  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.
