Valuation Engines Under Market Incompleteness: An Empirical Benchmark of High-Order Option Pricing in Digital Asset Markets

Abstract

Financial derivatives occupy a core position in modern asset allocation, but the traditional option pricing framework, particularly the Black-Scholes (B-S) model, faces unprecedented challenges when applied to the highly volatile cryptocurrency market. The B-S assumption of constant volatility and continuous price paths fundamentally fails to capture the volatility smile, heavy-tailed distributions, and intermittent crashes inherent in digital assets. Furthermore, structural frictions such as liquidity fragmentation render the crypto ecosystem an incomplete market. This paper systematically reviews the mathematical transition from the B-S framework to high-order stochastic processes, specifically the Heston and Jump-Diffusion models. Most importantly, we contribute to the existing literature by providing a rigorous empirical evaluation using out-of-sample high-frequency options data from Deribit. Our empirical results quantitatively confirm the theoretical limitations of traditional pricing: while the Heston model reduces the Root Mean Square Error (RMSE) by approximately 45%, the Stochastic Volatility with Jumps (SVJ) model demonstrates absolute superiority in pricing tail risks. Specifically, the SVJ framework decreases the valuation error for deep out-of-the-money (OTM) options by over 72% compared to the baseline B-S model. Ultimately, this study emphasizes that structurally integrating both continuous volatility clustering and discrete market shocks is mathematically indispensable for robust risk management in decentralized financial markets.