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

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