Finite Element Methods for Pricing Cryptocurrency Derivatives under Stochastic Volatility and Jump Dynamics: An Approximation-Space Perspective

This preprint explores the role of approximation-space mismatch in the numerical pricing of cryptocurrency derivatives under structurally irregular market conditions. The paper argues that many failures of classical pricing approaches in highly volatile and discontinuous markets arise not merely from parameter instability, but from inadequate functional representations of the solution itself. Using a variational finite element framework, the study examines how reduced Sobolev regularity, free-boundary effects, payoff singularities, stochastic volatility degeneracy, and jump-driven discontinuities influence convergence behaviour in numerical option pricing. Particular attention is given to adaptive mesh refinement, weighted Sobolev spaces, and local approximation strategies near singular regions of the state space. The paper combines concepts from computational mechanics, variational analysis, and financial mathematics, treating cryptocurrency derivatives as a stress-test environment for non-uniform approximation methods. Numerical experiments compare adaptive and uniform discretisation strategies under representative calibrated regimes and illustrate the role of local refinement in recovering stable convergence behaviour. This manuscript is released as a research preprint and represents part of an ongoing interdisciplinary investigation into structurally irregular systems, approximation theory, and numerical methods in financial PDEs.