In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

中文翻译：

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局部波动率函数期权定价的两点通量近似法和拟合两点通量近似法的收敛性


在本文中，我们采用数值近似来求解具有局部波动性的欧洲和美国期权定价的布莱克-斯科尔斯偏微分方程（PDE）。众所周知，该偏微分方程是退化的。局部波动率模型是波动率局部取决于股票价格和时间的模型。与欧式看涨期权的精确解的分析表示已知的恒定波动率或时间相关波动率模型相比，局部波动率没有分析解。空间离散化是使用经典的有限体积法和两点通量近似 (TPFA) 以及称为拟合两点通量近似 (FTPFA) 的新颖方案来执行的。拟合两点通量近似 (FTPFA) 结合了拟合有限体积法和标准 TPFA 方法。更准确地说，当股票价格接近零时，使用拟合有限体积方法，目的是处理 PDE 的简并性，而 TPFA 方法则用于其余空间域。这种组合产生了我们拟合的 TPFA 方案。时间离散化采用欧拉法。我们提供了两种完全离散方案的严格收敛证明。提供了支持理论结果的数值实验。