用于隐含波动率曲面的生成模型的隐空间无套利几何

中文摘要 AI 辅助

用于隐含波动率曲面的生成模型必须产生满足静态无套利约束的输出，我们在隐空间中研究这些约束。对于固定的生成器，我们根据生成曲面的无套利条件为每个隐代码分配一个标量余量，余量非负的代码构成可接受隐集。我们建立了严格可接受代码在小扰动下仍保持可接受的条件，且可接受集的边界以零余量为特征。对于正则边界分量，我们构建了一个水平集方程，其局部动力学指向零余量集。该分析将生成器视为从隐变量到曲面的映射，因此不限于特定架构，适用于变分自编码器（VAE）、生成对抗网络（GAN）及其他具有确定性实现映射的生成模型。数值测试在解析示例中恢复了已知边界；对在Heston曲面上训练的变分自编码器的实验表明，相似的重构误差可对应不同的可接受区域，且隐先验可能集中在该区域内，计算得到的边界还可用于修改生成违反约束曲面的隐代码。

英文摘要

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

发表机构

• Delft University of Technology（代尔夫特理工大学）
• Delft Institute of Applied Mathematics（代尔夫特应用数学研究所）

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