Hydrodynamics of Markets:Hidden Links Between Physics and Finance

TL;DR

This paper explores the deep connections between physics and financial markets, using Kelvin waves and affine equations to unify and solve complex models for pricing and risk management in finance.

Contribution

It introduces a unified mathematical framework employing Kelvin waves to analyze and solve diverse problems in financial modeling and derivatives pricing.

Findings

Unified approach to pricing models like Black-Scholes and Heston

Application of Kelvin waves to complex path-dependent options

Solution of hedging problems for cryptocurrency Automated Market Makers

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.