Variable time step operator splitting methods with stability and error estimates for pricing American options

In this work, three variable time step operator splitting (OS) methods, namely, BDF-OS, Crank–Nicolson (CN)-OS and midpoint (MP)-OS, are developed to solve the linear complementarity problem (LCP) in American option pricing. The stability and error estimates of the proposed methods are derived. The proposed methods are unconditionally stable for (σ2+d−3r)⩽0, where σ is the volatility, d is the dividend and r is the interest rate. Further, these are conditionally stable for (σ2+d−3r)>0, and conditions on time step size (kn) are derived in terms of σ, d and r. The bounds on the errors at each time step are also produced. The spatial differential operators are discretized with finite difference approximations, and algorithms are provided to solve the fully discrete problems. The numerical illustrations are supplemented to demonstrate the efficiency, stability and convergence of the proposed methods.