Abstract

We present a comparative analysis of CPU and GPU implementations for solving jump-diffusion Hamilton-Jacobi-Bellman (HJB) equations in high-frequency cryptocurrency market making. By formulating the market maker's decision problem as a stochastic optimal control problem, we derive optimal quoting strategies through HJB PDEs. Our key innovation is the integration of jump-diffusion processes that explicitly capture the discontinuous price movements characteristic of cryptocurrency markets, implemented with Gauss-Hermite quadrature for numerical stability. Jump-aware strategies achieve up to 43% higher profitability in volatile market conditions, with GPU implementation solving a 101×101 grid in approximately 0.89 ms.

Introduction

Market making is the continuous provision of liquidity through bid and ask quotes. In cryptocurrency markets, market makers face unique challenges: Bitcoin's annualized volatility frequently exceeds 80%, liquidity is fragmented across dozens of exchanges, large "whale" traders create significant adverse selection risks, and frequent large price jumps cannot be captured by continuous diffusion models alone. Traditional approaches from Avellaneda & Stoikov (2008) fail to address these dynamics adequately.

Motivation

Five factors motivate the jump-diffusion HJB approach:

1. Jump discontinuities: Crypto prices exhibit frequent large jumps requiring explicit jump-diffusion modeling beyond standard Brownian motion
2. Order flow toxicity: Real-time tracking of directional order flow imbalance enables dynamic market impact adjustment
3. Computational feasibility: GPU parallelization makes real-time HJB solving tractable for production deployment
4. Inventory risk: Explicit inventory penalty terms in the objective function enforce disciplined position management
5. Adverse selection: The toxicity-adjusted market impact parameter automatically widens spreads when order flow becomes toxic

Key Equations

Jump-diffusion mid-price process (Merton model):

Order execution intensities (execution probability vs. quote aggressiveness):

Order flow toxicity measure:

Toxicity-adaptive market impact:

Market maker's objective:

Hamilton-Jacobi-Bellman equation:

with terminal condition .

Gauss-Hermite jump integral approximation ( points):

Algorithm Blueprint

HJB Value Function Iteration — GPU Accelerated

Grid: Value function on state space

State points: grid points (solved in ~0.89ms on GPU)

Algorithm:

1. INITIALIZATION

   • Terminal condition: for all grid points

2. BACKWARD INDUCTION ( down to 0)

   • Launch CUDA kernel with blocks=(), threads=(32,32)
   • For each thread at grid point (i,j):
     • Compute state: ;
     • Load into shared memory (32x32 tile with halo region)
     • Estimate derivatives:
     • Diffusion contribution:
     • Jump term (Gauss-Hermite 5-point quadrature):
       • Points: [-2.020, -0.959, 0.0, 0.959, 2.020]
       • Weights: [0.088, 0.394, 0.945, 0.394, 0.088]
     • Control optimization over 5x5 bid/ask grid:
       • For bid_idx, ask_idx in [0..4]:
         • Candidate quotes:
         • If AND AND : compute candidate value
         • Execution intensity:
         • Store optimal $V[i,j]$ from best candidate

3. OPTIMAL QUOTES EXTRACTION

   • Bid quote:
   • Ask quote:

4. TOXICITY UPDATE (real-time on CPU)

   • Order flow toxicity:
   • Adaptive spread:
   • Re-solve HJB with updated market impact parameter

System Architecture

Results

| Strategy | Ann. Return | Sharpe | Max DD | Profitability vs Baseline | |---|---|---|---|---| | Simple spread heuristic | baseline | — | — | — | | Avellaneda-Stoikov (no jumps) | — | — | — | — | | Jump-diffusion HJB (CPU) | — | — | — | +43% volatile markets | | Jump-diffusion HJB (GPU) | — | — | — | +43% volatile markets |

GPU solves 101×101 grid in 0.89 ms vs CPU baseline, enabling real-time deployment. End-to-end latency including data transfer and order placement: 2–5 ms in production. Inventory risk reduction: significant vs non-inventory-aware strategies. The toxicity-adaptive automatically widens spreads during adverse order flow, reducing adverse selection losses.

Contributions

• Jump-diffusion extension to the HJB market-making framework, capturing discontinuous price movements characteristic of cryptocurrency markets
• Novel order flow toxicity measure with exponential decay weighting, directly modulating the market impact parameter in real time
• Massively parallel GPU implementation with shared memory tiling and coalesced access, achieving sub-millisecond HJB solving for 10,201 state points
• Gauss-Hermite quadrature (5-point) for accurate jump integral approximation with exponential convergence for smooth integrands
• Empirical validation on real Bitcoin market data demonstrating 43% higher profitability in volatile conditions vs non-jump-aware strategies

Paper

Author

Frankline Misango Oyolo Arithmax Research frankline@arithmax.com — Published: December 7, 2025