We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

中文翻译：

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时间分数布莱克-舒尔斯模型下的光谱准确期权定价

我们提出了勒让德-拉盖尔谱近似来为时间分数框架中的欧式和双壁垒期权定价。通过选择适当的基函数，谱离散化用于逼近时间分数 Black-Scholes 方程的空间导数。对于时间离散化，我们考虑流行的$L1$有限差分逼近，随阶收敛$\mathcal {O}((\Delta \tau )^{2-\alpha })$对于两次连续可微的函数。然而，当使用$L1$对于具有非光滑初始数据的问题的方案，只能在时间上达到一阶精度。这种低阶精度在解决时间分数 Black-Scholes European 和障碍期权定价问题时也可以观察到，这些问题的收益都是非平滑的。因此，为了提高时间收敛速度，我们考虑了 Richardson 外推法，该方法与空间中的谱近似相结合时，表现出更高阶的收敛性，从而在整个离散化网格上获得了高精度。与传统的有限差分方案相比，数值例子清楚地表明谱近似在少量网格点上呈指数收敛。此外，正如所证明的，使用外推方法可以在更少的时间步长内实现如此高的精度。