Hydrodynamics of Markets:Hidden Links Between Physics and Finance

Paper: arXiv 2403.09761

Authors: Alexander Lipton

Abstract

An intriguing link between a wide range of problems occurring in physics and financial engineering is presented. These problems include the evolution of small perturbations of linear flows in hydrodynamics, the movements of particles in random fields described by the Kolmogorov and Klein-Kramers equations, the Ornstein-Uhlenbeck and Feller processes, and their generalizations. They are reduced to affine differential and pseudo-differential equations and solved in a unified way by using Kelvin waves and developing a comprehensive math framework for calculating transition probabilities and expectations. Kelvin waves are instrumental for studying the well-known Black-Scholes, Heston, and Stein-Stein models and more complex path-dependent volatility models, as well as the pricing of Asian options, volatility and variance swaps, bonds, and bond options. Kelvin waves help to solve several cutting-edge problems, including hedging the impermanent loss of Automated Market Makers for cryptocurrency trading. This title is also available as Open Access on Cambridge Core.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly original and mathematically dense framework linking fluid dynamics (Kelvin waves) to financial stochastic processes and option pricing. While exceptionally strong theoretically, it focuses primarily on analytical derivations and theoretical modeling rather than extensive empirical backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Unify Financial Modeling <br>& Hydrodynamics via Physics"] --> B["Inputs: Financial Models & Stochastic Processes<br>(Black-Scholes, Heston, OU, Feller)"]
    B --> C["Mathematical Framework: Reduce to Affine &<br>Pseudo-Differential Equations"]
    C --> D["Core Method: Application of Kelvin Waves<br>& Integral Transforms"]
    D --> E["Calculations: Derive Transition Probabilities<br>& Expected Values"]
    E --> F["Outcomes: Unified Solution for Exotics<br>(Asian Options, Swaps) & AMM Impermanent Loss"]