Adaptive Shrinkage Calibration of Fractional Order and Volatility in the Time-Fractional Black–Scholes Model: Evaluation under Symmetric and Asymmetric Losses

<title>Abstract</title> <p> This paper develops a numerically verified framework for stabilizing maturity-specific calibration of fractional order and volatility in a dimensionally balanced time-fractional Black--Scholes model. The model uses an explicit time scale and a far-field call boundary derived from the associated fractional reaction equation, producing a Mittag--Leffler discount term. The forward problem is discretized by an $L1$ approximation of the Caputo derivative combined with Crank--Nicolson spatial averaging and is verified in the classical limit and against a fractional eigenmode with a known solution. Raw calibration is compared conceptually with fixed-target, causal dynamic-target, full-matrix adaptive state-space, and penalized smoothing estimators, with hyperparameters selected from validation prices. The Monte Carlo design covers multiple strike counts, three noise levels, and constant, smooth, jump, and oscillatory parameter paths under symmetric and asymmetric loss functions. The empirical application uses cleaned S\&amp;P~500 call prices observed on 11 May 2012. The recoverable evidence places the raw fractional-order estimates at the classical boundary and indicates strong local dependence between fractional order and volatility. The framework therefore emphasizes validation-controlled stabilization, transparent identification diagnostics, and a clear separation between parameter risk and held-out option-price performance. <bold>MSC 2020:</bold> 26A33; 62C12; 65M06; 91G20. </p>