渐近信息神经网络用于Black-Scholes隐含波动率计算
Asymptotically-informed neural networks for Black-Scholes implied volatility computation
- The University of Liverpool（利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出渐近信息神经网络架构，通过门控函数划分价格-对数货币性域并组合局部近似，高效计算Black-Scholes隐含波动率，在广泛参数域上精度远超标准前馈网络，并为Householder迭代提供高精度初值。
AI中文摘要：
Black-Scholes隐含波动率的计算是量化金融中的一项基本任务，支撑着期权估值、模型校准和风险管理。尽管隐含波动率在实践中被常规使用，但Black-Scholes定价公式的反演仍然是一个具有挑战性的数值问题，尤其是在对应于极端期权价格、执行价格或到期日的渐近区域中，此时逆映射对价格扰动变得高度敏感。在本文中，我们引入了一族新的渐近信息神经网络架构用于隐含波动率计算。利用Black-Scholes定价函数在不同波动率区域中的不同行为，我们提出了一族架构，通过门控函数系统学习价格-对数货币性域的可训练划分，并在每个区域内结合隐含波动率函数的专门局部近似。大量数值实验表明，所提出的模型在广泛的参数域上始终优于标准前馈神经网络，通常在相对精度上高出几个数量级，同时保持良好的泛化性能。此外，神经网络输出为三阶Householder格式提供了高精度的初始猜测，使得仅经过两次细化迭代即可实现接近机器精度的隐含波动率计算。
英文摘要：
The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.
