波动率曲面与方差期限结构的联合校准
Joint calibration of the volatility surface and variance term structure
AI总结：
本文提出一种联合校准框架，通过增加惩罚项同时拟合期权价格与方差期限结构，并在跳跃扩散模型及S&P 500期权数据上验证其有效性，解决了传统校准中方差期限结构误差大的问题。
摘要：
本文提出了一种针对复杂期权定价模型的校准框架，该框架能够同时拟合市场期权价格与方差期限结构。在传统目标函数（即Black-Scholes隐含波动率的平方误差之和）下校准的模型，可能产生与市场观察到的及期权价格隐含的方差期限结构存在较大误差的模型隐含方差期限结构。我证明，即使模型隐含的波动率曲面与市场观察到的波动率曲面高度吻合，这种情况也可能发生。所提出的联合校准通过将传统目标函数增加一个惩罚项来解决此问题，该惩罚项针对与观察到的方差期限结构的较大偏差。这一增强的目标函数具有一个超参数，用于控制波动率曲面与方差期限结构之间的相对权重。我在两个校准实验中，将该框架应用于一个具有随机波动率的跳跃扩散模型：第一个实验使用在Bates模型下生成的波动率曲面，第二个实验使用覆盖1996-2023年期间的S&P 500股指期权面板数据。我证明了所提出的方法能够很好地拟合观察到的期权价格，同时提供现实的方差期限结构。最后，我基于这些数值实验的结果，提供了关于超参数选择的指导。
英文摘要：
This article proposes a calibration framework for complex option pricing models that jointly fits market option prices and the term structure of variance. Calibrated models under the conventional objective function, the sum of squared errors in Black-Scholes implied volatilities, can produce model-implied variance term structures with large errors relative to those observed in the market and implied by option prices. I show that this can occur even when the model-implied volatility surface closely matches the volatility surface observed in the market. The proposed joint calibration addresses this issue by augmenting the conventional objective function with a penalty term for large deviations from the observed variance term structure. This augmented objective function features a hyperparameter that governs the relative weight placed on the volatility surface and the variance term structure. I test this framework on a jump-diffusion model with stochastic volatility in two calibration exercises: the first using volatility surfaces generated under a Bates model, and the second using a panel of S&P 500 equity index options covering the 1996-2023 period. I demonstrate that the proposed method is able to fit observed option prices well while delivering realistic term structures of variance. Finally, I provide guidance on the choice of hyperparameters based on the results of these numerical exercises.
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