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

Minmin Zeng; Yi Liu

cs.AIApr 5, 2026

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

Authors: Minmin Zeng, Yi Liu

Organizations: tsaftech.

Abstract

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 $\phi_{m}$ (bp)	Taker Fee $\phi_{t}$ (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
$\sigma$	Reference price volatility	$/√s	venue-dependent
$\Lambda$	Baseline fill rate	fills/s	$0.001$ – $0.1$
$k$	Fill-rate sensitivity	1/$	venue-dependent
$\gamma$	Risk aversion (CARA)	1/$	$10^{-4}$ – $10^{-2}$
$\phi_{m}$	Maker fee	bp	$0$ – $5$
$\phi_{t}^{\mathrm{CEX}}$	CEX taker fee (hedge cost)	bp	$3$ – $7.5$

Table 2: Model parameters

Fee Regime	$\phi_{m}$ (bp)	Spread	Fill Rate	Net Edge (bp)	APY
Zero fee (DEX-A)	0	$2\alpha+2/k$	$\bar{\lambda}_{0}$	$1/k$	$A_{0}$
Standard (CEX-B, low)	2	$+4$ bp wider	$0.82\bar{\lambda}_{0}$	$1/k-2$	$0.57A_{0}$
Standard (CEX-B, high)	5	$+10$ bp wider	$0.61\bar{\lambda}_{0}$	$1/k-5$	$0.21A_{0}$
Rebate (CEX-C)	$-1$	$-2$ bp tighter	$1.10\bar{\lambda}_{0}$	$1/k+1$	$1.15A_{0}$

Table 3: Theoretical APY comparison across fee regimes (representative parameters)

Figure 1: Hedge regime analysis. (a) Hedge viability in the

(\sigma,\phi_{t}^{\mathrm{CEX}})

plane: green region indicates hedging is beneficial (

\Gamma_{h}>1

), red indicates no-hedge regime (

\Gamma_{h}\leq 1

). Solid black line: hedge/no-hedge boundary. (b) Optimal hedge ratio

\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.

Symbol	Parameter	Baseline Value	Units
$\sigma$	Price volatility	$0.30\%$	per minute
$\Lambda$	Baseline fill rate	30	fills/hr
$k$	Fill sensitivity	200	$1/\%$
$\gamma$	Risk aversion	$10^{-3}$	$1/$$
$\alpha$	Adverse selection	3	bp
$\phi_{m}^{\mathrm{DEX}}$	DEX maker fee	0	bp

Table 4: Baseline parameter set for numerical simulations

Figure 2: APY phase diagram in $(\xi,\bar{\lambda})$ -space. Contour lines show APY $=\{0\%,50\%,100\%,200\%\}$ . The shaded region indicates $\mathrm{APY}<0$ (unprofitable). Baseline parameters from Table 4 .

Figure 3: APY vs. adverse selection cost

\alpha

. The vertical dashed line marks

\alpha^{*}=

critical adverse selection for zero APY. Zero-fee (solid) vs. CEX 2 bp fee (dashed).

Figure 4: APY vs. fill rate

\bar{\lambda}

(fills/hr). Solid: zero-fee DEX. Dashed: CEX with

\phi_{m}=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 $s^{*}=\delta^{b*}+\delta^{a*}$ as a function of adverse selection $\alpha$ 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

\zeta

for CEX taker fees

\phi_{t}^{\mathrm{CEX}}\in\{2,5,8\}

bp. The optimal hedge ratio decreases as hedging becomes more expensive. At

\phi_{t}^{\mathrm{CEX}}=8

bp, the optimal policy is no hedging (

\zeta^{*}=0

Figure 8: Expected PnL rate vs. cancel threshold

\theta

(bp). Small

\theta

: low fill rate dominates. Large

\theta

: high adverse selection dominates. The optimal

\theta^{*}

balances the two effects.

Figure 9: Simulated inventory path under the optimal policy with

\bar{q}=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

\Delta\tau

for three CEX taker fee levels. The optimal interval

\Delta\tau^{*}

(circles, dashed lines) balances delay risk cost (

\sim\Delta\tau^{2}

) against hedge transaction cost (

\sim 1/\Delta\tau

). Higher CEX fees shift the optimum to longer intervals. (b) Heatmap of the optimal interval

\Delta\tau^{*}

as a function of price volatility

\sigma

and fill rate

\bar{\lambda}

\phi_{t}^{\mathrm{CEX}}=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

|\delta^{\mathrm{FH}*}-\delta^{\mathrm{IP}*}|

(bp) vs. time for inventory levels

q\in\{0,5,10,20\}

with

\tau_{\eta}=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-\tau_{\eta}=2

h (red dotted line). (b) Relative PnL approximation error (%) vs. reference horizon

\tau_{\eta}

on a semi-log scale. The cubic convergence rate

O(\tau_{\eta}^{3})

is confirmed by the dotted reference line. At

\tau_{\eta}=2

h, the PnL error is below 0.1% for

q\leq 10

Figure 14: Latency–fill rate tradeoff analysis. (a) PnL rate (

/hr) vs. cancel threshold

\theta

for informed fractions

\pi\in{5%,15%,30%,50%}

. Circles mark the optimal threshold

\theta^{} $. Small$ \theta $: insufficient fill rate. Large$ \theta $: excessive adverse selection. Higher$ \pi $shifts the optimum left (tighter cancellation) and reduces PnL. (b) Optimal threshold$ \theta^{}

(blue) and resulting adverse selection

\alpha(\theta^{*})

(red) vs. informed fraction

\pi

. Green dashed: optimal PnL rate. As

\pi$ increases, the optimal threshold decreases (more aggressive cancellation) but total AS increases due to the growing informed component.

Figure 15: Robustness margin

\mathcal{R}

(\xi,\|\bm{w}\|_{2})

-space at the 95% confidence level. The solid black contour marks

\mathcal{R}=1

: above this line, the strategy may become unprofitable under worst-case parameter realization. The dashed navy contour marks

\mathcal{R}=2

(can absorb twice the estimated uncertainty). Parameters:

\rho_{\mathrm{inv}}=0.10

\rho_{\mathrm{hedge}}=0.05

\phi=0

(zero-fee regime).

Figure 16: Annualized Sharpe ratio in

(\alpha,\bar{\lambda})

-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

\bar{\lambda}

and half-spread

\bar{\delta}

have the largest impact, followed by capital

K

, order notional, and adverse selection

\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

) and adverse selection (

-0.32

) are the strongest negative drivers. The zero-fee advantage (eliminating

\phi_{m}

) has a dramatic effect, consistent with Theorem 6.1 .

Figure 18: Capital efficiency analysis. (a) Capital efficiency frontier in $(\mathrm{SR},\mathrm{APY})$ -space, parameterized by leverage $\ell$ (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 ). At baseline $\alpha=3$ bp, achieving $\mathrm{APY}>100\%$ requires $\ell\geq 5\times$ . (b) Asymptotic scaling laws. Blue: APY scales linearly in fill rate $\bar{\lambda}$ (at fixed $\alpha=3$ bp). Red: APY decays linearly in adverse selection $\alpha$ with critical threshold $\alpha^{*}\approx 5$ bp (at fixed $\bar{\lambda}=20$ /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

\approx 1.8\%

, baseline

\approx 3.2\%

, and adverse regime

\approx 7.5\%

. (b) VaR and CVaR of maximum drawdown as a function of adverse selection

\alpha

. 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

\alpha=5

bp, consistent with the APY zero-crossing at

\alpha^{*}

from Theorem 5.6 .

Figure 20: Zero-fee market expansion and entry–exit dynamics. (a) Tradeable market regions in the

(\alpha,\sigma)

-plane under three fee regimes: zero-fee (green boundary), low-fee

\phi_{m}=2

bp (blue dashed), and high-fee

\phi_{m}=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

\xi_{t}

with MM entry–exit dynamics following Theorem 6.10 . Green triangles mark entry events (

\xi_{t}<\xi_{\mathrm{entry}}

); red triangles mark exits (

\xi_{t}>\xi_{\mathrm{exit}}

). The orange-shaded hysteresis band between

\xi_{\mathrm{entry}}

and

\xi_{\mathrm{exit}}

prevents costly rapid switching.

Figure 21: Ergodic inventory distribution and Bayesian parameter estimation. (a) Stationary inventory density

\pi_{\infty}(q)

under the optimal spread policy for four values of the inventory penalization parameter

\eta

. Higher

\eta

leads to tighter inventory control (smaller

\sigma_{q}

), confirming the Gaussian approximation ( 60 ) with

\sigma_{q}^{2}=\bar{\lambda}^{*}/(2\eta k)

. (b) Bayesian sequential estimation of

\pi

from fill signals

Z_{i}

under three prior specifications: flat

\mathrm{Beta}(1,1)

, informative

\mathrm{Beta}(2,18)

, and misspecified

\mathrm{Beta}(5,5)

. All priors converge to the true

\pi=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

(\sigma_{\Pi},\mathrm{APY})

-space for

N\in\{1,2,3,5,10\}

trading pairs with heterogeneous parameters and pairwise correlation

\bar{\rho}=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

\mathrm{SR}_{N}/\mathrm{SR}_{1}

vs. number of pairs

N

for five correlation levels

\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 limits

1/\sqrt{\bar{\rho}}

from Corollary 5.26 . With

\bar{\rho}=0.3

, five pairs capture

\sim

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	$O(\tau_{\eta}^{3})$ 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– $\bar{\lambda}$ scaling	Param. sweep	$R^{2}>0.998$
Thm. 5.21	Exponential tail $e^{-\theta x}$	MC (50000 paths)	Tight in tail (Fig. 19 )

Table 5: Summary of numerical validation of theoretical results

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Price processes
Symbol	Description	Introduced
$S_{t}$	Reference (CEX-B) mid-price	§ 2
$\tilde{S}_{t}$	DEX-A mid-price, $\tilde{S}_{t}=S_{t}+\beta_{t}$	§ 2
$\beta_{t}$	DEX–CEX premium (OU process)	§ 2
$\sigma$	Reference price volatility	Assumption 2.1
$\sigma_{\beta}$	Premium volatility	Assumption 2.3

Appendix

Table 6: Summary of notation

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