Nonlinear Volatility Derivatives Under Generalized Mixed Fractional Brownian Dynamics with Long-Range Dependence: Distributional Analysis and Valuation

Abstract

This paper develops a distributional framework for discretely monitored volatility derivatives with nonlinear payoffs under generalized mixed fractional Brownian dynamics. Multiple fractional components with distinct Hurst parameters and positive weights generate heterogeneous temporal dependence while preserving a finite-dimensional Gaussian representation. The Brownian-equivalent semimartingale regime is distinguished from the selected Gaussian valuation law. Under the latter law, the normalized fractional exponential does not satisfy the classical discounted martingale property when the non-Brownian components are active in the regime considered. The resulting values are therefore initial-time Gaussian reference valuations and zero-expected-payoff strikes under this convention. They use an unconditional return law with trivial initial information; valuation after observing a history requires conditional Gaussian updating. Realized variance is represented as a positive quadratic form in correlated Gaussian returns and, by spectral decomposition, as a conic combination of independent noncentral chi-square variables. This yields Laguerre-series probability density and cumulative distribution functions. A sufficient scaling condition gives geometric coefficient bounds, absolute and uniform convergence of the density expansion on the nonnegative axis, and weighted absolute integrability that justifies the required sum–integral interchanges. These finite-grid results require a nondegenerate Gaussian return law and extend beyond the Brownian-equivalent Hurst regime. The formulas cover variance and volatility swaps, options, capped and floored contracts, knock-outs, and corridor-type payoffs. Numerical comparisons with Monte Carlo support distributional and valuation accuracy, including distributional tests outside the Brownian-equivalent Hurst regime. Deterministic scale tests show that minimizing the geometric convergence factor need not minimize the tested truncation order. Comparisons of balanced and Brownian-dominant weights further show payoff-dependent changes in valuations and truncation stability for the tested nonlinear quantities. These findings support direct checks of truncation and numerical precision alongside spectral scale selection.