Stochastics

Stochastic Thermodynamics of Price Impact

This repository contains the numerical implementation and validation code for the paper "A Stochastic Thermodynamics Approach to Price Impact and Round-Trip Arbitrage: Theory and Empirical Implications.https://arxiv.org/abs/2512.03123"

The code verifies the Financial Second Law and the Fluctuation Theorems derived in the paper by simulating trading trajectories under linear permanent impact and strictly convex temporary impact.

Repository Contents

• quant.py: A Python prototype for rapid verification and plotting.
• quant.cpp: A high-performance C++ implementation for production-grade simulation.

Key Features

1. Work Calculation ($W$): Computes dissipated work ($\eta \int v_t^2 dt$) numerically.
2. Variance Calculation ($V$): Computes position variance ($\int q_t^2 dt$) via Riemann sums.
3. Fluctuation Bound: Calculates the exponential bound $P(\Pi \ge 0) \le \exp(-W^2 / 2\sigma^2 V)$.
4. Strategies Verified:
   • Triangular Strategy: (Buy-then-sell linear/symmetric).
   • Ramp Strategy: (Smooth linear decay trading rate).

Prerequisites

Python

• Python 3.x
• NumPy (pip install numpy)

C++

• GCC (g++) or any standard C++11 compliant compiler.

How to Run

1. Python Simulation

Run the script to see the comparison between Numerical integration and Analytical formulas.

Expected Output

--- TRIANGULAR STRATEGY --- Work (Numerical): 100.0000 Work (Analytical): 100.0000 Var (Numerical): 8333.3333 Var (Analytical): 8333.3333 Fluctuation Bound: 1.499622e-07 Accuracy check: PASS

--- RAMP STRATEGY --- Work (Numerical): 33.3333 Work (Analytical): 33.3333 Var (Numerical): 3333.3333 Var (Analytical): 3333.3333 Fluctuation Bound: 1.888756e-02 Accuracy check: PASS

C++ Simulation

Compile and run the high-performance implementation.

g++ -o quant quant.cpp ./quant

Theoretical Alignment

This code strictly adheres to the corrected Linear Permanent Impact assumption ($\mathcal{I}(v) = \lambda v$). Under this framework, the permanent impact component integrates to zero over any closed round-trip cycle, meaning the dissipated work is governed solely by the temporary impact coefficient $\alpha = \eta$.

License

MIT License.

Free to use for academic and research purposes.