We develop a rigorous theoretical framework for optimal market making in perpetual futures markets with zero maker fees. 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 contributions include: (i) a PnL decomposition theorem separating revenue into spread income, adverse selection loss, inventory carrying cost, hedging friction, and funding rate exposure; (ii) the Hamilton-Jacobi-Bellman equation for the joint spread-inventory-hedging control problem under CARA utility with a verification theorem; (iii) High-APY Regime Theorems characterizing profitable regions via five dimensionless parameters, culminating in a Master APY Formula; (iv) analysis of zero-fee economics on decentralized perpetual exchanges with optimal entry-exit thresholds; (v) optimal cross-exchange hedging policies with funding rate dynamics and a hedge regime trichotomy; (vi) a robustness margin quantifying parameter uncertainty tolerance; (vii) exponential drawdown probability bounds and a universal APY-VaR identity; (viii) ergodic inventory distribution under optimal control with Bayesian adaptive estimation; (ix) Kelly-optimal leverage with ruin boundaries; and (x) multi-pair portfolio allocation with diversification saturation results. Numerical analysis with twenty-three figures reveals phase transitions between profitable and unprofitable regimes. Our framework unifies and extends the Avellaneda-Stoikov, Gueant-Lehalle-Fernandez-Tapia, and Glosten-Milgrom paradigms for modern decentralized venue microstructure.
Figures & tables
Venue Type
Maker Fee ϕm (bp)
Taker Fee ϕt (bp)
Net Round-Trip Cost
DEX-A (zero fee)
0
1–3
0 (maker–maker)
CEX-B (standard)
1.5–5
3–7.5
3 – 10 (maker–maker)
CEX-C (rebate)
−0.5 to −2
3–5
−1 to −4 (maker–maker)
Table 1: Representative fee structures across venue types (basis points)
Symbol
Description
Unit
Typical Range
σ
Reference price volatility
$/√s
venue-dependent
Λ
Baseline fill rate
fills/s
0.001 – 0.1
k
Fill-rate sensitivity
1/$
venue-dependent
γ
Risk aversion (CARA)
1/$
10−4 – 10−2
ϕm
Maker fee
bp
0 – 5
ϕtCEX
CEX taker fee (hedge cost)
bp
3 – 7.5
Table 2: Model parameters
Fee Regime
ϕm (bp)
Spread
Fill Rate
Net Edge (bp)
APY
Zero fee (DEX-A)
0
2α+2/k
λˉ0
1/k
A0
Standard (CEX-B, low)
2
+4 bp wider
0.82λˉ0
1/k−2
0.57A0
Standard (CEX-B, high)
5
+10 bp wider
0.61λˉ0
1/k−5
0.21A0
Rebate (CEX-C)
−1
−2 bp tighter
1.10λˉ0
1/k+1
1.15A0
Table 3: Theoretical APY comparison across fee regimes (representative parameters)
Figure 1: Hedge regime analysis. (a) Hedge viability in the (σ,ϕtCEX) plane: green region indicates hedging is beneficial ( Γh>1 ), red indicates no-hedge regime ( Γh≤1 ). Solid black line: hedge/no-hedge boundary. (b) Optimal hedge ratio ζ∗ 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.
Symbol
Parameter
Baseline Value
Units
σ
Price volatility
0.30%
per minute
Λ
Baseline fill rate
30
fills/hr
k
Fill sensitivity
200
1/%
γ
Risk aversion
10−3
$1/$$
α
Adverse selection
3
bp
ϕmDEX
DEX maker fee
0
bp
Table 4: Baseline parameter set for numerical simulations
Figure 2: APY phase diagram in (ξ,λˉ) -space. Contour lines show APY ={0%,50%,100%,200%} . The shaded region indicates APY<0 (unprofitable). Baseline parameters from Table 4 .
Figure 3: APY vs. adverse selection cost α . The vertical dashed line marks α∗= critical adverse selection for zero APY. Zero-fee (solid) vs. CEX 2 bp fee (dashed).
Figure 4: APY vs. fill rate λˉ (fills/hr). Solid: zero-fee DEX. Dashed: CEX with ϕm=2 bp.
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.
Figure 6: Optimal total spread s∗=δb∗+δa∗ as a function of adverse selection α and inventory q . The total spread is independent of inventory (Corollary 4.4 ); inventory affects only the bid–ask asymmetry.
Figure 7: APY vs. hedge ratio ζ for CEX taker fees ϕtCEX∈{2,5,8} bp. The optimal hedge ratio decreases as hedging becomes more expensive. At ϕtCEX=8 bp, the optimal policy is no hedging ( ζ∗=0 ).
Figure 8: Expected PnL rate vs. cancel threshold θ (bp). Small θ : low fill rate dominates. Large θ : high adverse selection dominates. The optimal θ∗ balances the two effects.
Figure 9: Simulated inventory path under the optimal policy with qˉ=100 contracts, showing spread-skew-induced mean reversion and side-gating at the boundaries.
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.
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.
Figure 12: Hedge interval sensitivity analysis. (a) Total cost rate vs. hedge re-evaluation interval Δτ for three CEX taker fee levels. The optimal interval Δτ∗ (circles, dashed lines) balances delay risk cost ( ∼Δτ2 ) against hedge transaction cost ( ∼1/Δτ ). Higher CEX fees shift the optimum to longer intervals. (b) Heatmap of the optimal interval Δτ∗ as a function of price volatility σ and fill rate λˉ at ϕtCEX=5 bp. High volatility and high fill rates require more frequent hedge re-evaluation.
Figure 13: Numerical verification of Theorem 4.11 . (a) Absolute spread approximation error ∣δFH∗−δIP∗∣ (bp) vs. time for inventory levels q∈{0,5,10,20} with τη=2 h. Solid lines: exact difference. Dashed lines: theoretical bound from ( 58 ). The bound is tight at t=0 and t=T , and the two policies coincide exactly at t=T−τη=2 h (red dotted line). (b) Relative PnL approximation error (%) vs. reference horizon τη on a semi-log scale. The cubic convergence rate O(τη3) is confirmed by the dotted reference line. At τη=2 h, the PnL error is below 0.1% for q≤10 .
Figure 14: Latency–fill rate tradeoff analysis. (a) PnL rate (/hr)vs.cancelthreshold\thetaforinformedfractions\pi\in{5%,15%,30%,50%}.Circlesmarktheoptimalthreshold\theta^{}.Small\theta:insufficientfillrate.Large\theta:excessiveadverseselection.Higher\pishiftstheoptimumleft(tightercancellation)andreducesPnL.(b)Optimalthreshold\theta^{}(blue)andresultingadverseselection\alpha(\theta^{*})(red)vs.informedfraction\pi.Greendashed:optimalPnLrate.As\pi$ increases, the optimal threshold decreases (more aggressive cancellation) but total AS increases due to the growing informed component.
Figure 15: Robustness margin R in (ξ,∥w∥2) -space at the 95% confidence level. The solid black contour marks R=1 : above this line, the strategy may become unprofitable under worst-case parameter realization. The dashed navy contour marks R=2 (can absorb twice the estimated uncertainty). Parameters: ρinv=0.10 , ρhedge=0.05 , ϕ=0 (zero-fee regime).
Figure 16: Annualized Sharpe ratio in (α,λˉ) -space. Contours at SR ={1,3,5,10} . The high-SR region corresponds to low adverse selection and high fill rates.
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 rate λˉ and half-spread δˉ have the largest impact, followed by capital K , order notional, and adverse selection α . 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 ) and adverse selection ( −0.32 ) are the strongest negative drivers. The zero-fee advantage (eliminating ϕm ) has a dramatic effect, consistent with Theorem 6.1 .
Figure 18: Capital efficiency analysis. (a) Capital efficiency frontier in (SR,APY) -space, parameterized by leverage ℓ (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 ). At baseline α=3 bp, achieving APY>100% requires ℓ≥5× . (b) Asymptotic scaling laws. Blue: APY scales linearly in fill rate λˉ (at fixed α=3 bp). Red: APY decays linearly in adverse selection α with critical threshold α∗≈5 bp (at fixed λˉ=20 /hr). Both confirm the Master APY Formula predictions.
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 . The bound is tight in the tail, confirming the Brownian approximation. At the 5% level (gray dashed), the favorable regime has VaR ≈1.8% , baseline ≈3.2% , and adverse regime ≈7.5% . (b) VaR and CVaR of maximum drawdown as a function of adverse selection α . The shaded regions show the gap between VaR 95% and VaR 99% (orange) and between VaR 99% and CVaR 95% (red). Drawdown risk increases sharply beyond α=5 bp, consistent with the APY zero-crossing at α∗ from Theorem 5.6 .
Figure 20: Zero-fee market expansion and entry–exit dynamics. (a) Tradeable market regions in the (α,σ) -plane under three fee regimes: zero-fee (green boundary), low-fee ϕm=2 bp (blue dashed), and high-fee ϕm=5 bp (red dotted). The yellow-shaded region represents the market expansion zone —markets profitable only under zero fees (Proposition 6.3 ). (b) Simulated adverse selection ratio ξt with MM entry–exit dynamics following Theorem 6.10 . Green triangles mark entry events ( ξt<ξentry ); red triangles mark exits ( ξt>ξexit ). The orange-shaded hysteresis band between ξentry and ξexit prevents costly rapid switching.
Figure 21: Ergodic inventory distribution and Bayesian parameter estimation. (a) Stationary inventory density π∞(q) under the optimal spread policy for four values of the inventory penalization parameter η . Higher η leads to tighter inventory control (smaller σq ), confirming the Gaussian approximation ( 60 ) with σq2=λˉ∗/(2ηk) . (b) Bayesian sequential estimation of π from fill signals Zi under three prior specifications: flat Beta(1,1) , informative Beta(2,18) , and misspecified Beta(5,5) . All priors converge to the true π=0.15 at rate 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.
Figure 22: Multi-pair portfolio allocation and diversification. (a) Efficient frontier in (σΠ,APY) -space for N∈{1,2,3,5,10} trading pairs with heterogeneous parameters and pairwise correlation ρˉ=0.2 . Circles mark the maximum Sharpe ratio portfolio for each N . Adding pairs shifts the frontier leftward (lower risk) and upward (access to higher APY through diversification). (b) Sharpe ratio improvement SRN/SR1 vs. number of pairs N for five correlation levels ρˉ∈{0,0.1,0.3,0.5,0.8} under equal-weight allocation of homogeneous pairs. Dashed horizontal lines show the theoretical limits 1/ρˉ from Corollary 5.26 . With ρˉ=0.3 , five pairs capture ∼ 85% of the diversifiable risk reduction.
Theorem
Key Prediction
Method
Agreement
Thm. 4.3
Inventory-independent total spread
MC (1000 paths)
<0.2 bp error
Thm. 4.11
O(τη3) PnL convergence rate
Grid search
Confirmed (Fig. 13 )
Thm. 5.6
5D APY phase boundary
Param. sweep
<3% rel. error
Thm. D.1
Master APY factored form
MC (3000 paths)
<1.5% rel. error
Prop. 5.18
Linear APY– λˉ scaling
Param. sweep
R2>0.998
Thm. 5.21
Exponential tail e−θx
MC (50000 paths)
Tight in tail (Fig. 19 )
Table 5: Summary of numerical validation of theoretical results
We describe an adaptive market-making architecture that preserves the analytical structure of the Avellaneda--Stoikov framework while introducing a successor measure-style adaptation mechanism. In our paper we keep Avellaneda--Stoikov fast Hamilton--Jacobi--Bellman structure and make it adaptive to changing market regimes and trading objectives. The central idea is to separate market dynamics from the trading objective. The market state determines a low-dimensional set of Avellaneda--Stoikov parameters, while recent realized rewards determine a low-dimensional objective vector. The HJB forward map then converts this objective into optimal bid and ask quotes through a scalarization of future reward features.
We propose a market-making game played among N≥2 strategic market makers and a single non-strategic representative trader. In a continuous quote space, standard finite-game arguments do not apply, while non-convex and discontinuous payoffs rule out classical continuous existence results and complicate learning. Nonetheless, we characterize pure-strategy Nash equilibria (PSNEs) and show that at least one PSNE always exists under minimal assumptions on the underlying asset value distribution, and that the equilibrium bid-ask spread widens strictly with asset value volatility. To obtain a tractable learning framework, we discretize prices and, under an additional regularity condition, show that the time-averaged play of no-internal-regret learners approaches the continuous Nash price envelope as the tick size vanishes. Finally, we empirically validate our theoretical results through numerical simulations.
Matteo Onger, Luigi Foscari
Department of Computer Science, University of Milan
Market makers keep trading efficient by setting bid and ask prices for buyers and sellers. In sequential market making, trades happen independently based on price thresholds, requiring the market maker to maintain balanced, low-risk inventory over T rounds. This paper studies how to maximize revenue with unknown buyer and seller valuations using regret minimization. When the market maker only sees trade success or failure, an algorithm achieves regret and inventory bounds proportional to T to the power of 2/3. When full order book data is available for failed trades, regret and inventory limits improve significantly to the square root of T. We also establish a fundamental trade-off via a lower bound, by proving that tighter round-by-round inventory limits strictly force higher minimum regret.
Davide Maran, Marcello Restelli, Francesco Emanuele Stradi