Mellin Transform Pricing of Binary Options in a PDE-Equivalent Gaussian Diffusion for a Sub-Mixed Fractional Vasicek Model

This paper develops closed-form prices for binary call options in a time-inhomogeneous Gaussian diffusion whose effective volatility and covariance coefficients are motivated by a sub-mixed fractional Vasicek model with H∈(3/4,1). We derive the pricing PDE by delta hedging and establish the equivalent martingale measure, market completeness, and the Feynman–Kac representation within this diffusion. Applying the Mellin transform—with an explicit treatment of the admissible transform strips and of the regularisation of the discontinuous payoffs—we obtain explicit analytical formulae for both cash-or-nothing call (CONC) and asset-or-nothing call (AONC) options. The closed-form formulae are validated independently by a finite-difference solution of the pricing PDE and by Monte Carlo simulation of the equivalent diffusion: the relative errors are below 0.1% with 106 simulated paths, and degenerate cases recover the classical Black–Scholes binary option prices up to machine precision. Within this diffusion, sensitivity analysis shows that the fractional parameter H has a state-dependent but quantitatively modest impact on prices (within about ±1.2% in our benchmark experiments), the effect being strongest for out-of-the-money options.