  This paper presents the Runge-Kutta-Legendre finite difference scheme,
allowing for an additional shift in its polynomial representation. A short
presentation of the stability region, comparatively to the
Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing
American options with the Runge-Kutta-Legendre scheme under the one factor
Black-Scholes and the two factor Heston stochastic volatility models, as well
as the pricing of butterfly spread and digital options under the uncertain
volatility model, where a Hamilton-Jacobi-Bellman partial differential equation
needs to be solved. We explore the order of convergence in these problems, as
well as the option greeks stability, compared to the literature and popular
schemes such as Crank-Nicolson, with Rannacher time-stepping.
