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Recent empirical evidence has highlighted the crucial role of jumps in both price and volatility within the cryptocurrency market. In this paper, we introduce an analytical model framework featuring fractional stochastic volatility, accommodating price--volatility co-jumps and volatility short-term dependency concurrently. We particularly focus on inverse options, including the emerging Quanto inverse options and their power-type generalizations, aimed at mitigating cryptocurrency exchange rate risk and adjusting inherent risk exposure. Characteristic function-based pricing--hedging formulas are derived for these inverse options. The general model framework is then applied to asymmetric Laplace jump-diffusions and Gaussian-mixed tempered stable-type processes, employing three types of fractional kernels, for an extensive empirical analysis involving model calibration on two independent Bitcoin options data sets, during and after the COVID-19 pandemic. Key insights from our theoretical analysis and empirical findings include: (1) the superior performance of fractional stochastic-volatility models compared to various benchmark models, including those incorporating jumps and stochastic volatility, (2) the practical necessity of jumps in both price and volatility, along with their co-jumps and rough volatility, in the cryptocurrency market, (3) stability of calibrated parameter values in line with stylized facts, and (4) the suggestion that a piecewise kernel offers much higher computational efficiency relative to the commonly used Riemann--Liouville kernel in constructing fractional models, yet maintaining the same accuracy level, thanks to its potential for obtaining explicit model characteristic functions.
Li et al. (Sun,) studied this question.