A compact finite difference scheme for fractional Black-Scholes option pricing model (Q2029151)

In this paper, the authors consider the following time-fractional Black-Scholes model \[ \begin{array}{l} \frac{\partial ^{\alpha} Y(S,\tau)}{\partial \tau ^{\alpha}}+0.5\sigma^2 S^2\frac{\partial ^2 Y(S,\tau)}{\partial S^2}+rS\frac{\partial Y(S,\tau)}{\partial S}-rY(S,\tau)=0,\\ [12pt] Y(0,\tau)=\tilde \phi (\tau),\: \: Y(\infty, \tau)=\tilde \omega(\tau)\\ \end{array} \] with terminal condition \(Y(S,T)=\psi (S)\), where \(Y\) denotes the price of a European option which depends on the asset price \(S\) and current time \(\tau\), \(\sigma\) is the volatility, \(r\) is the rick-free interest rate and \(\tau\) is the expiry time. The time fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied in Section 4. The effect of fractional order derivative on the solution profile corresponding to option price is also analyzed.