Time-Fractional Black-Scholes Model

• Time-fractional Black-Scholes model is a financial framework that uses fractional derivatives, such as the Caputo derivative, to incorporate memory effects and subdiffusive dynamics.
• It employs advanced mathematical formulations including inverse α-stable subordination and fractional Brownian motion to modify Greeks and enhance the analytic representation of option prices.
• High-order numerical methods like L1 schemes, B-spline collocation, and ANN-based solvers address computational challenges while ensuring stable and convergent solutions.

The time-fractional Black-Scholes model generalizes the classical Black-Scholes framework by incorporating memory effects and anomalous diffusion via fractional time-derivatives or non-Markovian noise. This formulation captures empirically observed subdiffusive and long-range dependence phenomena in financial time series, resulting in option-pricing models governed by time-fractional partial differential equations (PDEs). The mathematical foundation, analytic structure, numerical methodologies, and practical implications of such models span a wide spectrum, from pure stochastic processes (e.g., subordination and fractional Brownian motion) to fractional PDEs and high-order numerical algorithms.

1. Mathematical Formulation of the Time-Fractional Black-Scholes Model

The canonical form of the time-fractional Black–Scholes equation replaces the standard first-order time derivative with a Caputo derivative of order $\alpha\in(0,1)$: $${}^C D_t^\alpha V(S,t) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0, \quad S>0,\, 0<t<T,$$ where

$${}^C D_t^\alpha V(S,t) = \frac{1}{\Gamma(1-\alpha)}\int_{0}^{t} (t-\tau)^{-\alpha} \frac{\partial V}{\partial\tau}(S,\tau)d\tau.$$

This structure models subdiffusive dynamics and accounts for long waiting times or price stagnations, as frequently observed in financial markets (Krzyżanowski et al., 2019, Stanislavsky, 2011, Zhang et al., 13 Nov 2025, Kolokoltsov, 2011). Terminal and boundary conditions are typically chosen in analogy with the classical model: $$V(S,T) = \max(S-K,0) \quad (\text{call}), \qquad V(0,t) = 0, \qquad V(S\to\infty,t) \sim S-K e^{-r(T-t)}.$$ Extensions include tempered fractional derivatives to enforce finite moments (Zhou et al., 2023, Krzyżanowski et al., 2021), systems with fractional Brownian motion or subordination (Zhang et al., 13 Nov 2025, Shokrollahi, 2017), and multi-asset or mixed spatial-temporal fractional dynamics (Zakaria et al., 2020, Torres-Hernandez et al., 2020).

2. Stochastic Origins: Subdiffusion, Subordination, and Fractional Noise

The time-fractional Black-Scholes model emerges naturally from stochastic processes with heavy-tailed waiting times (continuous-time random walks), subordination, or self-similar Gaussian noise. Two principal mechanisms are:

• Inverse $\alpha$-stable subordination: The operational (trading) time is replaced by the inverse of a strictly increasing $\alpha$-stable subordinator, resulting in trajectories characterized by random plateaus ("trapping"). The corresponding asset price evolves as $Z_\alpha(t) = Z(S_\alpha(t))$, which under risk-neutral valuation leads directly to a fractional PDE with Caputo time-derivative (Stanislavsky, 2011, Krzyżanowski et al., 2019, Zhang et al., 13 Nov 2025, Kolokoltsov, 2011).
• Fractional Brownian motion: Non-Markovian price processes of the form $$S_t = S_0 \exp\bigl((r_d - r_f)t + \sigma B_H(t)\bigr)$$, with $B_H$ a fractional Brownian motion ($H > 1/2$), generate PDEs in which volatility scaling is altered according to the Hurst exponent, particularly in discrete-time market settings (Shokrollahi, 2017).

Both approaches yield "memory" effects and subdiffusive scaling at the level of marginal distributions and time-evolution of the option price, and can accommodate transaction costs and incomplete markets.

3. Analytical Structures, Closed-form Solutions, and Greeks

The presence of fractional time-derivatives fundamentally alters the analytic structure of solutions to the Black-Scholes equation:

• Integral representations: The price of European contingent claims often admits a representation as an integral over the distribution of the subordinator or via Fox $H$-functions and Mittag-Leffler kernels (Stanislavsky, 2011, Zhang et al., 13 Nov 2025).
• Modified Greeks: The computation of Greeks (delta, gamma, theta, vega, rho) incorporates the altered effective volatility and scaling; for example, under fractional Brownian motion plus transaction costs, the "effective" volatility is given by

$$\widehat{\sigma} = \sigma \sqrt{(\delta t)^{2H-1} + \frac{\alpha}{\sigma (\delta t)^{1-H}\sqrt{2/\pi}}}$$

which affects both the option value and the sensitivities (Shokrollahi, 2017).

• Laplace and eigenfunction expansions: Solution methods involve Laplace transforms in time, yielding solutions in terms of eigenfunction expansions with Mittag-Leffler or Wright-function time kernels (Kolokoltsov, 2011, Stanislavsky, 2011).
• Explicit formulas in tempered/subdiffusive models: For instance, in the sub-diffusive Black-Scholes model under a Girsanov transform, the price is

$$V_t = e^{-r(T-t)} \int_0^\infty \mathbb{E}_Z\bigl[\psi\bigl(x e^{r(T-t) - \frac{1}{2}\sigma^2 y + \sigma\sqrt{y}\,Z}\bigr)\bigr]\,\mathbb{P}\{L_{T-t-R_t} \in dy\},$$

where $L_{T-t-R_t}$ is the inverse subordinator (Zhang et al., 13 Nov 2025).

4. Numerical Methods for Time-Fractional Black-Scholes Equations

The nonlocality of Caputo and tempered fractional derivatives leads to increased algorithmic complexity. Key numerical strategies include:

• L1 and Alikhanov discretizations: Uniform and nonuniform L1-type difference approximations for the Caputo derivative yield global accuracy of $O(\tau^{2-\alpha})$ in time, with graded meshes mitigating initial singularities (Krzyżanowski et al., 2021, Song et al., 2021, V et al., 9 Aug 2025, Dimitrov et al., 2016).
• High-order spatial discretization: Fourth-order compact finite-difference (Song et al., 2021, Dimitrov et al., 2016, Zhou et al., 2023), exponential B-spline collocation (Singh et al., 2022, Garg et al., 30 Jan 2026), and modified cubic B-spline differential quadrature (V et al., 9 Aug 2025) methods achieve $O(h^2)$ to $O(h^4)$ spatial accuracy.
• Fast convolution via sum-of-exponentials (SOE): To overcome $O(N^2)$ time-complexity in history terms, SOE approximations enable $O(N)$ memory and per-step CPU with negligible loss of accuracy (Zhou et al., 2023, Song et al., 2021).
• Crank-Nicolson and weighted schemes: Generalization to $\theta$-methods yields stability and second-order accuracy at $\theta = \frac{2-2^{1-\alpha}}{3-2^{1-\alpha}}$ (Krzyżanowski et al., 2021, Krzyżanowski et al., 2019, Krzyżanowski et al., 2020).
• Meshless radial basis function collocation: Flexible RBF collocation schemes for multi-dimensional and space-fractional Black-Scholes combine analytic differentiation and preconditioning for stable, high-accuracy solutions in unstructured geometries (Torres-Hernandez et al., 2020).
• ANN-based solvers: Two-layer feedforward neural networks trained with Adam optimize over residuals of the discretized fractional PDE and enforce terminal and boundary conditions via loss penalization. Domain mapping addresses unbounded spatial domains and fine-tuning greatly accelerates convergence (Bajalan et al., 2021).
• Time-stepping on non-uniform meshes: Quadratic, Tavella-Randall, and other graded mesh techniques achieve higher order accuracy for weakly singular solutions (Dimitrov et al., 2016, Song et al., 2021).

5. Extensions: Transaction Costs, Tempered and Space-Fractional Models, Multi-Asset Options

The time-fractional paradigm admits extensive generalizations:

• Transaction costs: Incorporating proportional transaction fees into the self-financing, discrete-time hedge leads to a volatility inflation characterized by a "fractional Leland number" and recovers no-arbitrage pricing in settings where fractional Brownian motion would otherwise admit arbitrage (Shokrollahi, 2017).
• Tempered Caputo derivatives: Tempering (kernel regularization) ensures finite moments on the subordinator and avoids pathological price extremes. The resulting PDEs exhibit both memory and exponential decay, and have efficient compact-difference plus SOE-based numerical solvers (Zhou et al., 2023, Krzyżanowski et al., 2021).
• Space-fractional or nonlocal operators: Models driven by general Lévy processes and fractional spatial derivatives (fractional Laplacians, Riesz derivatives) arise in the context of jump diffusion and anomalous spatial diffusion (Torres-Hernandez et al., 2020).
• Multi-asset/multidimensional models: Fractional Black-Scholes equations with cross-derivative and correlation structure, as in two-asset options, admit fully analytic infinite-series solutions via the Samudu transform (Zakaria et al., 2020).
• American and barrier options: Linear complementarity and in-out parity in the finite-difference framework, together with memory-aware path generation for Monte Carlo (Longstaff-Schwartz) (Krzyżanowski et al., 2020), are directly compatible with fractional diffusion.

6. Empirical Validation, Calibration, and Practical Impact

Empirical studies provide evidence for the efficacy of time-fractional Black–Scholes models:

• Improved market fit: Fractional models yield option prices closer to observed quotes than classical Black–Scholes, especially in markets displaying volatility clustering, long-memory, and periods of illiquidity (Shokrollahi, 2017, Krzyżanowski et al., 2019).
• Estimation of model parameters: The Hurst exponent $H$ (in fBM models) is extracted via R/S analysis or Detrended Fluctuation Analysis; Caputo order $\alpha$ is commonly estimated via maximum-likelihood or implied volatility surface calibration (Shokrollahi, 2017).
• Impact on sensitivity and risk metrics: Option premiums, price sensitivities (Greeks), and hedging strategies exhibit marked dependence on the fractional parameters. Subdiffusive models generally increase option prices in the deep out-of-the-money region by effectively “stretching” time to maturity. In contrast, tempering mitigates this effect (Krzyżanowski et al., 2021, Zhang et al., 13 Nov 2025).
• Algorithmic feasibility: High-order finite-difference, spline, and ANN-based schemes yield computationally tractable, unconditionally stable, and convergent solvers even in the presence of full temporal nonlocality and complex boundary data (Song et al., 2021, Zhou et al., 2023, Garg et al., 30 Jan 2026, Bajalan et al., 2021).

7. Theoretical and Practical Considerations

Time-fractional Black–Scholes models raise several important theoretical and practical issues:

• Arbitrage and completeness: While continuous-time fractional Brownian motion models lack the semimartingale property and may admit arbitrage, discrete-time hedging with transaction costs restores “no-arbitrage-in-practice” pricing (Shokrollahi, 2017).
• Interpretability and model selection: The choice among subordination, fBM, pure Caputo, or tempered formulations depends on both empirical stylized facts and computational convenience.
• Boundary and initial condition specification: The nonlocal in time nature of Caputo derivatives preserves standard payoff (Dirichlet) initial conditions but requires careful attention to memory in numerical schemes, including for American and path-dependent options.
• Ill-posedness and stability: Algorithms must address reduced regularity (initial singularities), large condition numbers in high-dimensional RBF or spline systems (Torres-Hernandez et al., 2020, V et al., 9 Aug 2025), and the error amplification introduced by fractional time kernels.
• Practical recommendation: Optimal mesh selection, $\theta$-weighting in time, and the use of sum-of-exponential convolutions are essential for balancing computational load and achieving target accuracy in option pricing under time-fractional dynamics (Zhou et al., 2023, Krzyżanowski et al., 2019).

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In summary, the time-fractional Black-Scholes model is a mathematically rich and empirically justified extension of classical financial option pricing theory, integrating anomalous diffusion and memory within a rigorous PDE framework. Extensive recent research on arXiv has established the theoretical underpinnings, developed high-accuracy numerical algorithms, explored generalizations (e.g., transaction costs, multi-asset models), and validated the approach against real-world market data. The confluence of stochastic process theory, fractional calculus, and computational mathematics renders the time-fractional Black-Scholes framework a central object of current research in quantitative finance (Shokrollahi, 2017, Zhang et al., 13 Nov 2025, Song et al., 2021, Zhou et al., 2023, Krzyżanowski et al., 2019, Dimitrov et al., 2016, V et al., 9 Aug 2025, Torres-Hernandez et al., 2020, Singh et al., 2022, Krzyżanowski et al., 2021, Krzyżanowski et al., 2020, Garg et al., 30 Jan 2026, Bajalan et al., 2021, Zakaria et al., 2020, Kolokoltsov, 2011, Stanislavsky, 2011).