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We discover several surprising relationships between large classes ofseemingly unrelated foundational problems of financial engineering andfundamental problems of hydrodynamics and molecular physics. Solutions in allthese domains can be reduced to solving affine differential equations commonlyused in various mathematical and scientific disciplines to model dynamicsystems. We have identified connections in these seemingly disparate areas aswe link together small wave-like perturbations of linear flows in ideal andviscous fluids described in hydrodynamics by Kevin waves to motions of free andharmonically bound particles described in molecular physics by Klein-Kramersand Kolmogorov equations to Gaussian and non-Gaussian affine processes, e.g.,Ornstein-Uhlenbeck and Feller, arising in financial engineering. To furtheremphasize the parallels between these diverse fields, we build a coherentmathematical framework using Kevin waves to construct transition probabilitydensity functions for problems in hydrodynamics, molecular physics, andfinancial engineering. As one of the outcomes of our analysis, we discover thatthe original solution of the Kolmogorov equation contains an error, which wesubsequently correct. We apply our interdisciplinary approach to advance theunderstanding of various financial engineering topics, such as pricing of Asianoptions, volatility and variance swaps, options on stocks with path-dependentvolatility, bonds, and bond options. We also discuss further applications toother exciting problems of financial engineering.

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