Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results

Abstract: We discover several surprising relationships between large classes of seemingly unrelated foundational problems of financial engineering and fundamental problems of hydrodynamics and molecular physics. Solutions in all these domains can be reduced to solving affine differential equations commonly used in various mathematical and scientific disciplines to model dynamic systems. We have identified connections in these seemingly disparate areas as we link together small wave-like perturbations of linear flows in ideal and viscous fluids described in hydrodynamics by Kevin waves to motions of free and harmonically bound particles described in molecular physics by Klein-Kramers and Kolmogorov equations to Gaussian and non-Gaussian affine processes, e.g., Ornstein-Uhlenbeck and Feller, arising in financial engineering. To further emphasize the parallels between these diverse fields, we build a coherent mathematical framework using Kevin waves to construct transition probability density functions for problems in hydrodynamics, molecular physics, and financial engineering. As one of the outcomes of our analysis, we discover that the original solution of the Kolmogorov equation contains an error, which we subsequently correct. We apply our interdisciplinary approach to advance the understanding of various financial engineering topics, such as pricing of Asian options, volatility and variance swaps, options on stocks with path-dependent volatility, bonds, and bond options. We also discuss further applications to other exciting problems of financial engineering.