A Crank-Nicolson finite difference approach on the numerical estimation of rebate barrier option prices

In modelling financial derivatives, the pricing of barrier options are complicated as a result of their path-dependency and discontinuous payoffs. In the case of rebate knock-out barrier options, discount factors known as rebates are introduced, which are payable to the option holder when the barrier level is breached. The analytical closed-form solution for the vanilla options are known but the barrier options, owing to their discontinuous nature, can be obtained analytically using the extended Black-Scholes formula. This research work captures the solution of the corresponding option pricing partial differential equation on a discrete space-time grid. We employ the Crank-Nicolson finite difference scheme to estimate the prices of rebate barrier options, as well as to discuss the effect of rebate on barrier option values. This work will further investigate the spurious oscillations which arise from the sensitivity analysis of the Greeks of the barrier options using the Crank-Nicolson scheme. The theoretical convergence of the Crank-Nicolson discretisation scheme will be analysed. Furthermore, our research will compare the results from the extended Black-Scholes model based on continuous time monitoring, together with the finite difference results from the Crank-Nicolson method.