High-order accurate variable time step compact schemes for pricing vanilla and exotic options

In this paper, a fourth order accurate variable time step compact scheme is developed for pricing vanilla and exotic options. Variable time step midpoint discretization is considered for the temporal variable, and compact scheme is utilized for the spatial variable. The time-regularity estimates along with the existence and uniqueness results are obtained for the proposed temporal semi-discretization. The semi-discrete scheme is proved to be stable using the root condition for linear multi-step method. The stability condition is derived for the fully discrete problem using a difference equation based approach. The numerical illustrations are presented to validate the derived stability condition. Additionally, the log-log plot confirms that the proposed scheme is fourth order accurate in space variable and second order accurate in time variable.