Time-Fractional Black-Scholes PDE

• Time-Fractional Black-Scholes PDE is a fractional extension of the classical model that incorporates memory effects and subdiffusive dynamics via Caputo derivatives.
• Tempered variants add an exponential damping factor to the kernel, regularizing heavy-tail behavior and ensuring finite moments for asset price dynamics.
• Advanced numerical schemes, including high-order discretizations and neural network solvers, enable efficient and accurate option pricing under complex market conditions.

The time-fractional Black-Scholes partial differential equation (PDE) generalizes the classical Black-Scholes model to incorporate anomalous diffusion, memory effects, and subdiffusive asset dynamics via nonlocal-in-time, typically Caputo-type, fractional derivatives. The extension is motivated by empirical evidence of “trapping” in illiquid markets and heavy-tailed waiting times between asset price moves. Recent research has also developed robust algorithms and rigorous analyses for tempered fractional models—where the memory kernel decays exponentially at long times—leading to PDEs with tempered Caputo derivatives. These models, when paired with high-order spatial discretizations and modern time-stepping schemes, enable efficient and accurate option pricing under complex market dynamics.

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

The canonical form of the time-fractional Black-Scholes equation for a European option price $V(S,t)$, with $S$ the asset price and $t$ time, replaces the classical first-order time derivative by a Caputo derivative of order $\alpha\in(0,1)$: $$\frac{\partial^\alpha V(S,t)}{\partial t^\alpha} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0,$$ where the Caputo derivative is defined as

$$\frac{\partial^\alpha V}{\partial t^\alpha} = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha}\, \frac{\partial V}{\partial s}(S,s)\, ds.$$

Initial/terminal conditions are set by the option payoff, e.g., $V(S,0)=\max(S-K,0)$ for a European call. Boundary conditions are determined by financial asymptotics, e.g., $V(0,t)=0$ and $V(S\to\infty,t)\sim S-Ke^{-r(T-t)}$ (Krzyżanowski et al., 2019, Singh et al., 2022, Krzyżanowski et al., 2021).

The tempered time-fractional Black-Scholes PDE introduces a tempering parameter $\lambda > 0$ into the kernel,

$${}^C_0D_t^{\alpha,\lambda}V(S,t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha}e^{-\lambda(t-s)}\,\frac{\partial V}{\partial s}(S,s)\, ds - \lambda^\alpha V(S,t),$$

resulting in a model that regularizes the heavy tails of the pure stable kernel and restores finiteness of all moments in the associated subordinator (Krzyżanowski et al., 2021, Zhou et al., 2023).

2. Stochastic Subdiffusive Models and Risk-Neutral Pricing

Subdiffusive generalizations originate from modeling the asset as a time-changed geometric Brownian motion,

$$Z_{\alpha,\lambda}(t) = Z(S_{\alpha,\lambda}(t)),\quad Z(s) = Z_0 \exp(\mu s + \sigma B(s)),$$

where $S_{\alpha,\lambda}(t)$ is the inverse of a tempered $\alpha$-stable subordinator $W_{\alpha,\lambda}(\tau)$. Under the risk-neutral measure, option prices involve integrating the classical price kernel against the density of the random clock $S_{\alpha,\lambda}(t)$. Through Laplace transform, the resulting price function $w(z,t)$ satisfies a tempered fractional PDE with the same spatial structure as classical Black-Scholes but with the tempered Caputo derivative in time (Krzyżanowski et al., 2021).

A similar subordination and risk-neutral argument holds for the pure (non-tempered) stable case, with the Caputo derivative of order $\alpha < 1$ governing the price PDE. Explicit mixture formulas and eigenfunction expansions are available in both cases, providing convergence to the classical Black-Scholes formula as $\alpha \to 1$, $\lambda \to 0$ (Zhang et al., 13 Nov 2025, Kolokoltsov, 2011).

3. Analytical Structure and Effect of Fractional Order and Tempering

Key features distinguishing the time-fractional Black-Scholes PDE from the classical model include:

• Nonlocality in time: The option price evolution depends on the entire past, encoded by the fractional memory kernel. This leads to Mittag–Leffler functions replacing exponentials in the solution semigroup.
• Subdiffusive effect: Parameter $\alpha<1$ produces slower-than-exponential relaxation, modeling financial markets with “trapping” or periods of inactivity. Option prices are depressed relative to the classical model, with stronger memory as $\alpha$ decreases (Krzyżanowski et al., 2019, Kolokoltsov, 2011).
• Tempering: The parameter $\lambda>0$ in the tempered Caputo derivative exponentially suppresses far-tail memory, ensuring that the market clock possesses moments of all orders (unlike the pure stable case), and modulates the effect of “aging” in the dynamics (Krzyżanowski et al., 2021, Zhou et al., 2023).
• Classical limit: As $\alpha \to 1$ and $\lambda \to 0$, the model reduces smoothly to the classical Black-Scholes PDE (Krzyżanowski et al., 2021).

4. Numerical Discretization: High-Order and Fast Schemes

Several classes of robust, high-order schemes have been established for both pure and tempered time-fractional Black-Scholes PDEs:

• Finite Difference Weighted Schemes: The weighted $\theta$-scheme generalizes the Crank-Nicolson method to fractional order, with the Caputo derivative discretized via the L1 formula and central differences for spatial terms. Time accuracy is $O(\Delta t^{2-\alpha})$; spatial order is $O(\Delta x^2)$ (Krzyżanowski et al., 2019).
• High-Order Meshes and Compact Schemes: Spatial discretization on non-uniform, graded or Tavella–Randall meshes achieves fourth-order spatial accuracy, optimizing resolution near singularities and payoff kinks. Compatibility with the fractional temporal discretization yields global error $O(h^4 + \Delta t^{2-\alpha})$ (Dimitrov et al., 2016).
• Spline and DQM Methodologies: Modified cubic B-spline based differential quadrature methods and exponential B-spline collocation deliver unconditional stability, order $O(h^4)$ or $O(h^2)$ in space, and $O(\Delta t^{2-\alpha})$ in time (V et al., 9 Aug 2025, Singh et al., 2022).
• Tempered Fractional, Graded Time, and SOE Acceleration: Graded meshes in time address initial singularity, restoring leading-order convergence. The sum-of-exponentials (SOE) approximation compresses the memory kernel, reducing computational cost from $O(N^2)$ to $O(NP)$, where $P$ is the number of exponentials (Zhou et al., 2023).
• Meshless RBF Methods: For multidimensional fractional Black-Scholes PDEs, radial basis function collocation achieves high flexibility with QR-based preconditioning to control condition number, and is compatible with multidimensional spatial fractional operators (Torres-Hernandez et al., 2020).

Representative numerical results confirm that these schemes achieve their predicted rates, with unconditional stability under essentially all reasonable mesh and parameter regimes.

5. Semi-Analytical and Neural Approaches

• Eigenfunction/Mixture Expansions: Solutions can be represented by inverse Laplace transforms, eigenfunction expansions with Mittag–Leffler kernels, or convolutions against inverse subordinator densities (Zhang et al., 13 Nov 2025, Kolokoltsov, 2011). For the two-asset time-fractional Black-Scholes PDE, the Samudu transform yields an explicit convergent series in time (Zakaria et al., 2020).
• Neural Network Solvers: Two-layer feed-forward artificial neural networks, employing domain-mapped spatial input and Adam optimization, have been developed to solve both integer- and fractional-order Black-Scholes PDEs. The discretization reduces the PDE to a sequence of ODEs at each time-step, which are then approximated by the neural net. The approach is mesh-free in space, achieving errors $10^{-4}$–$10^{-6}$, with performance enhanced by weight fine-tuning across time levels (Bajalan et al., 2021).

6. Applications and Extensions

The time-fractional and tempered time-fractional Black-Scholes models are directly applicable to:

• Pricing of European, American, and Barrier Options: The finite-difference and meshless schemes extend to options with free boundaries (American) and absorbing boundary conditions (barrier), retaining stability and accuracy (Krzyżanowski et al., 2020, Singh et al., 2022).
• Modeling incomplete and subdiffusive markets: Fractional models encode deviations from the Markovian paradigm, capture incomplete-market risk, and can be extended to nonlinear settings arising in robust control/game-theoretic finance (Kolokoltsov, 2011).
• Higher-Dimensional and Multi-Asset Derivatives: The meshless RBF framework and series solutions based on the Samudu transform and spectral methods generalize naturally to high-dimensional option pricing problems (Torres-Hernandez et al., 2020, Zakaria et al., 2020).
• Market Calibration: The additional parameters $(\alpha,\lambda)$ permit greater flexibility in fitting empirical implied volatility surfaces and non-exponential decay of memory (Krzyżanowski et al., 2021).

7. Outlook and Future Directions

Active research continues in several areas:

• Extension to double-fractional (space-time) models and non-constant orders to capture further nonlocality and memory effects (Torres-Hernandez et al., 2020).
• Optimization of fast-time stepping with SOE and Alikhanov-type variable-step approaches for large-scale computations (Zhou et al., 2023, Song et al., 2021).
• Rigorous analysis for convergence in the presence of non-smooth payoff data and singularities near $t=0$ with graded meshes and adaptive time-stepping (Dimitrov et al., 2016, Song et al., 2021).
• Integration of robust control, game-theoretic, and stochastic subordination approaches for risk management and hedging in incomplete and anomalous markets (Kolokoltsov, 2011, Zhang et al., 13 Nov 2025).
• Numerical benchmarks and open-source implementations to further consolidate these methods for practical quantitative finance.

These advances collectively position the time-fractional and tempered time-fractional Black-Scholes PDEs as powerful tools for modeling, analysis, and computation in modern quantitative finance, particularly in regimes where memory, subdiffusion, and heavy-tailed phenomena are dominant (Krzyżanowski et al., 2021, Zhou et al., 2023, Zhang et al., 13 Nov 2025, Krzyżanowski et al., 2019, V et al., 9 Aug 2025, Dimitrov et al., 2016, Torres-Hernandez et al., 2020, Kolokoltsov, 2011).