In this paper, we propose a robust and accurate numerical algorithm to reconstruct a local volatility function using the Black–Scholes (BS) partial differential equation (PDE). Using the BS PDE and given market data, option prices at strike prices and expiry times, a time-dependent local volatility function is computed. The proposed algorithm consists of the following steps: (1) The time-dependent volatility function is computed using a recently developed method; (2) A Monte Carlo simulation technique is used to find the effective region which has a strong influence on option prices; and we partition the effective domain into several sub-regions and define a local volatility function based on the time-dependent volatility function on the sub-regions; and (3) Finally, we calibrate the local volatility function using the fully implicit finite difference method and the conjugate gradient optimization algorithm. We demonstrate the robustness and accuracy of the proposed local volatility reconstruction algorithm using manufactured volatility surface and real market data.

在本文中，我们提出了一种稳健而准确的数值算法，以使用Black-Scholes（BS）偏微分方程（PDE）来重建局部波动率函数。使用BS PDE和给定的市场数据，行使价和到期时间的期权价格，可以计算时间相关的本地波动率函数。所提出的算法包括以下步骤：（1）使用最新开发的方法来计算时间相关的波动率函数；（2）采用蒙特卡罗模拟技术寻找对期权价格影响较大的有效区域。然后将有效域划分为几个子区域，并基于子区域上与时间相关的波动率函数定义局部波动率函数；（3）最后，我们使用完全隐式有限差分法和共轭梯度优化算法来校准局部波动率函数。我们使用制造的波动面和实际市场数据证明了所提出的局部波动率重建算法的鲁棒性和准确性。