Abstract
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.
Sampling-based algorithm for filtering using Markov chain approximations