This paper presents the Runge–Kutta–Legendre (RKL) 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 RKL 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.

中文翻译：

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使用 RUNGE-KUTTA-LEGENDRE 有限差分方案定价美式期权

本文介绍了 Runge-Kutta-Legendre (RKL) 有限差分方案，允许对其多项式表示进行额外的转换。与 Runge-Kutta-Chebyshev 方案相比，稳定区域的简短介绍如下。然后，我们探讨了单因子 Black-Scholes 和两因子 Heston 随机波动率模型下 RKL 方案的美式期权定价问题，以及不确定波动率模型下的蝶形价差和数字期权定价问题，其中 Hamilton- Jacobi-Bellman 偏微分方程需要求解。与文献和流行的方案（如 Crank-Nicolson，Rannacher 时间步长）相比，我们探讨了这些问题的收敛顺序，以及选项希腊稳定性。