基于维纳混沌的martingale建模与隐含波动率校准方法
A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration
AI总结：
本文提出了一种基于维纳混沌的martingale建模方法，用于高效校准隐含波动率表面。
摘要：
对期权价格表面的校准需要指定一个足够灵活的martingale模型，用于在风险中性测度下折扣资产价格。假设布朗噪声和均方可积性，我们基于martingale表示定理构建了一个过参数化的模型。特别是，我们通过截断维纳-混沌展开近似martingale的终端值，并通过计算相应的条件期望来恢复中间动态。利用维纳混沌的赫尔米特多项式形式，我们得到易于实现的表达式，从而能够快速校准到目标隐含波动率表面。我们通过在模拟和真实市场数据上的数值实验，展示了所得到模型的灵活性和表现力。
英文摘要：
Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.
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