Analytical Pricing of Volatility-Linked Financial Derivatives Under the Sub-Mixed Fractional Brownian Motion Framework in a No-Arbitrage Complete Market

This paper develops a unified analytical approach for pricing a broad class of volatility-linked financial derivatives under the sub-mixed fractional geometric Brownian motion model. The proposed framework captures key empirical features of financial markets, including correlated non-stationary Gaussian increments and long-memory dependence, while preserving the semimartingale property required for arbitrage-free pricing. We present the exact distribution of the realized variance as a quadratic form of correlated non-stationary Gaussian increments, which leads to a closed-form expression for the cumulative distribution function via a Laguerre-series expansion. These distributional results enable analytical pricing formulas for an extensive family of volatility-linked derivatives. Monte Carlo simulations confirm the accuracy and computational efficiency of the proposed formulas, while numerical investigations illustrate the significant impact of non-stationarity, long-memory effects, and the Hurst parameter on derivative values. These results contribute to a deeper theoretical understanding and more effective computational methods for pricing nonlinear volatility derivatives in markets characterized by persistent temporal dependence and non-stationary stochastic dynamics.