Crypto Inverse-Power Options and Fractional Stochastic Volatility

Recent empirical evidence has highlighted the crucial role of jumps in both price and volatility within the cryptocurrency market. In this paper, we integrate price--volatility co-jumps and volatility short-term dependency into a coherent model framework, featuring fractional stochastic volatility. We particularly focus on inverse options, including the emerging Quanto inverse options and their power-type generalizations, aiming at mitigating cryptocurrency exchange rate risk and adjusting inherent risk exposure. Characteristic function-based pricing--hedging formulas are derived for these inverse options. The model framework is 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, along with high computational efficiency when utilizing a piecewise kernel, (2) the practical necessity of considering jumps in both price and volatility, along with rough volatility, in pricing and hedging cryptocurrency options, (3) stability of calibrated parameter values in line with stylized facts.