Optimal convergence rate of the explicit finite difference scheme for American option valuation

An optimal convergence rate O(Δx)O(Δx) for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition σ2ΔtΔx2⩽1 using completely PDE methods. As a corollary, a binomial tree scheme of an American put option (where σ2ΔtΔx2=1) is convergent unconditionally with the rate O((Δt)1/2)O((Δt)1/2).