发表机构
School of Mathematics, Georgia Institute of Technology（佐治亚理工学院数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出NNLCI神经网络，通过局部校正粗细网格的期权定价解，以少量高保真数据训练，将细网格解的RMSE降低4-12倍，可高效用于高维期权定价，降低计算需求。
AI 中文摘要
我们提出了具有局部收敛输入的神经网络（NNLCI）的新应用，以提高现有多资产期权定价数值方法的效率。已引入NNLCI最简洁的输入格式，提供了极大的便利性和效率。NNLCI使用神经网络对粗网格和细网格（相对于粗网格）的解进行局部校正，仅需要最少的高保真训练数据。我们在一维、二维和三维空间维度下，针对Black-Scholes方程下的现金或无期权，以及Heston随机波动率模型下的单资产向下敲入障碍看涨期权（其定价偏微分方程在现货价格S和瞬时方差v上为二维），对该方法进行了验证。在每种情况下，即使神经网络仅在少量参数组合子集上训练，NNLCI也能将细网格数值解的均方根误差（RMSE）在测试集上降低约4至12倍。这些结果表明，NNLCI显著降低了高维问题在实时期权交易和风险管理中的计算需求，具有低训练成本和强泛化能力。
英文摘要
We present a novel application of Neural Networks with Local Converging Inputs (NNLCI) to improve the efficiency of existing numerical methods for pricing multi-asset options. The most concise input format for NNLCI has been introduced, offering substantial convenience and efficiency. NNLCI uses a neural network to locally correct solutions from a coarse mesh and a refined mesh (relative to the coarse one), requiring only a minimal amount of high-fidelity training data. We demonstrate this approach on cash-or-nothing options under the Black-Scholes equation in one, two, and three spatial dimensions, and on single-asset down-and-out barrier call options under the Heston stochastic-volatility model (whose pricing PDE is two-dimensional in the spot price $S$ and the instantaneous variance $v$). In each case, NNLCI reduces the root-mean-square error (RMSE) of the refined-mesh numerical solution by a factor of approximately 4-12 on test sets, even when the neural network is trained on only a small subset of parameter combinations. These results demonstrate that NNLCI significantly reduces computational requirements for high-dimensional problems in real-time options trading and risk management, offering low training costs and strong generalization ability.
Comments15 pages
