Abstract

This study introduces a comprehensive analytical framework for valuing nonlinear payoff volatility derivatives under a mixed fractional geometric Brownian motion model with Hurst parameter H∈(34,1). The model ensures no-arbitrage pricing within a complete-market environment while capturing both the stochastic features of standard Brownian motion and the long-memory characteristics of fractional Brownian motion. A central contribution is the derivation of the probability distribution of discretely sampled realized variance, addressing a longstanding challenge in the analytical pricing of volatility-linked derivatives. By expressing realized variance as a quadratic form of mixed fractional Brownian motion increments, we obtain a Laguerre series expansion for its probability density function and a generalized series representation for its conditional moments. These results enable closed-form evaluation of expectations involving nonlinear functions of the square root of realized variance, leading to tractable pricing formulas for a wide range of volatility-linked instruments, including swaps, options, capped or floored variants, and contracts with knock-out or corridor features. Monte Carlo simulations based on the exact Gaussian structure implied by the model demonstrate strong internal consistency and computational efficiency of the proposed pricing formulas within the simulated setting, and reveal notable sensitivity of fair strike prices to the Hurst parameter. Overall, this study advances the theoretical foundations of volatility derivative valuation by providing a tractable approach for incorporating long-memory dynamics under a no-arbitrage framework, while noting that empirical validation remains an avenue for future research.