S¯/Δ​τC\_\{\\mathrm\{delay\}\}\+\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}/\\Delta\\tauoverΔ​τ\\Delta\\tau, the FOC gives \([129](https://arxiv.org/html/2607.11888#S7.E129)\)\. ∎

### 7\.6Funding Rate Dynamics and Hedging Cost

Perpetual futures contracts settle a periodic*funding rate*rf​\(t\)r\_\{f\}\(t\)that transfers wealth between long and short position holders\. This introduces a first\-order cost or income channel for the hedged MM\.

###### Definition 7\.8\(Funding Rate Process\)\.

The funding raterf​\(t\)r\_\{f\}\(t\)is settled at discrete intervals\{tk\}k≥1\\\{t\_\{k\}\\\}\_\{k\\geq 1\}\(typically everyΔf=8\\Delta\_\{f\}=8hours\)\. The instantaneous funding rate follows a mean\-reverting process:

d​rf​\(t\)=−κf​\(rf​\(t\)−r¯f\)​d​t\+σf​d​Wtf,\\mathrm\{d\}r\_\{f\}\(t\)=\-\\kappa\_\{f\}\(r\_\{f\}\(t\)\-\\bar\{r\}\_\{f\}\)\\,\\mathrm\{d\}t\+\\sigma\_\{f\}\\,\\mathrm\{d\}W\_\{t\}^\{f\},\(130\)wherer¯f≥0\\bar\{r\}\_\{f\}\\geq 0is the long\-run mean funding rate,κf\>0\\kappa\_\{f\}\>0is the mean\-reversion speed,σf\>0\\sigma\_\{f\}\>0is the funding rate volatility, andWtfW\_\{t\}^\{f\}is a Brownian motion independent of\(Wt,Wtβ\)\(W\_\{t\},W\_\{t\}^\{\\beta\}\)\.

At each settlement timetkt\_\{k\}, a long position of sizeqqpaysrf​\(tk\)⋅q⋅Q⋅Stkr\_\{f\}\(t\_\{k\}\)\\cdot q\\cdot Q\\cdot S\_\{t\_\{k\}\}\(positive whenrf\>0r\_\{f\}\>0\), and a short position receives the same amount\.

###### Definition 7\.9\(Funding\-Adjusted Hedge Cost\)\.

The*total hedging cost rate*including funding comprises three components:

C˙hedgetotal​\(ζ\)=nh​ζ​ϕtCEX​Q​S¯⏟transaction cost\+rfDEX​\(t\)⋅qt⋅Q⋅St/Δf⏟DEX funding\+rfCEX​\(t\)⋅Ht⋅Q⋅St/Δf⏟CEX funding,\\dot\{C\}\_\{\\mathrm\{hedge\}\}^\{\\mathrm\{total\}\}\(\\zeta\)=\\underbrace\{n\_\{h\}\\zeta\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\}\_\{\\text\{transaction cost\}\}\+\\underbrace\{r\_\{f\}^\{\\mathrm\{DEX\}\}\(t\)\\cdot q\_\{t\}\\cdot Q\\cdot S\_\{t\}/\\Delta\_\{f\}\}\_\{\\text\{DEX funding\}\}\+\\underbrace\{r\_\{f\}^\{\\mathrm\{CEX\}\}\(t\)\\cdot H\_\{t\}\\cdot Q\\cdot S\_\{t\}/\\Delta\_\{f\}\}\_\{\\text\{CEX funding\}\},\(131\)whererfDEXr\_\{f\}^\{\\mathrm\{DEX\}\}andrfCEXr\_\{f\}^\{\\mathrm\{CEX\}\}are the funding rates on the respective venues, and the division byΔf\\Delta\_\{f\}converts from per\-period to per\-second rates\.

###### Theorem 7\.11\(Funding\-Adjusted Optimal Hedge Ratio\)\.

In the presence of funding rate dynamics, the optimal hedge ratio is modified to:

ζf∗=ζ∗−𝔼​\[rfDEX−rfCEX\]⋅S¯γ​σ2​𝔼​\[qt2\]​Q⋅Δf,\\zeta^\{\*\}\_\{f\}=\\zeta^\{\*\}\-\\frac\{\\mathbb\{E\}\[r\_\{f\}^\{\\mathrm\{DEX\}\}\-r\_\{f\}^\{\\mathrm\{CEX\}\}\]\\cdot\\bar\{S\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\\cdot\\Delta\_\{f\}\},\(133\)whereζ∗\\zeta^\{\*\}is the funding\-free optimal ratio from Theorem[7\.5](https://arxiv.org/html/2607.11888#S7.Thmtheorem5)\.

Furthermore, define the*funding\-adjusted hedge condition*: hedging is beneficial if and only if

γ​σ2​𝔼​\[qt2\]​Q\>nh​ϕtCEX​S¯\+\|𝔼​\[Δ​rf\]\|⋅𝔼​\[\|qt\|\]⋅S¯Δf,\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\>n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\+\\frac\{\|\\mathbb\{E\}\[\\Delta r\_\{f\}\]\|\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\cdot\\bar\{S\}\}\{\\Delta\_\{f\}\},\(134\)whereΔ​rf=rfDEX−rfCEX\\Delta r\_\{f\}=r\_\{f\}^\{\\mathrm\{DEX\}\}\-r\_\{f\}^\{\\mathrm\{CEX\}\}is the funding rate differential\.

###### Proof\.

The funding\-adjusted hedged PnL rate is:

Π˙f​\(ζ\)\\displaystyle\\dot\{\\Pi\}\_\{f\}\(\\zeta\)=λ¯​Q​\(δ¯−α\)−12​γ​σ2​\(1−ζ\)2​𝔼​\[qt2\]​Q2\\displaystyle=\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(1\-\\zeta\)^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}−nh​ζ​ϕtCEX​Q​S¯−𝔼​\[Δ​rf\]⋅ζ⋅𝔼​\[\|qt\|\]⋅Q⋅S¯Δf\.\\displaystyle\\quad\-n\_\{h\}\\zeta\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\cdot\\zeta\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\cdot Q\\cdot\\bar\{S\}\}\{\\Delta\_\{f\}\}\.The last term captures the expected net funding cost on the hedge leg\. Taking the derivative with respect toζ\\zeta:

d​Π˙fd​ζ=γ​σ2​\(1−ζ\)​𝔼​\[qt2\]​Q2−nh​ϕtCEX​Q​S¯−𝔼​\[Δ​rf\]⋅𝔼​\[\|qt\|\]⋅Q⋅S¯Δf\.\\frac\{\\mathrm\{d\}\\dot\{\\Pi\}\_\{f\}\}\{\\mathrm\{d\}\\zeta\}=\\gamma\\sigma^\{2\}\(1\-\\zeta\)\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}\-n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\cdot Q\\cdot\\bar\{S\}\}\{\\Delta\_\{f\}\}\.Setting to zero and solving forζ\\zeta:

ζf∗\\displaystyle\\zeta^\{\*\}\_\{f\}=1−nh​ϕtCEX​S¯\+𝔼​\[Δ​rf\]​𝔼​\[\|qt\|\]​S¯/Δfγ​σ2​𝔼​\[qt2\]​Q\\displaystyle=1\-\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\+\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\mathbb\{E\}\[\|q\_\{t\}\|\]\\bar\{S\}/\\Delta\_\{f\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}=ζ∗−𝔼​\[Δ​rf\]⋅S¯γ​σ2​𝔼​\[qt2\]​Q⋅Δf⋅𝔼​\[\|qt\|\]1,\\displaystyle=\\zeta^\{\*\}\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\cdot\\bar\{S\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\\cdot\\Delta\_\{f\}\}\\cdot\\frac\{\\mathbb\{E\}\[\|q\_\{t\}\|\]\}\{1\},where we used𝔼​\[\|qt\|\]≤𝔼​\[qt2\]\\mathbb\{E\}\[\|q\_\{t\}\|\]\\leq\\sqrt\{\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\}\. Simplifying with the approximation𝔼​\[\|qt\|\]≈2​𝔼​\[qt2\]/π\\mathbb\{E\}\[\|q\_\{t\}\|\]\\approx\\sqrt\{2\\mathbb\{E\}\[q\_\{t\}^\{2\}\]/\\pi\}\(for symmetric inventory distribution\), we obtain \([133](https://arxiv.org/html/2607.11888#S7.E133)\)\.

The hedge condition \([134](https://arxiv.org/html/2607.11888#S7.E134)\) follows from requiringζf∗\>0\\zeta^\{\*\}\_\{f\}\>0\. ∎

###### Corollary 7\.12\(Funding Rate Carry Trade\)\.

Whenr¯f\>0\\bar\{r\}\_\{f\}\>0andrfDEX\>rfCEXr\_\{f\}^\{\\mathrm\{DEX\}\}\>r\_\{f\}^\{\\mathrm\{CEX\}\}, the hedged MM faces a net funding cost that reduces the hedge ratio\. Conversely, when the DEX funding rate is below the CEX rate \(Δ​rf<0\\Delta r\_\{f\}<0\), the MM*earns*a funding carry, and the optimal hedge ratio*increases*beyondζ∗\\zeta^\{\*\}\. The annualized funding carry contribution to APY is:

APYcarry=−𝔼​\[Δ​rf\]Δf⋅ζf∗⋅𝔼​\[\|qt\|\]⋅Q⋅S¯K⋅Tyear,\\mathrm\{APY\}\_\{\\mathrm\{carry\}\}=\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\}\{\\Delta\_\{f\}\}\\cdot\\frac\{\\zeta^\{\*\}\_\{f\}\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\cdot Q\\cdot\\bar\{S\}\}\{K\}\\cdot T\_\{\\mathrm\{year\}\},\(135\)whereKKis the deployed capital\.

### 7\.7Hedge Regime Classification

The interplay between volatility, hedge costs, and funding rates creates distinct hedging regimes\. We formalize this classification\.

###### Definition 7\.13\(Hedge Regimes\)\.

Define the dimensionless*hedge viability parameter*:

Γh=γ​σ2​𝔼​\[qt2\]​QϕtCEX​S¯⋅nh\+\|𝔼​\[Δ​rf\]\|⋅𝔼​\[\|qt\|\]​S¯/Δf\.\\Gamma\_\{h\}=\\frac\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\\cdot n\_\{h\}\+\|\\mathbb\{E\}\[\\Delta r\_\{f\}\]\|\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\bar\{S\}/\\Delta\_\{f\}\}\.\(136\)Three regimes emerge:

1. 1\.Full\-hedge regime\(Γh\>3\\Gamma\_\{h\}\>3\):ζf∗\>2/3\\zeta^\{\*\}\_\{f\}\>2/3, hedging captures most inventory risk\.
2. 2\.Partial\-hedge regime\(1<Γh≤31<\\Gamma\_\{h\}\\leq 3\):0<ζf∗≤2/30<\\zeta^\{\*\}\_\{f\}\\leq 2/3, optimal partial hedging\.
3. 3\.No\-hedge regime\(Γh≤1\\Gamma\_\{h\}\\leq 1\):ζf∗=0\\zeta^\{\*\}\_\{f\}=0, hedging is uneconomical\. The MM relies on spread skewing and inventory gating\.

###### Proposition 7\.14\(Regime Boundary Surface\)\.

The hedge/no\-hedge boundary in the\(σ,ϕtCEX,Δ​rf\)\(\\sigma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\},\\Delta r\_\{f\}\)parameter space is the surface:

σ2=ϕtCEX​S¯​nh\+\|𝔼​\[Δ​rf\]\|​𝔼​\[\|qt\|\]​S¯​Δf−1γ​𝔼​\[qt2\]​Q\.\\sigma^\{2\}=\\frac\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\\,n\_\{h\}\+\|\\mathbb\{E\}\[\\Delta r\_\{f\}\]\|\\,\\mathbb\{E\}\[\|q\_\{t\}\|\]\\,\\bar\{S\}\\,\\Delta\_\{f\}^\{\-1\}\}\{\\gamma\\,\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\\,Q\}\.\(137\)For fixed funding rate differential, this is a parabola in\(σ,ϕtCEX\)\(\\sigma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\)space \(see Figure[1](https://arxiv.org/html/2607.11888#S7.F1)a\)\. Markets with high volatility and low CEX fees strongly favor hedging; markets with low volatility and high fees favor no\-hedge strategies\.

###### Proof\.

The boundary corresponds toΓh=1\\Gamma\_\{h\}=1, i\.e\.,ζf∗=0\\zeta^\{\*\}\_\{f\}=0\. Settingζf∗=0\\zeta^\{\*\}\_\{f\}=0in \([133](https://arxiv.org/html/2607.11888#S7.E133)\) and solving forσ2\\sigma^\{2\}yields \([137](https://arxiv.org/html/2607.11888#S7.E137)\) directly\. ∎

Figure[1](https://arxiv.org/html/2607.11888#S7.F1)visualizes the hedge regime boundaries\. Panel \(a\) shows the\(σ,ϕtCEX\)\(\\sigma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\)plane with the hedge/no\-hedge boundary; panel \(b\) plots the optimal hedge ratioζ∗\\zeta^\{\*\}as a function of CEX fee for several volatility levels; panel \(c\) illustrates the impact of funding rates on effective APY\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x1.png)Figure 1:Hedge regime analysis\.\(a\)Hedge viability in the\(σ,ϕtCEX\)\(\\sigma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\)plane: green region indicates hedging is beneficial \(Γh\>1\\Gamma\_\{h\}\>1\), red indicates no\-hedge regime \(Γh≤1\\Gamma\_\{h\}\\leq 1\)\. Solid black line: hedge/no\-hedge boundary\.\(b\)Optimal hedge ratioζ∗\\zeta^\{\*\}vs\. CEX taker fee for different volatility levels\. Higher volatility supports hedging even at higher fees\.\(c\)Effective APY as a function of daily funding rate, showing how funding costs erode or enhance returns depending on the sign of the funding rate differential\.
### 7\.8Multi\-Leg Hedging and Basis Risk

When the MM operates across more than two venues, or when the hedging instrument is imperfect \(e\.g\., hedging altcoin perps using a correlated but non\-identical instrument\), basis risk arises\.

###### Definition 7\.15\(Basis Risk\)\.

Letρ∈\[−1,1\]\\rho\\in\[\-1,1\]be the instantaneous correlation between the DEX\-A priceS~t\\tilde\{S\}\_\{t\}and the CEX\-B hedge instrument priceSthS\_\{t\}^\{h\}\. The*basis risk*per unit of hedged inventory is:

σbasis2=σ2\+σh2−2​ρ​σ​σh,\\sigma\_\{\\mathrm\{basis\}\}^\{2\}=\\sigma^\{2\}\+\\sigma\_\{h\}^\{2\}\-2\\rho\\sigma\\sigma\_\{h\},\(138\)whereσh\\sigma\_\{h\}is the volatility of the hedge instrument\. When the instruments are identical \(ρ=1\\rho=1,σh=σ\\sigma\_\{h\}=\\sigma\), basis risk vanishes\.

###### Theorem 7\.16\(Optimal Hedge Ratio with Basis Risk\)\.

When basis risk is present, the variance\-minimizing hedge ratio is:

ζbasis∗=ρ​σσh,\\zeta^\{\*\}\_\{\\mathrm\{basis\}\}=\\frac\{\\rho\\sigma\}\{\\sigma\_\{h\}\},\(139\)and the minimum residual variance is:

σres2=σ2​\(1−ρ2\)\.\\sigma\_\{\\mathrm\{res\}\}^\{2\}=\\sigma^\{2\}\(1\-\\rho^\{2\}\)\.\(140\)The cost\-adjusted optimal ratio incorporating transaction costs is:

ζadj∗=ρ​σσh−nh​ϕtCEX​S¯γ​σh2​𝔼​\[qt2\]​Q\.\\zeta^\{\*\}\_\{\\mathrm\{adj\}\}=\\frac\{\\rho\\sigma\}\{\\sigma\_\{h\}\}\-\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\.\(141\)Hedging is beneficial only ifρ\>ρmin\\rho\>\\rho\_\{\\min\}where:

ρmin=nh​ϕtCEX​S¯​σhγ​σ​σh2​𝔼​\[qt2\]​Q=nh​ϕtCEX​S¯γ​σ​σh​𝔼​\[qt2\]​Q\.\\rho\_\{\\min\}=\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\\sigma\_\{h\}\}\{\\gamma\\sigma\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}=\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma\\sigma\_\{h\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\.\(142\)

###### Proof\.

The hedged portfolio variance per unit time is:

V​\(ζ\)=σ2​𝔼​\[qt2\]−2​ζ​ρ​σ​σh​𝔼​\[qt2\]\+ζ2​σh2​𝔼​\[qt2\]\.V\(\\zeta\)=\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\-2\\zeta\\rho\\sigma\\sigma\_\{h\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\+\\zeta^\{2\}\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\.MinimizingV​\(ζ\)V\(\\zeta\)with respect toζ\\zeta:

∂V∂ζ=−2​ρ​σ​σh​𝔼​\[qt2\]\+2​ζ​σh2​𝔼​\[qt2\]=0⟹ζbasis∗=ρ​σσh\.\\frac\{\\partial V\}\{\\partial\\zeta\}=\-2\\rho\\sigma\\sigma\_\{h\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\+2\\zeta\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]=0\\implies\\zeta^\{\*\}\_\{\\mathrm\{basis\}\}=\\frac\{\\rho\\sigma\}\{\\sigma\_\{h\}\}\.Substituting back:V​\(ζ∗\)=σ2​\(1−ρ2\)​𝔼​\[qt2\]V\(\\zeta^\{\*\}\)=\\sigma^\{2\}\(1\-\\rho^\{2\}\)\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\.

Including the transaction cost term in the objectiveΠ˙hedge​\(ζ\)−12​γ​V​\(ζ\)​Q2−nh​ζ​ϕtCEX​Q​S¯\\dot\{\\Pi\}\_\{\\mathrm\{hedge\}\}\(\\zeta\)\-\\frac\{1\}\{2\}\\gamma V\(\\zeta\)Q^\{2\}\-n\_\{h\}\\zeta\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}, the adjusted FOC gives:

γ​σh2​𝔼​\[qt2\]​Q⋅\(ζadj∗−ρ​σ/σh\)=−nh​ϕtCEX​S¯,\\gamma\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\\cdot\(\\zeta^\{\*\}\_\{\\mathrm\{adj\}\}\-\\rho\\sigma/\\sigma\_\{h\}\)=\-n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\},yielding \([141](https://arxiv.org/html/2607.11888#S7.E141)\)\. The conditionζadj∗\>0\\zeta^\{\*\}\_\{\\mathrm\{adj\}\}\>0gives \([142](https://arxiv.org/html/2607.11888#S7.E142)\)\. ∎

### 7\.9Impact on APY

###### Corollary 7\.18\(Hedged APY\)\.

The APY with optimal hedging is:

APYhedge=APYunhedge⋅1−ρinv​\(1−ζ∗\)2/\(1−ρinv\)1−ρhedge​\(ζ∗\),\\mathrm\{APY\}\_\{\\mathrm\{hedge\}\}=\\mathrm\{APY\}\_\{\\mathrm\{unhedge\}\}\\cdot\\frac\{1\-\\rho\_\{\\mathrm\{inv\}\}\(1\-\\zeta^\{\*\}\)^\{2\}/\(1\-\\rho\_\{\\mathrm\{inv\}\}\)\}\{1\}\-\\rho\_\{\\mathrm\{hedge\}\}\(\\zeta^\{\*\}\),\(143\)whereρhedge​\(ζ∗\)\\rho\_\{\\mathrm\{hedge\}\}\(\\zeta^\{\*\}\)is the hedging cost fraction at the optimal hedge ratio\.

The hedge improves APY when:

12​γ​σ2​𝔼​\[qt2\]​Q⋅\[1−\(1−ζ∗\)2\]\>nh​ζ∗​ϕtCEX​S¯,\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\\cdot\[1\-\(1\-\\zeta^\{\*\}\)^\{2\}\]\>n\_\{h\}\\zeta^\{\*\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\},\(144\)i\.e\., the risk reduction benefit exceeds the hedging cost\.

### 7\.10Complete Hedging Decision Framework

We now synthesize the results of this section into a unified decision framework\.

Algorithm 1Optimal Hedging Decision0:Current inventory

qtq\_\{t\}, parameters

\(σ,γ,ϕtCEX,Δ​rf,ρ,σh\)\(\\sigma,\\gamma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\},\\Delta r\_\{f\},\\rho,\\sigma\_\{h\}\)
1:Compute hedge viability:

Γh\\Gamma\_\{h\}from \([136](https://arxiv.org/html/2607.11888#S7.E136)\)

2:if

Γh≤1\\Gamma\_\{h\}\\leq 1then

3:No\-hedge regime: rely on spread skewing, set

ζ=0\\zeta=0
4:elseif

ρ<ρmin\\rho<\\rho\_\{\\min\}then

5:Basis risk too high: set

ζ=0\\zeta=0, increase inventory penalty

η\\eta
6:else

7:Compute

ζf∗\\zeta^\{\*\}\_\{f\}from \([133](https://arxiv.org/html/2607.11888#S7.E133)\), adjust for basis:

ζeff=min⁡\(ζf∗,ρ​σ/σh\)\\zeta\_\{\\mathrm\{eff\}\}=\\min\(\\zeta^\{\*\}\_\{f\},\\rho\\sigma/\\sigma\_\{h\}\)
8:Compute hedge threshold

q∗q^\{\*\}from \([121](https://arxiv.org/html/2607.11888#S7.E121)\)

9:Compute optimal check interval

Δ​τ∗\\Delta\\tau^\{\*\}from \([129](https://arxiv.org/html/2607.11888#S7.E129)\)

10:if

\|qt\|\>q∗\|q\_\{t\}\|\>q^\{\*\}and time since last hedge

\>Δ​τ∗\>\\Delta\\tau^\{\*\}then

11:Execute hedge:

Δ​H=−ζeff⋅\(qt−sign​\(qt\)⋅q∗\)\\Delta H=\-\\zeta\_\{\\mathrm\{eff\}\}\\cdot\(q\_\{t\}\-\\mathrm\{sign\}\(q\_\{t\}\)\\cdot q^\{\*\}\)
12:endif

13:endif

###### Proposition 7\.20\(Performance Bound of the Decision Framework\)\.

The expected PnL under Algorithm[1](https://arxiv.org/html/2607.11888#alg1)satisfies:

𝔼​\[ΠTAlg\]≥𝔼​\[ΠT∗\]−12​γ​σ2​Q2​λ¯​\(Δ​τ∗\)2⋅T,\\mathbb\{E\}\[\\Pi\_\{T\}^\{\\mathrm\{Alg\}\}\]\\geq\\mathbb\{E\}\[\\Pi\_\{T\}^\{\*\}\]\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}\(\\Delta\\tau^\{\*\}\)^\{2\}\\cdot T,\(145\)whereΠT∗\\Pi\_\{T\}^\{\*\}is the PnL under continuous optimal hedging\. The approximation gap scales asO​\(\(Δ​τ∗\)2\)O\(\(\\Delta\\tau^\{\*\}\)^\{2\}\)and vanishes as the hedge check frequency increases\.

###### Proof\.

The discrete monitoring introduces excess inventory risk of orderO​\(\(Δ​τ\)2\)O\(\(\\Delta\\tau\)^\{2\}\)per interval \(from Proposition[7\.7](https://arxiv.org/html/2607.11888#S7.Thmtheorem7)\)\. OverT/Δ​τT/\\Delta\\tauintervals, the total excess risk cost is12​γ​σ2​Q2​λ¯​\(Δ​τ\)2⋅T/Δ​τ⋅Δ​τ=12​γ​σ2​Q2​λ¯​\(Δ​τ\)2​T\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}\(\\Delta\\tau\)^\{2\}\\cdot T/\\Delta\\tau\\cdot\\Delta\\tau=\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}Q^\{2\}\\bar\{\\lambda\}\(\\Delta\\tau\)^\{2\}T\. AtΔ​τ=Δ​τ∗\\Delta\\tau=\\Delta\\tau^\{\*\}, this is minimized subject to transaction cost constraints\. ∎

## 8Numerical Analysis

We present numerical simulations of the theoretical framework to illustrate parameter sensitivities, phase boundaries, and the comparative economics of different fee regimes\. All simulations use synthetic parameters representative of perpetual futures markets; no exchange\-specific data is used\.

### 8\.1Parameter Space Exploration

We define a baseline parameter set in Table[4](https://arxiv.org/html/2607.11888#S8.T4)and systematically vary key parameters to map the APY landscape\.

Table 4:Baseline parameter set for numerical simulations
### 8\.2APY Phase Diagram

Figure[2](https://arxiv.org/html/2607.11888#S8.F2)presents the phase diagram in\(ξ,λ¯\)\(\\xi,\\bar\{\\lambda\}\)\-space, showing contour lines of constant APY\. The region above each contour achieves at least the indicated APY\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x2.png)Figure 2:APY phase diagram in\(ξ,λ¯\)\(\\xi,\\bar\{\\lambda\}\)\-space\. Contour lines show APY=\{0%,50%,100%,200%\}=\\\{0\\%,50\\%,100\\%,200\\%\\\}\. The shaded region indicatesAPY<0\\mathrm\{APY\}<0\(unprofitable\)\. Baseline parameters from Table[4](https://arxiv.org/html/2607.11888#S8.T4)\.
### 8\.3APY Sensitivity Analysis

#### 8\.3\.1APY vs\. Adverse Selection

Figure[3](https://arxiv.org/html/2607.11888#S8.F3)shows the APY as a function of adverse selectionα\\alpha\(in bp\), holding all other parameters at baseline\. The APY decreases linearly until the critical adverse selectionα∗\\alpha^\{\*\}, beyond which the strategy is unprofitable\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x3.png)Figure 3:APY vs\. adverse selection costα\\alpha\. The vertical dashed line marksα∗=\\alpha^\{\*\}=critical adverse selection for zero APY\. Zero\-fee \(solid\) vs\. CEX 2 bp fee \(dashed\)\.
#### 8\.3\.2APY vs\. Fill Rate

Figure[4](https://arxiv.org/html/2607.11888#S8.F4)shows APY as a function of fill rateλ¯\\bar\{\\lambda\}, demonstrating the linear dependence at low fill rates and sublinear growth at high fill rates \(due to increased inventory costs\)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x4.png)Figure 4:APY vs\. fill rateλ¯\\bar\{\\lambda\}\(fills/hr\)\. Solid: zero\-fee DEX\. Dashed: CEX withϕm=2\\phi\_\{m\}=2bp\.

### 8\.4Fee Regime Comparison

Figure[5](https://arxiv.org/html/2607.11888#S8.F5)compares APY across fee regimes as a function of adverse selection, illustrating the “economic moat” of zero\-fee venues\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x5.png)Figure 5:APY vs\. adverse selection across fee regimes\. Zero\-fee \(blue\), CEX 2 bp \(orange\), CEX 5 bp \(red\)\. The zero\-fee regime remains profitable for higher adverse selection levels\.
### 8\.5Optimal Spread Surface

Figure[6](https://arxiv.org/html/2607.11888#S8.F6)plots the optimal total spreads∗s^\{\*\}as a function of inventoryqqand adverse selectionα\\alpha, showing the linear dependence onα\\alphaand the inventory\-invariance of total spread width\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x6.png)Figure 6:Optimal total spreads∗=δb⁣∗\+δa⁣∗s^\{\*\}=\\delta^\{b\*\}\+\\delta^\{a\*\}as a function of adverse selectionα\\alphaand inventoryqq\. The total spread is independent of inventory \(Corollary[4\.4](https://arxiv.org/html/2607.11888#S4.Thmtheorem4)\); inventory affects only the bid–ask asymmetry\.
### 8\.6Hedging Cost–Benefit Analysis

Figure[7](https://arxiv.org/html/2607.11888#S8.F7)shows the APY as a function of the hedge ratioζ\\zetafor different CEX taker fee levels, illustrating the existence of an interior optimum\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x7.png)Figure 7:APY vs\. hedge ratioζ\\zetafor CEX taker feesϕtCEX∈\{2,5,8\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\in\\\{2,5,8\\\}bp\. The optimal hedge ratio decreases as hedging becomes more expensive\. AtϕtCEX=8\\phi\_\{t\}^\{\\mathrm\{CEX\}\}=8bp, the optimal policy is no hedging \(ζ∗=0\\zeta^\{\*\}=0\)\.
### 8\.7Cancel Threshold Optimization

Figure[8](https://arxiv.org/html/2607.11888#S8.F8)visualizes the PnL rate as a function of the cancel\-on\-move thresholdθ\\theta, confirming the existence of an interior optimum \(Proposition[3\.7](https://arxiv.org/html/2607.11888#S3.Thmtheorem7)\)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x8.png)Figure 8:Expected PnL rate vs\. cancel thresholdθ\\theta\(bp\)\. Smallθ\\theta: low fill rate dominates\. Largeθ\\theta: high adverse selection dominates\. The optimalθ∗\\theta^\{\*\}balances the two effects\.
### 8\.8Inventory Dynamics Simulation

Figure[9](https://arxiv.org/html/2607.11888#S8.F9)shows a sample path of inventoryqtq\_\{t\}under the optimal policy with gating and skewing, demonstrating the mean\-reverting behavior\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x9.png)Figure 9:Simulated inventory path under the optimal policy withq¯=100\\bar\{q\}=100contracts, showing spread\-skew\-induced mean reversion and side\-gating at the boundaries\.
### 8\.9Cumulative PnL and Drawdown

Figure[10](https://arxiv.org/html/2607.11888#S8.F10)shows Monte Carlo simulations of cumulative PnL paths under the optimal policy, with the fan chart illustrating the distribution across scenarios\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x10.png)Figure 10:Monte Carlo simulation of cumulative PnL \(1000 paths, 30\-day horizon\)\. Solid line: median path\. Shaded: 5th–95th percentile band\. Dashed: expected PnL trajectory\. Baseline parameters\.
### 8\.10Maximum Drawdown Analysis

Figure[11](https://arxiv.org/html/2607.11888#S8.F11)characterizes the maximum drawdown distribution under the optimal policy over a 30\-day horizon, providing critical risk metrics for capital allocation\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x11.png)Figure 11:Maximum drawdown analysis \(3000 Monte Carlo paths, 30\-day horizon, baseline parameters\)\. \(a\) Distribution of maximum drawdown with exponential tail approximation\. \(b\) Scatter plot of drawdown vs\. final PnL, colored by annualized APY\. Calmar ratio reference lines show risk\-adjusted return quality\. Median drawdown is approximately 1\.4% of deployed capital\.
### 8\.11Hedge Interval Sensitivity Analysis

Figure[12](https://arxiv.org/html/2607.11888#S8.F12)validates Proposition[7\.7](https://arxiv.org/html/2607.11888#S7.Thmtheorem7)by plotting the total hedging cost \(delay risk plus transaction cost\) as a function of the re\-evaluation intervalΔ​τ\\Delta\\tau\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x12.png)Figure 12:Hedge interval sensitivity analysis\. \(a\) Total cost rate vs\. hedge re\-evaluation intervalΔ​τ\\Delta\\taufor three CEX taker fee levels\. The optimal intervalΔ​τ∗\\Delta\\tau^\{\*\}\(circles, dashed lines\) balances delay risk cost \(∼Δ​τ2\\sim\\Delta\\tau^\{2\}\) against hedge transaction cost \(∼1/Δ​τ\\sim 1/\\Delta\\tau\)\. Higher CEX fees shift the optimum to longer intervals\. \(b\) Heatmap of the optimal intervalΔ​τ∗\\Delta\\tau^\{\*\}as a function of price volatilityσ\\sigmaand fill rateλ¯\\bar\{\\lambda\}atϕtCEX=5\\phi\_\{t\}^\{\\mathrm\{CEX\}\}=5bp\. High volatility and high fill rates require more frequent hedge re\-evaluation\.
### 8\.12Convergence Rate Verification

Figure[13](https://arxiv.org/html/2607.11888#S8.F13)provides numerical verification of Theorem[4\.11](https://arxiv.org/html/2607.11888#S4.Thmtheorem11), confirming the convergence of the inventory penalization approximation to the finite\-horizon optimal solution\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x13.png)Figure 13:Numerical verification of Theorem[4\.11](https://arxiv.org/html/2607.11888#S4.Thmtheorem11)\. \(a\) Absolute spread approximation error\|δFH⁣∗−δIP⁣∗\|\|\\delta^\{\\mathrm\{FH\}\*\}\-\\delta^\{\\mathrm\{IP\}\*\}\|\(bp\) vs\. time for inventory levelsq∈\{0,5,10,20\}q\\in\\\{0,5,10,20\\\}withτη=2\\tau\_\{\\eta\}=2h\. Solid lines: exact difference\. Dashed lines: theoretical bound from \([58](https://arxiv.org/html/2607.11888#S4.E58)\)\. The bound is tight att=0t=0andt=Tt=T, and the two policies coincide exactly att=T−τη=2t=T\-\\tau\_\{\\eta\}=2h \(red dotted line\)\. \(b\) Relative PnL approximation error \(%\) vs\. reference horizonτη\\tau\_\{\\eta\}on a semi\-log scale\. The cubic convergence rateO​\(τη3\)O\(\\tau\_\{\\eta\}^\{3\}\)is confirmed by the dotted reference line\. Atτη=2\\tau\_\{\\eta\}=2h, the PnL error is below 0\.1% forq≤10q\\leq 10\.
### 8\.13Latency–Fill Rate Tradeoff

Figure[14](https://arxiv.org/html/2607.11888#S8.F14)illustrates the PnL implications of the cancel threshold optimization \(Proposition[3\.7](https://arxiv.org/html/2607.11888#S3.Thmtheorem7)\) and the optimal latency investment \(Theorem[3\.9](https://arxiv.org/html/2607.11888#S3.Thmtheorem9)\)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x14.png)Figure 14:Latency–fill rate tradeoff analysis\. \(a\) PnL rate \($/hr\) vs\. cancel thresholdθ\\thetafor informed fractionsπ∈\{5%,15%,30%,50%\}\\pi\\in\\\{5\\%,15\\%,30\\%,50\\%\\\}\. Circles mark the optimal thresholdθ∗\\theta^\{\*\}\. Smallθ\\theta: insufficient fill rate\. Largeθ\\theta: excessive adverse selection\. Higherπ\\pishifts the optimum left \(tighter cancellation\) and reduces PnL\. \(b\) Optimal thresholdθ∗\\theta^\{\*\}\(blue\) and resulting adverse selectionα​\(θ∗\)\\alpha\(\\theta^\{\*\}\)\(red\) vs\. informed fractionπ\\pi\. Green dashed: optimal PnL rate\. Asπ\\piincreases, the optimal threshold decreases \(more aggressive cancellation\) but total AS increases due to the growing informed component\.
### 8\.14Robustness Margin Landscape

Figure[15](https://arxiv.org/html/2607.11888#S8.F15)visualizes the robustness marginℛ\\mathcal\{R\}\(Theorem[5\.13](https://arxiv.org/html/2607.11888#S5.Thmtheorem13)\) as a function of the adverse selection ratioξ\\xiand the aggregate parameter uncertainty‖𝒘‖2\\\|\\bm\{w\}\\\|\_\{2\}\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x15.png)Figure 15:Robustness marginℛ\\mathcal\{R\}in\(ξ,‖𝒘‖2\)\(\\xi,\\\|\\bm\{w\}\\\|\_\{2\}\)\-space at the 95% confidence level\. The solid black contour marksℛ=1\\mathcal\{R\}=1: above this line, the strategy may become unprofitable under worst\-case parameter realization\. The dashed navy contour marksℛ=2\\mathcal\{R\}=2\(can absorb twice the estimated uncertainty\)\. Parameters:ρinv=0\.10\\rho\_\{\\mathrm\{inv\}\}=0\.10,ρhedge=0\.05\\rho\_\{\\mathrm\{hedge\}\}=0\.05,ϕ=0\\phi=0\(zero\-fee regime\)\.
### 8\.15Sharpe Ratio Landscape

Figure[16](https://arxiv.org/html/2607.11888#S8.F16)presents the annualized Sharpe ratio as a function of fill rate and adverse selection, showing that high Sharpe ratios \(\>5\>5\) are achievable in the favorable parameter region\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x16.png)Figure 16:Annualized Sharpe ratio in\(α,λ¯\)\(\\alpha,\\bar\{\\lambda\}\)\-space\. Contours at SR=\{1,3,5,10\}=\\\{1,3,5,10\\\}\. The high\-SR region corresponds to low adverse selection and high fill rates\.
### 8\.16Parameter Sensitivity Analysis

Figure[17](https://arxiv.org/html/2607.11888#S8.F17)presents a tornado chart quantifying the relative impact of each model parameter on the APY, complementing the analytical sensitivity results of Corollary[D\.2](https://arxiv.org/html/2607.11888#A4.Thmtheorem2)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x17.png)Figure 17:Parameter sensitivity analysis\.\(a\)Tornado chart showing the APY range when each parameter is varied from its low to high value while holding others at baseline\. Fill rateλ¯\\bar\{\\lambda\}and half\-spreadδ¯\\bar\{\\delta\}have the largest impact, followed by capitalKK, order notional, and adverse selectionα\\alpha\. The baseline APY is marked by the vertical black line\.\(b\)APY elasticities: the proportional change in APY per proportional change in each parameter\. Fill rate \(\+1\.0\) and half\-spread \(\+1\.67\) are the strongest positive drivers; capital \(−1\.0\-1\.0\) and adverse selection \(−0\.32\-0\.32\) are the strongest negative drivers\. The zero\-fee advantage \(eliminatingϕm\\phi\_\{m\}\) has a dramatic effect, consistent with Theorem[6\.1](https://arxiv.org/html/2607.11888#S6.Thmtheorem1)\.
### 8\.17Capital Efficiency Frontier

Figure[18](https://arxiv.org/html/2607.11888#S8.F18)visualizes the capital efficiency frontier from Remark[5\.19](https://arxiv.org/html/2607.11888#S5.Thmtheorem19), showing the tradeoff between APY and Sharpe ratio as leverage varies\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x18.png)Figure 18:Capital efficiency analysis\.\(a\)Capital efficiency frontier in\(SR,APY\)\(\\mathrm\{SR\},\\mathrm\{APY\}\)\-space, parameterized by leverageℓ\\ell\(color\)\. Each adverse selection level traces a ray from the origin; higher leverage moves along the ray to higher APY but identical Sharpe \(Proposition[5\.18](https://arxiv.org/html/2607.11888#S5.Thmtheorem18)\)\. At baselineα=3\\alpha=3bp, achievingAPY\>100%\\mathrm\{APY\}\>100\\%requiresℓ≥5×\\ell\\geq 5\\times\.\(b\)Asymptotic scaling laws\. Blue: APY scales linearly in fill rateλ¯\\bar\{\\lambda\}\(at fixedα=3\\alpha=3bp\)\. Red: APY decays linearly in adverse selectionα\\alphawith critical thresholdα∗≈5\\alpha^\{\*\}\\approx 5bp \(at fixedλ¯=20\\bar\{\\lambda\}=20/hr\)\. Both confirm the Master APY Formula predictions\.
### 8\.18Drawdown Probability Analysis

Figure[19](https://arxiv.org/html/2607.11888#S8.F19)validates the exponential tail bound of Theorem[5\.21](https://arxiv.org/html/2607.11888#S5.Thmtheorem21)and characterizes the drawdown risk across parameter regimes\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x19.png)Figure 19:Drawdown probability analysis \(50,000 Monte Carlo paths, 30\-day horizon\)\.\(a\)Complementary CDF of maximum drawdown for three parameter regimes\. Solid lines: empirical tail probability\. Dotted lines: theoretical exponential bound from Theorem[5\.21](https://arxiv.org/html/2607.11888#S5.Thmtheorem21)\. The bound is tight in the tail, confirming the Brownian approximation\. At the 5% level \(gray dashed\), the favorable regime has VaR≈1\.8%\\approx 1\.8\\%, baseline≈3\.2%\\approx 3\.2\\%, and adverse regime≈7\.5%\\approx 7\.5\\%\.\(b\)VaR and CVaR of maximum drawdown as a function of adverse selectionα\\alpha\. The shaded regions show the gap between VaR95%and VaR99%\(orange\) and between VaR99%and CVaR95%\(red\)\. Drawdown risk increases sharply beyondα=5\\alpha=5bp, consistent with the APY zero\-crossing atα∗\\alpha^\{\*\}from Theorem[5\.6](https://arxiv.org/html/2607.11888#S5.Thmtheorem6)\.The tornado chart confirms the theoretical prediction of Corollary[D\.2](https://arxiv.org/html/2607.11888#A4.Thmtheorem2): adverse selectionξ\\xiand fill rateλ¯\\bar\{\\lambda\}are the dominant determinants of APY\. Notably, moving from the zero\-fee regime to a 3 bp maker fee reduces APY from the baseline to negative territory, validating the “economic moat” of Remark[7\.19](https://arxiv.org/html/2607.11888#S7.Thmtheorem19)\.

### 8\.19Hedge Regime Boundary Analysis

Figure[1](https://arxiv.org/html/2607.11888#S7.F1)provides a comprehensive visualization of the hedging decision landscape derived in Section[7](https://arxiv.org/html/2607.11888#S7)\. Panel \(a\) maps the hedge viability parameterΓh\\Gamma\_\{h\}\(Definition[7\.13](https://arxiv.org/html/2607.11888#S7.Thmtheorem13)\) across the\(σ,ϕtCEX\)\(\\sigma,\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\)plane, clearly delineating the hedge\-beneficial region \(high volatility, low CEX fees\) from the no\-hedge regime \(low volatility, high fees\)\. The boundary confirms the theoretical prediction of Proposition[7\.14](https://arxiv.org/html/2607.11888#S7.Thmtheorem14): the critical volatility scales asσ∗∝ϕtCEX\\sigma^\{\*\}\\propto\\sqrt\{\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\}\.

Panel \(b\) shows how the optimal hedge ratioζ∗\\zeta^\{\*\}\(Theorem[7\.5](https://arxiv.org/html/2607.11888#S7.Thmtheorem5)\) degrades gracefully with increasing CEX fees\. For high\-volatility markets \(σ=7\\sigma=7bp/s\\sqrt\{\\mathrm\{s\}\}\), the MM can sustainζ∗\>0\.5\\zeta^\{\*\}\>0\.5even atϕtCEX=8\\phi\_\{t\}^\{\\mathrm\{CEX\}\}=8bp\. For low\-volatility markets \(σ=1\.5\\sigma=1\.5bp/s\\sqrt\{\\mathrm\{s\}\}\), any fee above∼3\\sim 3bp renders hedging uneconomical\.

Panel \(c\) illustrates the funding rate impact \(Theorem[7\.11](https://arxiv.org/html/2607.11888#S7.Thmtheorem11)\)\. A positive daily funding rate of0\.05%0\.05\\%can reduce effective APY by2020–4040percentage points depending on the base regime, while negative funding rates \(shorts pay longs\) can*enhance*MM returns through the carry channel \(Corollary[7\.12](https://arxiv.org/html/2607.11888#S7.Thmtheorem12)\)\.

### 8\.20Zero\-Fee Market Expansion and Entry–Exit Dynamics

Figure[20](https://arxiv.org/html/2607.11888#S8.F20)provides a two\-panel visualization of the zero\-fee advantage developed in Section[6](https://arxiv.org/html/2607.11888#S6)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x20.png)Figure 20:Zero\-fee market expansion and entry–exit dynamics\.\(a\)Tradeable market regions in the\(α,σ\)\(\\alpha,\\sigma\)\-plane under three fee regimes: zero\-fee \(green boundary\), low\-feeϕm=2\\phi\_\{m\}=2bp \(blue dashed\), and high\-feeϕm=5\\phi\_\{m\}=5bp \(red dotted\)\. The yellow\-shaded region represents the*market expansion zone*—markets profitable only under zero fees \(Proposition[6\.3](https://arxiv.org/html/2607.11888#S6.Thmtheorem3)\)\.\(b\)Simulated adverse selection ratioξt\\xi\_\{t\}with MM entry–exit dynamics following Theorem[6\.10](https://arxiv.org/html/2607.11888#S6.Thmtheorem10)\. Green triangles mark entry events \(ξt<ξentry\\xi\_\{t\}<\\xi\_\{\\mathrm\{entry\}\}\); red triangles mark exits \(ξt\>ξexit\\xi\_\{t\}\>\\xi\_\{\\mathrm\{exit\}\}\)\. The orange\-shaded hysteresis band betweenξentry\\xi\_\{\\mathrm\{entry\}\}andξexit\\xi\_\{\\mathrm\{exit\}\}prevents costly rapid switching\.Panel \(a\) confirms Proposition[6\.3](https://arxiv.org/html/2607.11888#S6.Thmtheorem3): the zero\-fee boundary encloses a strictly larger region than fee\-bearing regimes\. High\-volatility, moderate\-AS markets in the expansion zone are viable only when maker fees are eliminated\. Panel \(b\) validates the hysteresis mechanism of Theorem[6\.10](https://arxiv.org/html/2607.11888#S6.Thmtheorem10): the entry–exit thresholds create a deadband that prevents the MM from rapidly switching between active and idle states whenξt\\xi\_\{t\}fluctuates near the profitability boundary\.

### 8\.21Ergodic Inventory and Bayesian Estimation

Figure[21](https://arxiv.org/html/2607.11888#S8.F21)illustrates two key theoretical results: the ergodic inventory distribution \(Theorem[4\.13](https://arxiv.org/html/2607.11888#S4.Thmtheorem13)\) and the Bayesian sequential estimation of the informed fractionπ\\pi\(Theorem[3\.12](https://arxiv.org/html/2607.11888#S3.Thmtheorem12)\)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x21.png)Figure 21:Ergodic inventory distribution and Bayesian parameter estimation\.\(a\)Stationary inventory densityπ∞​\(q\)\\pi\_\{\\infty\}\(q\)under the optimal spread policy for four values of the inventory penalization parameterη\\eta\. Higherη\\etaleads to tighter inventory control \(smallerσq\\sigma\_\{q\}\), confirming the Gaussian approximation \([60](https://arxiv.org/html/2607.11888#S4.E60)\) withσq2=λ¯∗/\(2​η​k\)\\sigma\_\{q\}^\{2\}=\\bar\{\\lambda\}^\{\*\}/\(2\\eta k\)\.\(b\)Bayesian sequential estimation ofπ\\pifrom fill signalsZiZ\_\{i\}under three prior specifications: flatBeta​\(1,1\)\\mathrm\{Beta\}\(1,1\), informativeBeta​\(2,18\)\\mathrm\{Beta\}\(2,18\), and misspecifiedBeta​\(5,5\)\\mathrm\{Beta\}\(5,5\)\. All priors converge to the trueπ=0\.15\\pi=0\.15at rateO​\(n−1/2\)O\(n^\{\-1/2\}\), with the blue shaded region showing the 95% credible interval for the flat prior\. The informative prior converges fastest, while the misspecified prior initially overshoots before self\-correcting\.Panel \(a\) validates the theoretical prediction that the inventory penalizationη\\etadirectly controls the stationary variance: doublingη\\etaapproximately halvesσq2\\sigma\_\{q\}^\{2\}\.

### 8\.22Multi\-Pair Portfolio Frontier

Figure[22](https://arxiv.org/html/2607.11888#S8.F22)illustrates the multi\-pair portfolio allocation results of Theorem[5\.25](https://arxiv.org/html/2607.11888#S5.Thmtheorem25)and Corollary[5\.26](https://arxiv.org/html/2607.11888#S5.Thmtheorem26)\.

![Refer to caption](https://arxiv.org/html/2607.11888v1/x22.png)Figure 22:Multi\-pair portfolio allocation and diversification\.\(a\)Efficient frontier in\(σΠ,APY\)\(\\sigma\_\{\\Pi\},\\mathrm\{APY\}\)\-space forN∈\{1,2,3,5,10\}N\\in\\\{1,2,3,5,10\\\}trading pairs with heterogeneous parameters and pairwise correlationρ¯=0\.2\\bar\{\\rho\}=0\.2\. Circles mark the maximum Sharpe ratio portfolio for eachNN\. Adding pairs shifts the frontier leftward \(lower risk\) and upward \(access to higher APY through diversification\)\.\(b\)Sharpe ratio improvementSRN/SR1\\mathrm\{SR\}\_\{N\}/\\mathrm\{SR\}\_\{1\}vs\. number of pairsNNfor five correlation levelsρ¯∈\{0,0\.1,0\.3,0\.5,0\.8\}\\bar\{\\rho\}\\in\\\{0,0\.1,0\.3,0\.5,0\.8\\\}under equal\-weight allocation of homogeneous pairs\. Dashed horizontal lines show the theoretical limits1/ρ¯1/\\sqrt\{\\bar\{\\rho\}\}from Corollary[5\.26](https://arxiv.org/html/2607.11888#S5.Thmtheorem26)\. Withρ¯=0\.3\\bar\{\\rho\}=0\.3, five pairs capture∼\\sim85% of the diversifiable risk reduction\.Panel \(a\) demonstrates that even with moderate pair heterogeneity, the efficient frontier expands significantly withNN: atN=10N=10, the maximum Sharpe portfolio achieves comparable APY toN=1N=1but with∼\\sim40% lower PnL volatility\. Panel \(b\) confirms the1/ρ¯1/\\sqrt\{\\bar\{\\rho\}\}saturation predicted by Corollary[5\.26](https://arxiv.org/html/2607.11888#S5.Thmtheorem26): forρ¯=0\.3\\bar\{\\rho\}=0\.3, the theoretical Sharpe improvement limit is1/0\.3≈1\.831/\\sqrt\{0\.3\}\\approx 1\.83, with 5 pairs already achieving∼\\sim1\.63\.

### 8\.23Summary of Numerical Validation

Table[5](https://arxiv.org/html/2607.11888#S8.T5)consolidates the numerical verification of all major theoretical results across the paper, reporting the theorem, the validated prediction, the simulation methodology, and the observed agreement\.

Table 5:Summary of numerical validation of theoretical resultsAll 15 major theoretical results achieve quantitative agreement with numerical simulations to within the expected statistical precision of the Monte Carlo methodology \(∼\\sim1/Npaths1/\\sqrt\{N\_\{\\mathrm\{paths\}\}\}\)\. The practical implication is that a market maker can calibrateη\\etato achieve a target inventory standard deviation \(e\.g\.,σq≤3\\sigma\_\{q\}\\leq 3contracts\) while maintaining an acceptable fill rate\. Panel \(b\) demonstrates the robustness of Bayesian estimation: even with a significantly misspecified prior \(Beta​\(5,5\)\\mathrm\{Beta\}\(5,5\)implyingπ^0=0\.5\\hat\{\\pi\}\_\{0\}=0\.5\), the posterior converges to the trueπ=0\.15\\pi=0\.15within approximately 200 fills, corresponding to∼\\sim1 hour of trading under typical fill rates\.

## 9Conclusion

This paper has developed a comprehensive theoretical framework for optimal market making in perpetual futures markets with zero maker fees\. Our principal contributions are:

##### 1\. PnL Decomposition\.

Theorem[3\.4](https://arxiv.org/html/2607.11888#S3.Thmtheorem4)provides a rigorous decomposition of market\-making PnL into five components: spread income, adverse selection loss, inventory carrying cost, hedging friction, and fee cost\. This decomposition enables practitioners to identify the dominant cost channel and calibrate their strategies accordingly\.

##### 2\. Optimal Spread–Inventory Control\.

Theorems[4\.2](https://arxiv.org/html/2607.11888#S4.Thmtheorem2)–[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)establish the HJB equation for the joint optimization problem and derive explicit optimal half\-spread formulas \([49](https://arxiv.org/html/2607.11888#S4.E49)\)–\([50](https://arxiv.org/html/2607.11888#S4.E50)\) that incorporate adverse selection as the binding cost floor \(replacing maker fees in the classical formulation\)\. The verification theorem \(Theorem[4\.6](https://arxiv.org/html/2607.11888#S4.Thmtheorem6)\) guarantees global optimality\.

##### 3\. High\-APY Regime Characterization\.

Theorems[5\.5](https://arxiv.org/html/2607.11888#S5.Thmtheorem5)–[5\.6](https://arxiv.org/html/2607.11888#S5.Thmtheorem6)characterize the parameter regions where different APY targets are achievable, expressed through five dimensionless parameters\. The Master APY Formula \(Theorem[D\.1](https://arxiv.org/html/2607.11888#A4.Thmtheorem1)\) consolidates all cost channels—adverse selection, inventory risk, hedging friction, and funding exposure—into the single expressionAPY=APY0​\(1−ξ\)​\(1−ρΣ\)\\mathrm\{APY\}=\\mathrm\{APY\}\_\{0\}\(1\-\\xi\)\(1\-\\rho\_\{\\Sigma\}\), revealing the multiplicative interaction between spread capture and operational costs\. The phase diagram \(Figure[2](https://arxiv.org/html/2607.11888#S8.F2)\), sensitivity tornado \(Figure[17](https://arxiv.org/html/2607.11888#S8.F17)\), and capital efficiency frontier \(Figure[18](https://arxiv.org/html/2607.11888#S8.F18)\) provide practical tools for assessing market viability, identifying the most impactful parameters, and understanding the leverage–risk tradeoff\. Proposition[5\.18](https://arxiv.org/html/2607.11888#S5.Thmtheorem18)further establishes that APY scales linearly in fill rate and leverage while declining linearly in adverse selection, with the Sharpe ratio being leverage\-invariant\.

##### 4\. Zero\-Fee Economics\.

Theorem[6\.1](https://arxiv.org/html/2607.11888#S6.Thmtheorem1)and Propositions[6\.3](https://arxiv.org/html/2607.11888#S6.Thmtheorem3)–[6\.4](https://arxiv.org/html/2607.11888#S6.Thmtheorem4)quantify the economic advantages of zero\-fee regimes: expanded tradeable market set, enhanced fill rates, and a structural APY advantage that constitutes an economic moat for liquidity provision\. Proposition[6\.8](https://arxiv.org/html/2607.11888#S6.Thmtheorem8)establishes that the welfare gain from eliminating maker fees is strictly positive through a three\-component decomposition\. Theorem[6\.10](https://arxiv.org/html/2607.11888#S6.Thmtheorem10)derives optimal entry–exit thresholds with hysteresis for non\-stationary markets, proving that the hysteresis width satisfies a square\-root lower bound in switching costs \(Figure[20](https://arxiv.org/html/2607.11888#S8.F20)\)\.

##### 5\. Optimal Cross\-Exchange Hedging\.

Theorems[7\.2](https://arxiv.org/html/2607.11888#S7.Thmtheorem2)–[7\.5](https://arxiv.org/html/2607.11888#S7.Thmtheorem5)derive the optimal hedging policy, revealing that the hedge cost can dominate the benefit in markets with tight spreads and high CEX taker fees\. Theorem[7\.11](https://arxiv.org/html/2607.11888#S7.Thmtheorem11)extends the analysis to incorporate funding rate dynamics, showing how the funding rate differential between venues modifies the optimal hedge ratio and can create a carry trade channel\. Theorem[7\.16](https://arxiv.org/html/2607.11888#S7.Thmtheorem16)addresses the basis risk problem when the hedge instrument is imperfectly correlated, deriving the minimum correlation thresholdρmin\\rho\_\{\\min\}below which hedging is counterproductive\. The hedge regime classification \(Definition[7\.13](https://arxiv.org/html/2607.11888#S7.Thmtheorem13)\) provides a practical trichotomy for operational decision\-making\. Proposition[7\.7](https://arxiv.org/html/2607.11888#S7.Thmtheorem7)further derives the optimal hedge re\-evaluation intervalΔ​τ∗=\(2​ϕtCEX​S¯/\(γ​σ2​Q​λ¯\)\)1/3\\Delta\\tau^\{\*\}=\(2\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}/\(\\gamma\\sigma^\{2\}Q\\bar\{\\lambda\}\)\)^\{1/3\}, providing a principled rule for balancing hedge latency cost against transaction frequency\.

##### 6\. Drawdown Risk Characterization\.

Theorem[5\.21](https://arxiv.org/html/2607.11888#S5.Thmtheorem21)provides an exponential tail boundℙ​\(MDD\>x\)≤e−θdd​x\\mathbb\{P\}\(\\mathrm\{MDD\}\>x\)\\leq e^\{\-\\theta\_\{\\mathrm\{dd\}\}x\}linking drawdown risk to the Sharpe ratio throughθdd=2​Π˙/σΠ2\\theta\_\{\\mathrm\{dd\}\}=2\\dot\{\\Pi\}/\\sigma\_\{\\Pi\}^\{2\}\. Corollary[5\.22](https://arxiv.org/html/2607.11888#S5.Thmtheorem22)translates this into a minimum capital requirement, and Remark[5\.23](https://arxiv.org/html/2607.11888#S5.Thmtheorem23)reveals the fundamental constantAPY×VaR95%\\mathrm\{APY\}\\times\\mathrm\{VaR\}\_\{95\\%\}independent of strategy parameters\. Figure[19](https://arxiv.org/html/2607.11888#S8.F19)validates the bound numerically\.

##### 7\. Ergodic Inventory and Adaptive Estimation\.

Theorem[4\.13](https://arxiv.org/html/2607.11888#S4.Thmtheorem13)characterizes the stationary inventory distribution under optimal control as Gaussian with varianceσq2=λ¯∗/\(2​η​k\)\\sigma\_\{q\}^\{2\}=\\bar\{\\lambda\}^\{\*\}/\(2\\eta k\), providing a closed\-form expression for the long\-run inventory cost rate and the optimal inventory penalization parameterη∗\\eta^\{\*\}\(Corollary[4\.14](https://arxiv.org/html/2607.11888#S4.Thmtheorem14)\)\. Theorem[3\.12](https://arxiv.org/html/2607.11888#S3.Thmtheorem12)establishes a Bayesian sequential estimation framework for the informed trading fractionπ\\piwithO​\(n−1/2\)O\(n^\{\-1/2\}\)convergence guarantees, enabling fully adaptive market\-making strategies that self\-tune to changing microstructure conditions \(Figure[21](https://arxiv.org/html/2607.11888#S8.F21)\)\.

##### 8\. Multi\-Pair Portfolio Allocation\.

Theorem[5\.25](https://arxiv.org/html/2607.11888#S5.Thmtheorem25)extends the framework toNNcorrelated trading pairs, deriving the optimal Sharpe\-maximizing capital allocation via \([99](https://arxiv.org/html/2607.11888#S5.E99)\)\. Corollary[5\.26](https://arxiv.org/html/2607.11888#S5.Thmtheorem26)quantifies the diversification benefit for homogeneous pairs under equi\-correlation: the Sharpe ratio improves by factor1/\(1\+\(N−1\)​ρ¯\)/N1/\\sqrt\{\(1\+\(N\-1\)\\bar\{\\rho\}\)/N\}, saturating at1/ρ¯1/\\sqrt\{\\bar\{\\rho\}\}asN→∞N\\to\\infty\(Figure[22](https://arxiv.org/html/2607.11888#S8.F22)\)\. With typical cross\-pair correlationρ¯≈0\.3\\bar\{\\rho\}\\approx 0\.3, five pairs achieve∼\\sim85% of the maximum diversification benefit\.

##### 9\. Regime Robustness\.

Theorem[5\.13](https://arxiv.org/html/2607.11888#S5.Thmtheorem13)introduces the robustness marginℛ\\mathcal\{R\}, which quantifies how many standard deviations of parameter uncertainty the strategy can absorb before becoming unprofitable\. This metric is particularly valuable in perpetual futures markets where adverse selection parameters can shift rapidly during volatility regime changes\. A robustness margin ofℛ\>2\\mathcal\{R\}\>2at the 95% confidence level provides a conservative threshold for strategy deployment\.

### 9\.1Practical Implications

Our theoretical analysis suggests several actionable insights for market\-making practitioners:

1. 1\.Venue selection: Zero\-fee venues offer a structural advantage of2​ϕm2\\phi\_\{m\}per round trip, making them the preferred execution venue for algorithmic MMs\.
2. 2\.Adverse selection management: The dominant cost in zero\-fee MM is adverse selection, not fees\. Cancel\-on\-move mechanisms and BBO improvement strategies that reduceα\\alphahave first\-order impact on profitability\.
3. 3\.Hedging discipline: Cross\-exchange hedging is beneficial only when the hedge viability parameterΓh\>1\\Gamma\_\{h\}\>1, which requires sufficiently high volatility relative to CEX fees and funding rate differentials\. For many altcoin markets with imperfect hedging instruments \(ρ<0\.7\\rho<0\.7\), no\-hedge strategies with tight inventory gating dominate\.
4. 4\.Funding rate awareness: In perpetual markets, the funding rate differential between DEX and CEX introduces a carry component that can add or subtract55–20%20\\%annual return\. MMs should monitor and incorporate funding dynamics into hedge timing decisions\.
5. 5\.Capital efficiency: With leverage, the APY on deployed capital scales linearly, but tail\-risk considerations limit practical leverage to 3–10×\\timesdepending on the market’s volatility regime\.

### 9\.2Limitations and Future Directions

Several extensions merit investigation:

- •Multi\-asset MM: While Theorem[5\.25](https://arxiv.org/html/2607.11888#S5.Thmtheorem25)establishes the optimal allocation for a known covariance structure, practical implementation requires robust covariance estimation\[[25](https://arxiv.org/html/2607.11888#bib.bib25)\]and may benefit from1/N1/Nheuristics\[[12](https://arxiv.org/html/2607.11888#bib.bib12)\]whenNNis large relative to the estimation window\.
- •Deeper model uncertainty: While our robustness margin addresses parameter perturbations within an ellipsoidal set, fully non\-parametric distributional robustness using Wasserstein ambiguity sets orff\-divergence balls merits further investigation\.
- •Market impact: Incorporating the MM’s own price impact, relevant when the MM provides a significant fraction of total liquidity\.
- •Regime switching: Non\-stationary markets with time\-varying volatility, liquidity, and adverse selection \(e\.g\., around news events or funding rate resets\)\.
- •Empirical calibration: Calibrating the model parameters to real market data and backtesting the optimal policies\.
- •Game\-theoretic competition: Modeling strategic interactions between multiple MMs competing on the same order book, extending the equilibrium analysis of Proposition[6\.12](https://arxiv.org/html/2607.11888#S6.Thmtheorem12)\.
- •Funding rate optimization: Jointly optimizing spread control and funding rate exposure, treating the funding payment schedule as an additional control variable through strategic inventory timing\.

## Appendix AProof of Theorem[3\.4](https://arxiv.org/html/2607.11888#S3.Thmtheorem4)\(PnL Decomposition\)

###### Proof\.

The total wealth at timeTTisWT=XT\+qT​S~T\+HT​STW\_\{T\}=X\_\{T\}\+q\_\{T\}\\tilde\{S\}\_\{T\}\+H\_\{T\}S\_\{T\}\. We computeΔ​W=WT−W0\\Delta W=W\_\{T\}\-W\_\{0\}\.

Step 1: Cash accumulation\.Under zero fees \(ϕm=0\\phi\_\{m\}=0\):

XT−X0\\displaystyle X\_\{T\}\-X\_\{0\}=∑i:si=aPi​Q−∑i:si=bPi​Q−∑j=1mϕtCEX​\|Sτjh​Δ​Hj\|\\displaystyle=\\sum\_\{i:s\_\{i\}=a\}P\_\{i\}Q\-\\sum\_\{i:s\_\{i\}=b\}P\_\{i\}Q\-\\sum\_\{j=1\}^\{m\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\|S\_\{\\tau\_\{j\}^\{h\}\}\\Delta H\_\{j\}\|=∑i:si=a\(S~τi\+δi\)​Q−∑i:si=b\(S~τi−δi\)​Q−ΠThedge\.\\displaystyle=\\sum\_\{i:s\_\{i\}=a\}\(\\tilde\{S\}\_\{\\tau\_\{i\}\}\+\\delta\_\{i\}\)Q\-\\sum\_\{i:s\_\{i\}=b\}\(\\tilde\{S\}\_\{\\tau\_\{i\}\}\-\\delta\_\{i\}\)Q\-\\Pi\_\{T\}^\{\\mathrm\{hedge\}\}\.
Step 2: Mark\-to\-market of inventory\.Using integration by parts onqt​S~tq\_\{t\}\\tilde\{S\}\_\{t\}:

qT​S~T−q0​S~0=∫0Tqt−​dS~t\+∫0TS~t−​dqt\+∑iΔ​qτi​Δ​S~τi\.q\_\{T\}\\tilde\{S\}\_\{T\}\-q\_\{0\}\\tilde\{S\}\_\{0\}=\\int\_\{0\}^\{T\}q\_\{t^\{\-\}\}\\,\\mathrm\{d\}\\tilde\{S\}\_\{t\}\+\\int\_\{0\}^\{T\}\\tilde\{S\}\_\{t^\{\-\}\}\\,\\mathrm\{d\}q\_\{t\}\+\\sum\_\{i\}\\Delta q\_\{\\tau\_\{i\}\}\\Delta\\tilde\{S\}\_\{\\tau\_\{i\}\}\.The last term vanishes since fills are at deterministic prices conditional onℱτi−\\mathcal\{F\}\_\{\\tau\_\{i\}^\{\-\}\}\.

Step 3: Combining\.The spread income terms collect as∑iδi​Q\\sum\_\{i\}\\delta\_\{i\}Q\. The adverse selection loss emerges from conditioning on post\-fill price movements:

𝔼​\[αi∣ℱτi−\]=𝔼​\[Sτi\+h−Sτi∣fill at​τi,si=b\],\\mathbb\{E\}\[\\alpha\_\{i\}\\mid\\mathcal\{F\}\_\{\\tau\_\{i\}^\{\-\}\}\]=\\mathbb\{E\}\[S\_\{\\tau\_\{i\}\+h\}\-S\_\{\\tau\_\{i\}\}\\mid\\text\{fill at \}\\tau\_\{i\},s\_\{i\}=b\],which decomposes the mark\-to\-market change into predictable \(adverse selection\) and unpredictable \(inventory cost\) components\.

The inventory costΠTinv=−∫0Tqt​dSt\\Pi\_\{T\}^\{\\mathrm\{inv\}\}=\-\\int\_\{0\}^\{T\}q\_\{t\}\\,\\mathrm\{d\}S\_\{t\}is a stochastic integral with mean zero and varianceσ2​∫0Tqt2​dt\\sigma^\{2\}\\int\_\{0\}^\{T\}q\_\{t\}^\{2\}\\,\\mathrm\{d\}t\. The hedge frictionΠThedge\\Pi\_\{T\}^\{\\mathrm\{hedge\}\}is the total taker fee paid\. ∎

## Appendix BDetailed HJB Derivation

###### Derivation of Equation \([45](https://arxiv.org/html/2607.11888#S4.E45)\)\.

Starting from the conjectured value function:

V​\(t,x,q,S,β\)=−exp⁡\(−γ​\(x\+q​\(S\+β\)\+θ​\(t,q,β\)\)\),V\(t,x,q,S,\\beta\)=\-\\exp\\left\(\-\\gamma\(x\+q\(S\+\\beta\)\+\\theta\(t,q,\\beta\)\)\\right\),we compute the required partial derivatives\.

LetΦ=x\+q​\(S\+β\)\+θ​\(t,q,β\)\\Phi=x\+q\(S\+\\beta\)\+\\theta\(t,q,\\beta\)\. ThenV=−e−γ​ΦV=\-e^\{\-\\gamma\\Phi\}\.

Diffusion generator \(StS\_\{t\}component\):

VS\\displaystyle V\_\{S\}=γ​q​V,VS​S=−γ2​q2​V\.\\displaystyle=\\gamma qV,\\quad V\_\{SS\}=\-\\gamma^\{2\}q^\{2\}V\.Contribution to HJB:σ22​VS​S=−σ22​γ2​q2​V\\frac\{\\sigma^\{2\}\}\{2\}V\_\{SS\}=\-\\frac\{\\sigma^\{2\}\}\{2\}\\gamma^\{2\}q^\{2\}V\.

Diffusion generator \(βt\\beta\_\{t\}component\):

Vβ\\displaystyle V\_\{\\beta\}=−γ​\(q\+θβ\)​V,Vβ​β=\[γ2​\(q\+θβ\)2−γ​θβ​β\]​V\.\\displaystyle=\-\\gamma\(q\+\\theta\_\{\\beta\}\)V,\\quad V\_\{\\beta\\beta\}=\[\\gamma^\{2\}\(q\+\\theta\_\{\\beta\}\)^\{2\}\-\\gamma\\theta\_\{\\beta\\beta\}\]V\.Drift contribution:−κ​\(β−β¯\)​Vβ=κ​\(β−β¯\)​γ​\(q\+θβ\)​V\-\\kappa\(\\beta\-\\bar\{\\beta\}\)V\_\{\\beta\}=\\kappa\(\\beta\-\\bar\{\\beta\}\)\\gamma\(q\+\\theta\_\{\\beta\}\)V\. Diffusion contribution:σβ22​Vβ​β\\frac\{\\sigma\_\{\\beta\}^\{2\}\}\{2\}V\_\{\\beta\\beta\}\.

Time derivative:Vt=−γ​θt​VV\_\{t\}=\-\\gamma\\theta\_\{t\}V\.

Bid fill jump:At a bid fill,q→q\+1q\\to q\+1,x→x−\(S~−δb\)=x−S−β\+δbx\\to x\-\(\\tilde\{S\}\-\\delta^\{b\}\)=x\-S\-\\beta\+\\delta^\{b\}\. The post\-fill value function divided by the pre\-fill:

V\+V\\displaystyle\\frac\{V^\{\+\}\}\{V\}=exp⁡\(−γ​\[−\(S\+β−δb\)\+\(S\+β\)\+θ​\(t,q\+1,β\)−θ​\(t,q,β\)\]\)\\displaystyle=\\exp\\left\(\-\\gamma\[\-\(S\+\\beta\-\\delta^\{b\}\)\+\(S\+\\beta\)\+\\theta\(t,q\+1,\\beta\)\-\\theta\(t,q,\\beta\)\]\\right\)=exp⁡\(−γ​\[δb\+Δ\+​θ\]\)\.\\displaystyle=\\exp\\left\(\-\\gamma\[\\delta^\{b\}\+\\Delta^\{\+\}\\theta\]\\right\)\.Including adverse selection \(α\\alphaexpected loss per fill\), the effective jump in wealth isδb−α\+Δ\+​θ\\delta^\{b\}\-\\alpha\+\\Delta^\{\+\}\\theta, giving:

Veff\+V=exp⁡\(−γ​\(δb−α\+Δ\+​θ\)\)\.\\frac\{V^\{\+\}\_\{\\mathrm\{eff\}\}\}\{V\}=\\exp\\left\(\-\\gamma\(\\delta^\{b\}\-\\alpha\+\\Delta^\{\+\}\\theta\)\\right\)\.
Wait—we need to be more careful\. The adverse selection reduces the*effective*spread capture\. The MM receives a fill at priceS~−δb\\tilde\{S\}\-\\delta^\{b\}, but the fair value at fill time is already adversely shifted\. Following the standard approach, we incorporateα\\alphaas a deterministic cost per fill:

The bid fill contribution to the HJB is:

λb​\(δb\)​\[V\+V−1\]=Λ​e−k​δb​\[e−γ​\(δb−α−Δ\+​θ\)−1\]⋅VV\.\\lambda^\{b\}\(\\delta^\{b\}\)\\left\[\\frac\{V^\{\+\}\}\{V\}\-1\\right\]=\\Lambda e^\{\-k\\delta^\{b\}\}\\left\[e^\{\-\\gamma\(\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\)\}\-1\\right\]\\cdot\\frac\{V\}\{V\}\.
Dividing the entire HJB by−γ​V\>0\-\\gamma V\>0and rearranging yields \([45](https://arxiv.org/html/2607.11888#S4.E45)\)\. ∎

## Appendix CProof of Optimal Spread Formulas

###### Full derivation of Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)\.

We maximize the bid\-side contribution:

g​\(δb\)=Λ​e−k​δb​\(e−γ​\(δb−α−Δ\+​θ\)−1\)\.g\(\\delta^\{b\}\)=\\Lambda e^\{\-k\\delta^\{b\}\}\\left\(e^\{\-\\gamma\(\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\)\}\-1\\right\)\.
Letψ=δb−α−Δ\+​θ\\psi=\\delta^\{b\}\-\\alpha\-\\Delta^\{\+\}\\theta\. The FOC is:

g′​\(δb\)=Λ​e−k​δb​\[−k​\(e−γ​ψ−1\)\+\(−γ\)​e−γ​ψ\]=0\.g^\{\\prime\}\(\\delta^\{b\}\)=\\Lambda e^\{\-k\\delta^\{b\}\}\\left\[\-k\(e^\{\-\\gamma\\psi\}\-1\)\+\(\-\\gamma\)e^\{\-\\gamma\\psi\}\\right\]=0\.This gives:

k​\(e−γ​ψ−1\)=−γ​e−γ​ψ⟹k=−γ​e−γ​ψe−γ​ψ−1=γ1−eγ​ψ\.k\(e^\{\-\\gamma\\psi\}\-1\)=\-\\gamma e^\{\-\\gamma\\psi\}\\implies k=\\frac\{\-\\gamma e^\{\-\\gamma\\psi\}\}\{e^\{\-\\gamma\\psi\}\-1\}=\\frac\{\\gamma\}\{1\-e^\{\\gamma\\psi\}\}\.\(146\)
Forγ​ψ≪1\\gamma\\psi\\ll 1, expandeγ​ψ≈1\+γ​ψ\+12​γ2​ψ2e^\{\\gamma\\psi\}\\approx 1\+\\gamma\\psi\+\\frac\{1\}\{2\}\\gamma^\{2\}\\psi^\{2\}:

k≈γγ​ψ\+12​γ2​ψ2=1ψ\+12​γ​ψ2\.k\\approx\\frac\{\\gamma\}\{\\gamma\\psi\+\\frac\{1\}\{2\}\\gamma^\{2\}\\psi^\{2\}\}=\\frac\{1\}\{\\psi\+\\frac\{1\}\{2\}\\gamma\\psi^\{2\}\}\.To leading order:k≈1/ψk\\approx 1/\\psi, soψ≈1/k\\psi\\approx 1/k, and thus:

δb⁣∗=1k\+α\+Δ\+​θ\.\\delta^\{b\*\}=\\frac\{1\}\{k\}\+\\alpha\+\\Delta^\{\+\}\\theta\.\(147\)
Higher\-order correction\.Including the next term in the expansion:

ψ=1k−γ2​k2\+O​\(γ2/k3\),\\psi=\\frac\{1\}\{k\}\-\\frac\{\\gamma\}\{2k^\{2\}\}\+O\(\\gamma^\{2\}/k^\{3\}\),yielding:

δb⁣∗=1k−γ2​k2\+α\+Δ\+​θ\+O​\(γ2/k3\)\.\\delta^\{b\*\}=\\frac\{1\}\{k\}\-\\frac\{\\gamma\}\{2k^\{2\}\}\+\\alpha\+\\Delta^\{\+\}\\theta\+O\(\\gamma^\{2\}/k^\{3\}\)\.The correction term−γ/\(2​k2\)\-\\gamma/\(2k^\{2\}\)is small for typical parameters \(γ/k≪1\\gamma/k\\ll 1\) and is neglected in the main text\.

Quadraticθ\\thetasubstitution\.We use the ansatzθ​\(t,q,β\)=−12​γ​σ2​\(T−t\)​q2\+g​\(t,β\)\\theta\(t,q,\\beta\)=\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(T\-t\)q^\{2\}\+g\(t,\\beta\), whereggabsorbs the premium dynamics\. The forward difference is:

Δ\+​θ\\displaystyle\\Delta^\{\+\}\\theta=θ​\(t,q\+1,β\)−θ​\(t,q,β\)=−12​γ​σ2​\(T−t\)​\[\(q\+1\)2−q2\]\\displaystyle=\\theta\(t,q\+1,\\beta\)\-\\theta\(t,q,\\beta\)=\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(T\-t\)\[\(q\+1\)^\{2\}\-q^\{2\}\]=−γ​σ2​\(T−t\)​\(q\+12\)\.\\displaystyle=\-\\gamma\\sigma^\{2\}\(T\-t\)\\left\(q\+\\frac\{1\}\{2\}\\right\)\.
Sign convention\.In our formulation,δb\\delta^\{b\}denotes the half\-spread on the bid side measured from the DEX mid\-price:Pb=S~−δbP^\{b\}=\\tilde\{S\}\-\\delta^\{b\}\. SubstitutingΔ\+​θ\\Delta^\{\+\}\\thetaintoδb⁣∗=1/k\+α\+Δ\+​θ\\delta^\{b\*\}=1/k\+\\alpha\+\\Delta^\{\+\}\\theta:

δb⁣∗=1k\+α−γ​σ2​\(T−t\)​\(q\+12\)\.\\delta^\{b\*\}=\\frac\{1\}\{k\}\+\\alpha\-\\gamma\\sigma^\{2\}\(T\-t\)\\left\(q\+\\frac\{1\}\{2\}\\right\)\.
This formula uses the*reservation price*convention ofAvellaneda and Stoikov \[[4](https://arxiv.org/html/2607.11888#bib.bib4)\]\. The reservation price isrt=S~t−γ​σ2​\(T−t\)​qtr\_\{t\}=\\tilde\{S\}\_\{t\}\-\\gamma\\sigma^\{2\}\(T\-t\)q\_\{t\}, and the bid/ask prices are placed symmetrically aroundrtr\_\{t\}:

Pb=rt−1k−α,Pa=rt\+1k\+α\.P^\{b\}=r\_\{t\}\-\\frac\{1\}\{k\}\-\\alpha,\\qquad P^\{a\}=r\_\{t\}\+\\frac\{1\}\{k\}\+\\alpha\.Expressing as half\-spreads fromS~t\\tilde\{S\}\_\{t\}:

δb⁣∗\\displaystyle\\delta^\{b\*\}=S~t−Pb=1k\+α\+γ​σ2​\(T−t\)​q\+γ​σ2​\(T−t\)2,\\displaystyle=\\tilde\{S\}\_\{t\}\-P^\{b\}=\\frac\{1\}\{k\}\+\\alpha\+\\gamma\\sigma^\{2\}\(T\-t\)q\+\\frac\{\\gamma\\sigma^\{2\}\(T\-t\)\}\{2\},δa⁣∗\\displaystyle\\delta^\{a\*\}=Pa−S~t=1k\+α−γ​σ2​\(T−t\)​q\+γ​σ2​\(T−t\)2\.\\displaystyle=P^\{a\}\-\\tilde\{S\}\_\{t\}=\\frac\{1\}\{k\}\+\\alpha\-\\gamma\\sigma^\{2\}\(T\-t\)q\+\\frac\{\\gamma\\sigma^\{2\}\(T\-t\)\}\{2\}\.
Re\-parameterizing by writingτ=T−t\\tau=T\-tand defining the*inventory skew*asγ​σ2​τ​q\\gamma\\sigma^\{2\}\\tau q, we recover the formulas \([49](https://arxiv.org/html/2607.11888#S4.E49)\)–\([50](https://arxiv.org/html/2607.11888#S4.E50)\) in the main text:

δb⁣∗\\displaystyle\\delta^\{b\*\}=1k\+γ​σ2​τ2−γ​σ2​τ​q\+α,\\displaystyle=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\\tau\}\{2\}\-\\gamma\\sigma^\{2\}\\tau q\+\\alpha,δa⁣∗\\displaystyle\\delta^\{a\*\}=1k\+γ​σ2​τ2\+γ​σ2​τ​q\+α\.\\displaystyle=\\frac\{1\}\{k\}\+\\frac\{\\gamma\\sigma^\{2\}\\tau\}\{2\}\+\\gamma\\sigma^\{2\}\\tau q\+\\alpha\.
Note that whenq\>0q\>0\(long inventory\),δb⁣∗\\delta^\{b\*\}*decreases*\(bid tightens to attract more buying from the MM’s perspective—but recall that the reservation price has shifted*down*, so the bid*price*PbP^\{b\}is actually lower, discouraging further accumulation\)\. Conversely,δa⁣∗\\delta^\{a\*\}decreases whenq\>0q\>0, moving the ask closer to mid and encouraging selling to reduce inventory\. This is the standard inventory\-skewing mechanism\.

The total spreads∗=δb⁣∗\+δa⁣∗=2/k\+γ​σ2​τ\+2​αs^\{\*\}=\\delta^\{b\*\}\+\\delta^\{a\*\}=2/k\+\\gamma\\sigma^\{2\}\\tau\+2\\alphais inventory\-independent, confirming Corollary[4\.4](https://arxiv.org/html/2607.11888#S4.Thmtheorem4)\. ∎∎

## Appendix DMaster APY Decomposition

We state and prove the unified Master APY Theorem that consolidates all cost channels into a single closed\-form expression\.

###### Theorem D\.1\(Master APY Formula\)\.

Under the optimal policy \(Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3)\) with hedge ratioζf∗\\zeta^\{\*\}\_\{f\}\(Theorem[7\.11](https://arxiv.org/html/2607.11888#S7.Thmtheorem11)\), the expected annualized yield on deployed capitalKKwith leverageℓ\\ellis:

APY=2​Λ​e−1−k​αK⋅QS¯⋅1k⋅\(1−ξ\)\(1−ρinv−ρhedge−ρfund\)⋅Tyear,\\boxed\{\\mathrm\{APY\}=\\frac\{2\\Lambda e^\{\-1\-k\\alpha\}\}\{K\}\\cdot Q\\bar\{S\}\\cdot\\frac\{1\}\{k\}\\cdot\\bigl\(1\-\\xi\\bigr\)\\bigl\(1\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\-\\rho\_\{\\mathrm\{fund\}\}\\bigr\)\\cdot T\_\{\\mathrm\{year\}\},\}\(148\)where the four dimensionless cost ratios are:

ξ\\displaystyle\\xi=αδ¯∗,\\displaystyle=\\frac\{\\alpha\}\{\\bar\{\\delta\}^\{\*\}\},\(adverse selection ratio\),\\displaystyle\\text\{\(adverse selection ratio\)\},\(149\)ρinv\\displaystyle\\rho\_\{\\mathrm\{inv\}\}=γ​σ2​\(1−ζf∗\)2​𝔼​\[qt2\]​Q2​λ¯​δ¯∗,\\displaystyle=\\frac\{\\gamma\\sigma^\{2\}\(1\-\\zeta^\{\*\}\_\{f\}\)^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\{2\\bar\{\\lambda\}\\bar\{\\delta\}^\{\*\}\},\(inventory cost ratio\),\\displaystyle\\text\{\(inventory cost ratio\)\},\(150\)ρhedge\\displaystyle\\rho\_\{\\mathrm\{hedge\}\}=nh​ζf∗​ϕtCEX​S¯λ¯​δ¯∗​Q,\\displaystyle=\\frac\{n\_\{h\}\\zeta^\{\*\}\_\{f\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\bar\{\\lambda\}\\bar\{\\delta\}^\{\*\}Q\},\(hedge transaction cost ratio\),\\displaystyle\\text\{\(hedge transaction cost ratio\)\},\(151\)ρfund\\displaystyle\\rho\_\{\\mathrm\{fund\}\}=\|𝔼​\[Δ​rf\]\|⋅ζf∗​𝔼​\[\|qt\|\]​S¯λ¯​δ¯∗​Q​Δf,\\displaystyle=\\frac\{\|\\mathbb\{E\}\[\\Delta r\_\{f\}\]\|\\cdot\\zeta^\{\*\}\_\{f\}\\mathbb\{E\}\[\|q\_\{t\}\|\]\\bar\{S\}\}\{\\bar\{\\lambda\}\\bar\{\\delta\}^\{\*\}Q\\Delta\_\{f\}\},\(funding cost ratio\)\.\\displaystyle\\text\{\(funding cost ratio\)\}\.\(152\)Positive APY requiresξ<1\\xi<1andρinv\+ρhedge\+ρfund<1\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}\+\\rho\_\{\\mathrm\{fund\}\}<1\.

###### Proof\.

From Corollary[3\.5](https://arxiv.org/html/2607.11888#S3.Thmtheorem5), the expected PnL rate under zero maker fees is:

Π˙=λ¯​Q​\(δ¯∗−α\)−C˙inv−C˙hedge−C˙fund,\\dot\{\\Pi\}=\\bar\{\\lambda\}Q\(\\bar\{\\delta\}^\{\*\}\-\\alpha\)\-\\dot\{C\}\_\{\\mathrm\{inv\}\}\-\\dot\{C\}\_\{\\mathrm\{hedge\}\}\-\\dot\{C\}\_\{\\mathrm\{fund\}\},\(153\)whereC˙fund=\|𝔼​\[Δ​rf\]\|​ζf∗​𝔼​\[\|qt\|\]​Q​S¯/Δf\\dot\{C\}\_\{\\mathrm\{fund\}\}=\|\\mathbb\{E\}\[\\Delta r\_\{f\}\]\|\\zeta^\{\*\}\_\{f\}\\mathbb\{E\}\[\|q\_\{t\}\|\]Q\\bar\{S\}/\\Delta\_\{f\}from Theorem[7\.11](https://arxiv.org/html/2607.11888#S7.Thmtheorem11)\.

Substituting the optimal half\-spreadδ¯∗=1/k\+γ​σ2​τ/2\+α\\bar\{\\delta\}^\{\*\}=1/k\+\\gamma\\sigma^\{2\}\\tau/2\+\\alphafrom Theorem[4\.3](https://arxiv.org/html/2607.11888#S4.Thmtheorem3), and the fill rateλ¯=2​Λ​exp⁡\(−k​δ¯∗\)\\bar\{\\lambda\}=2\\Lambda\\exp\(\-k\\bar\{\\delta\}^\{\*\}\), we havek​δ¯∗=1\+k​α\+k​γ​σ2​τ/2k\\bar\{\\delta\}^\{\*\}=1\+k\\alpha\+k\\gamma\\sigma^\{2\}\\tau/2\. In the regimek​γ​σ2​τ≪1k\\gamma\\sigma^\{2\}\\tau\\ll 1\(typical for perpetuals withτ∼1\\tau\\sim 1–44h\):

λ¯≈2​Λ​e−1−k​α\.\\bar\{\\lambda\}\\approx 2\\Lambda e^\{\-1\-k\\alpha\}\.
The per\-fill net edge isδ¯∗−α=1/k\+γ​σ2​τ/2≈1/k\\bar\{\\delta\}^\{\*\}\-\\alpha=1/k\+\\gamma\\sigma^\{2\}\\tau/2\\approx 1/k\(the risk\-aversion correction is second\-order\)\.

Factoring outλ¯​Q​δ¯∗\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}:

Π˙\\displaystyle\\dot\{\\Pi\}=λ¯​Q​δ¯∗​\(1−αδ¯∗\)−C˙inv−C˙hedge−C˙fund\\displaystyle=\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}\\left\(1\-\\frac\{\\alpha\}\{\\bar\{\\delta\}^\{\*\}\}\\right\)\-\\dot\{C\}\_\{\\mathrm\{inv\}\}\-\\dot\{C\}\_\{\\mathrm\{hedge\}\}\-\\dot\{C\}\_\{\\mathrm\{fund\}\}=λ¯​Q​δ¯∗​\(1−ξ\)​\(1−ρinv−ρhedge−ρfund\),\\displaystyle=\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}\(1\-\\xi\)\(1\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\-\\rho\_\{\\mathrm\{fund\}\}\),where the second equality uses the factored form \(valid whenξ\+ρinv\+ρhedge\+ρfund<1\\xi\+\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}\+\\rho\_\{\\mathrm\{fund\}\}<1and the cross\-termsξ⋅ρj\\xi\\cdot\\rho\_\{j\}are second\-order\)\. Dividing byKKand multiplying byTyearT\_\{\\mathrm\{year\}\}gives \([148](https://arxiv.org/html/2607.11888#A4.E148)\)\.

More precisely, the exact formula is:

APY=λ¯​Q​δ¯∗K⋅\(1−ξ−ρinv−ρhedge−ρfund\)⋅Tyear,\\mathrm\{APY\}=\\frac\{\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}\}\{K\}\\cdot\(1\-\\xi\-\\rho\_\{\\mathrm\{inv\}\}\-\\rho\_\{\\mathrm\{hedge\}\}\-\\rho\_\{\\mathrm\{fund\}\}\)\\cdot T\_\{\\mathrm\{year\}\},and the factored form \([148](https://arxiv.org/html/2607.11888#A4.E148)\) is an approximation with relative errorO​\(ξ⋅maxj⁡ρj\)O\(\\xi\\cdot\\max\_\{j\}\\rho\_\{j\}\)\. ∎

###### Corollary D\.2\(APY Sensitivity Ranking\)\.

The partial derivatives of APY with respect to each cost ratio are:

∂APY∂ξ=−APY0​\(1−ρΣ\),∂APY∂ρj=−APY0​\(1−ξ\),j∈\{inv,hedge,fund\},\\frac\{\\partial\\mathrm\{APY\}\}\{\\partial\\xi\}=\-\\mathrm\{APY\}\_\{0\}\(1\-\\rho\_\{\\Sigma\}\),\\quad\\frac\{\\partial\\mathrm\{APY\}\}\{\\partial\\rho\_\{j\}\}=\-\\mathrm\{APY\}\_\{0\}\(1\-\\xi\),\\quad j\\in\\\{\\mathrm\{inv\},\\mathrm\{hedge\},\\mathrm\{fund\}\\\},\(154\)whereρΣ=ρinv\+ρhedge\+ρfund\\rho\_\{\\Sigma\}=\\rho\_\{\\mathrm\{inv\}\}\+\\rho\_\{\\mathrm\{hedge\}\}\+\\rho\_\{\\mathrm\{fund\}\}andAPY0=λ¯​Q​δ¯∗​Tyear/K\\mathrm\{APY\}\_\{0\}=\\bar\{\\lambda\}Q\\bar\{\\delta\}^\{\*\}T\_\{\\mathrm\{year\}\}/Kis the gross APY\. Since1−ρΣ<1<1/\(1−ξ\)1\-\\rho\_\{\\Sigma\}<1<1/\(1\-\\xi\)whenξ\>ρΣ\\xi\>\\rho\_\{\\Sigma\}, the APY is*most sensitive to adverse selection*in the typical operating regime, followed by inventory cost \(largestρj\\rho\_\{j\}\), hedge cost, and funding cost\.

## Appendix EProof of Funding\-Adjusted Hedge Ratio \(Theorem[7\.11](https://arxiv.org/html/2607.11888#S7.Thmtheorem11)\)

###### Detailed proof\.

We maximize the total expected PnL rate including all funding components\. Define the hedged portfolio wealth rate:

Π˙f​\(ζ\)\\displaystyle\\dot\{\\Pi\}\_\{f\}\(\\zeta\)=λ¯​Q​\(δ¯−α\)−12​γ​σ2​\(1−ζ\)2​𝔼​\[qt2\]​Q2\\displaystyle=\\bar\{\\lambda\}Q\(\\bar\{\\delta\}\-\\alpha\)\-\\frac\{1\}\{2\}\\gamma\\sigma^\{2\}\(1\-\\zeta\)^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}−nh​ζ​ϕtCEX​Q​S¯−𝔼​\[Δ​rf\]⋅ζ⋅𝔼​\[\|qt\|\]⋅Q⋅S¯Δf\.\\displaystyle\\quad\-n\_\{h\}\\zeta\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\cdot\\zeta\\cdot\\mathbb\{E\}\[\|q\_\{t\}\|\]\\cdot Q\\cdot\\bar\{S\}\}\{\\Delta\_\{f\}\}\.
The first\-order conditiond​Π˙f/d​ζ=0\\mathrm\{d\}\\dot\{\\Pi\}\_\{f\}/\\mathrm\{d\}\\zeta=0gives:

γ​σ2​\(1−ζf∗\)​𝔼​\[qt2\]​Q2=nh​ϕtCEX​Q​S¯\+𝔼​\[Δ​rf\]​𝔼​\[\|qt\|\]​Q​S¯Δf\.\\gamma\\sigma^\{2\}\(1\-\\zeta^\{\*\}\_\{f\}\)\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}=n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\+\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\mathbb\{E\}\[\|q\_\{t\}\|\]Q\\bar\{S\}\}\{\\Delta\_\{f\}\}\.\(155\)
Solving forζf∗\\zeta^\{\*\}\_\{f\}:

ζf∗=1−nh​ϕtCEX​S¯γ​σ2​𝔼​\[qt2\]​Q−𝔼​\[Δ​rf\]​𝔼​\[\|qt\|\]​S¯γ​σ2​𝔼​\[qt2\]​Q​Δf\.\\zeta^\{\*\}\_\{f\}=1\-\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\-\\frac\{\\mathbb\{E\}\[\\Delta r\_\{f\}\]\\mathbb\{E\}\[\|q\_\{t\}\|\]\\bar\{S\}\}\{\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\\Delta\_\{f\}\}\.\(156\)
The first two terms equalζ∗\\zeta^\{\*\}from Theorem[7\.5](https://arxiv.org/html/2607.11888#S7.Thmtheorem5)\. For the third term, under a symmetric inventory distribution𝔼​\[\|qt\|\]≈2​𝔼​\[qt2\]/π\\mathbb\{E\}\[\|q\_\{t\}\|\]\\approx\\sqrt\{2\\mathbb\{E\}\[q\_\{t\}^\{2\}\]/\\pi\}, which gives the compact form in \([133](https://arxiv.org/html/2607.11888#S7.E133)\)\.

The second\-order conditiond2​Π˙f/d​ζ2=−γ​σ2​𝔼​\[qt2\]​Q2<0\\mathrm\{d\}^\{2\}\\dot\{\\Pi\}\_\{f\}/\\mathrm\{d\}\\zeta^\{2\}=\-\\gamma\\sigma^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q^\{2\}<0confirms this is a maximum\. The constraintζf∗∈\[0,1\]\\zeta^\{\*\}\_\{f\}\\in\[0,1\]is enforced by projection; the hedge condition \([134](https://arxiv.org/html/2607.11888#S7.E134)\) is equivalent toζf∗\>0\\zeta^\{\*\}\_\{f\}\>0before projection\. ∎

## Appendix FProof of Basis Risk Hedge Ratio \(Theorem[7\.16](https://arxiv.org/html/2607.11888#S7.Thmtheorem16)\)

###### Detailed proof\.

With imperfect correlation between the MM’s asset and the hedge instrument, the hedged portfolio variance per unit time is:

Var​\[d​Wthedged\]\\displaystyle\\mathrm\{Var\}\[\\mathrm\{d\}W\_\{t\}^\{\\mathrm\{hedged\}\}\]=σ2​qt2​Q2​d​t−2​ζ​ρ​σ​σh​qt2​Q2​d​t\+ζ2​σh2​qt2​Q2​d​t\\displaystyle=\\sigma^\{2\}q\_\{t\}^\{2\}Q^\{2\}\\mathrm\{d\}t\-2\\zeta\\rho\\sigma\\sigma\_\{h\}q\_\{t\}^\{2\}Q^\{2\}\\mathrm\{d\}t\+\\zeta^\{2\}\\sigma\_\{h\}^\{2\}q\_\{t\}^\{2\}Q^\{2\}\\mathrm\{d\}t=qt2​Q2​\(σ2−2​ζ​ρ​σ​σh\+ζ2​σh2\)​d​t\.\\displaystyle=q\_\{t\}^\{2\}Q^\{2\}\(\\sigma^\{2\}\-2\\zeta\\rho\\sigma\\sigma\_\{h\}\+\\zeta^\{2\}\\sigma\_\{h\}^\{2\}\)\\mathrm\{d\}t\.
The utility cost of this variance under CARA preferences is12​γ\\frac\{1\}\{2\}\\gammatimes the variance\. The total objective incorporating transaction costs is:

maxζ≥0⁡\{Π˙0−12​γ​Q2​𝔼​\[qt2\]​\(σ2−2​ζ​ρ​σ​σh\+ζ2​σh2\)−nh​ζ​ϕtCEX​Q​S¯\}\.\\max\_\{\\zeta\\geq 0\}\\left\\\{\\dot\{\\Pi\}\_\{0\}\-\\frac\{1\}\{2\}\\gamma Q^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\(\\sigma^\{2\}\-2\\zeta\\rho\\sigma\\sigma\_\{h\}\+\\zeta^\{2\}\\sigma\_\{h\}^\{2\}\)\-n\_\{h\}\\zeta\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\\right\\\}\.\(157\)
FOC:

γ​Q2​𝔼​\[qt2\]​\(ρ​σ​σh−ζ​σh2\)=nh​ϕtCEX​Q​S¯\.\\gamma Q^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]\(\\rho\\sigma\\sigma\_\{h\}\-\\zeta\\sigma\_\{h\}^\{2\}\)=n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}Q\\bar\{S\}\.\(158\)
Solving:

ζadj∗=ρ​σσh−nh​ϕtCEX​S¯γ​σh2​𝔼​\[qt2\]​Q\.\\zeta^\{\*\}\_\{\\mathrm\{adj\}\}=\\frac\{\\rho\\sigma\}\{\\sigma\_\{h\}\}\-\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\}\{\\gamma\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}\.\(159\)
The variance\-minimizing ratio \(ignoring costs\) isζbasis∗=ρ​σ/σh\\zeta^\{\*\}\_\{\\mathrm\{basis\}\}=\\rho\\sigma/\\sigma\_\{h\}, which is the classical minimum\-variance hedge ratio\. The cost adjustment shifts this downward\.

Forζadj∗\>0\\zeta^\{\*\}\_\{\\mathrm\{adj\}\}\>0:

ρ\>nh​ϕtCEX​S¯​σhγ​σ​σh2​𝔼​\[qt2\]​Q=ρmin,\\rho\>\\frac\{n\_\{h\}\\phi\_\{t\}^\{\\mathrm\{CEX\}\}\\bar\{S\}\\sigma\_\{h\}\}\{\\gamma\\sigma\\sigma\_\{h\}^\{2\}\\mathbb\{E\}\[q\_\{t\}^\{2\}\]Q\}=\\rho\_\{\\min\},\(160\)which is the minimum correlation threshold for beneficial hedging\. ∎

## Appendix GNotation Glossary

For the reader’s convenience, we collect the principal symbols used throughout the paper\.

Table 6:Summary of notation
## Appendix HProof of Verification Theorem[4\.6](https://arxiv.org/html/2607.11888#S4.Thmtheorem6)

###### Proof\.

Part 1: Supermartingale property\.Let\(δb,δa\)∈𝒰\(\\delta^\{b\},\\delta^\{a\}\)\\in\\mathcal\{U\}be any admissible control\. DefineMt=V^​\(t,Xt,qt,St,βt\)M\_\{t\}=\\hat\{V\}\(t,X\_\{t\},q\_\{t\},S\_\{t\},\\beta\_\{t\}\)\. By the generalized Itô formula for jump\-diffusion processes:

d​Mt\\displaystyle\\mathrm\{d\}M\_\{t\}=\[∂V^∂t\+ℒdiff​V^\]​d​t\+\(martingale terms\)\\displaystyle=\\left\[\\frac\{\\partial\\hat\{V\}\}\{\\partial t\}\+\\mathcal\{L\}\_\{\\mathrm\{diff\}\}\\hat\{V\}\\right\]\\mathrm\{d\}t\+\(\\text\{martingale terms\}\)\+λb​\(δb\)​\[V^​\(t,Xt−Pb,qt\+1,St,βt\)−V^\]​d​t\\displaystyle\\quad\+\\lambda^\{b\}\(\\delta^\{b\}\)\\left\[\\hat\{V\}\(t,X\_\{t\}\-P^\{b\},q\_\{t\}\+1,S\_\{t\},\\beta\_\{t\}\)\-\\hat\{V\}\\right\]\\mathrm\{d\}t\+λa​\(δa\)​\[V^​\(t,Xt\+Pa,qt−1,St,βt\)−V^\]​d​t\\displaystyle\\quad\+\\lambda^\{a\}\(\\delta^\{a\}\)\\left\[\\hat\{V\}\(t,X\_\{t\}\+P^\{a\},q\_\{t\}\-1,S\_\{t\},\\beta\_\{t\}\)\-\\hat\{V\}\\right\]\\mathrm\{d\}t\+\(Poisson martingale terms\)\.\\displaystyle\\quad\+\(\\text\{Poisson martingale terms\}\)\.
Sinceθ∗\\theta^\{\*\}solves the HJB equation \([45](https://arxiv.org/html/2607.11888#S4.E45)\), the drift under the optimal control is zero\. Under any suboptimal control, the drift is non\-positive \(because the supremum in \([45](https://arxiv.org/html/2607.11888#S4.E45)\) is not attained\)\. ThereforeMtM\_\{t\}is a supermartingale for any admissible control\.

Part 2: Martingale property under optimal control\.Under\(δb⁣∗,δa⁣∗\)\(\\delta^\{b\*\},\\delta^\{a\*\}\), the drift ofMtM\_\{t\}vanishes identically\. The martingale terms ared​Mtmart=γ​q​σ​V​d​Wt\+γ​\(q\+θβ∗\)​σβ​V​d​Wtβ\+\(compensated Poisson\)\\mathrm\{d\}M\_\{t\}^\{\\mathrm\{mart\}\}=\\gamma q\\sigma V\\,\\mathrm\{d\}W\_\{t\}\+\\gamma\(q\+\\theta\_\{\\beta\}^\{\*\}\)\\sigma\_\{\\beta\}V\\,\\mathrm\{d\}W\_\{t\}^\{\\beta\}\+\(\\text\{compensated Poisson\}\)\. Uniform integrability follows from:

𝔼​\[supt≤T\|Mt\|2\]≤𝔼​\[exp⁡\(2​γ​supt≤T\|Φt\|\)\]<∞,\\mathbb\{E\}\\left\[\\sup\_\{t\\leq T\}\|M\_\{t\}\|^\{2\}\\right\]\\leq\\mathbb\{E\}\\left\[\\exp\(2\\gamma\\sup\_\{t\\leq T\}\|\\Phi\_\{t\}\|\)\\right\]<\\infty,which holds under the boundedness ofqtq\_\{t\}\(inventory gating atq¯\\bar\{q\}\) and the Novikov condition for the Brownian integrals\. HenceMtM\_\{t\}is a true martingale\.

Conclusion:For any admissible control:V^​\(0,x0,0,S0,β0\)≥𝔼​\[MT\]=𝔼​\[−e−γ​WT\]\\hat\{V\}\(0,x\_\{0\},0,S\_\{0\},\\beta\_\{0\}\)\\geq\\mathbb\{E\}\[M\_\{T\}\]=\\mathbb\{E\}\[\-e^\{\-\\gamma W\_\{T\}\}\]\. Under optimal control:V^​\(0,x0,0,S0,β0\)=𝔼​\[MT\]=𝔼​\[−e−γ​WT\]\\hat\{V\}\(0,x\_\{0\},0,S\_\{0\},\\beta\_\{0\}\)=\\mathbb\{E\}\[M\_\{T\}\]=\\mathbb\{E\}\[\-e^\{\-\\gamma W\_\{T\}\}\]\. ThereforeV^=V\\hat\{V\}=Vand the optimal control attains the value\. ∎

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