Pricing stock loans under the L $\acute{e}$vy- $\alpha$-stable process with jumps

In this paper, the pricing of stock loans when the underlying follows a L $\acute{e}$vy- $\alpha$-stable process with jumps is considered. Under this complicated model, the stock loan value satisfies a fractional-partial-integro-differential equation (FPIDE) with a free boundary. The difficulties in solving the resulting FPIDE system are caused by the non-localness of the fractional-integro differential operator, together with the nonlinearity resulting from the early exercise opportunity of stock loans. Despite so many difficulties, we have managed to propose a preconditioned conjugate gradient normal residual (PCGNR) method to price efficiently the stock loan under such a complicated model. In the proposed approach, the moving pricing domain is successfully dealt with by introducing a penalty term, however, the semi-globalness of the fractional-integro operator is elegantly handled by the PCGNR method together with the fast Fourier transform (FFT) technique. Remarkably, we show both theoretically and numerically that the solution determined from the fixed domain problem by the current method is always above the intrinsic value of the corresponding option. Numerical experiments suggest the accuracy and advantage of the current approach over other methods that can be compared. Based on the numerical results, a quantitative discussion on the impacts of key parameters is also provided.