Analyzing cryptocurrency risk with a stochastic volatility normal tempered stable process via hybrid optimization

Stochastic volatility models are employed to measure and manage financial risks in cryptocurrency markets. In this study, a subordinated Lévy process, driven by a univariate integrated square-root Cox–Ingersoll–Ross process is examined. The resulting stochastic volatility normal tempered stable (SVNTS) process effectively captures key stylized facts such as heavy tails, asymmetry and volatility clustering. Leveraging the fast Fourier transform (FFT), a numerical approximation of the return probability density is derived, facilitating the computation of tail risk measures. The SVNTS model is calibrated to a range of cryptocurrency data sets, with its goodness-of- fit rigorously evaluated. In addition, a novel parameter estimation framework is proposed for the SVNTS distribution. This approach integrates maximum likelihood estimation based on the FFT-derived density, augmented by a Bayesian optimization scheme employing Gaussian regression and expected improvement for initial parameter selection. Final refinement is achieved through particle swarm optimization, a technique rarely used in comparable studies. To capture the intricate dependencies among assets, a Student t-copula is employed, enabling the separation of marginal distributions from their joint dependency structure. This distinction is essential for accurately modeling tail dependence, particularly under conditions of market stress. Based on a robust multivariate simulation framework, portfolio optimization is performed by minimizing conditional value-at-risk across a selection of cryptocurrencies. Finally, the accuracy of the resulting value-at-risk and conditional value-at-risk estimates is validated through extensive backtesting procedures.