在机制转换测度变换下 rBergomi 模型中的 VIX 定价
VIX pricing in the rBergomi model under a regime switching change of measure
AI总结：
针对 rBergomi 模型确定性测度变换导致 VIX 微笑平坦的问题，本文提出机制转换随机测度变换，并用半解析 Monte Carlo、VIX 近似与方差缩减实现高效定价和市场校准。
摘要：
在物理测度下，rBergomi 模型将对数方差建模为截断 Brownian 半平稳过程。随后，应用一个确定性的测度变换。rBergomi 模型能够用很少的参数再现观察到的 SP500 市场波动率微笑，但由于采用确定性测度变换，它会产生平坦的 VIX 微笑，这与市场中观察到的向上倾斜的微笑形成对比。我们利用某一非齐次分数阶 Ornstein–Uhlenbeck 方程的精确解，为 rBergomi 模型构建了一种机制转换的随机测度变换；该变换既能产生向上倾斜的 VIX 微笑，又配备了高效的半解析 Monte Carlo 方法来为 VIX 期权定价。该模型还允许对 VIX 进行近似，从而显著降低 VIX 期权和期货定价的计算成本。一种基于底层连续时间 Markov 链的方差缩减技术使我们能够进一步降低计算成本。我们通过将模型校准到观察到的市场微笑来验证其能力，并讨论了结果。
英文摘要：
The rBergomi model under the physical measure consists of modeling the log-variance as a truncated Brownian semi-stationary process. Then, a deterministic change of measure is applied. The rBergomi model is able to reproduce observed market SP500 smiles with few parameters, but by virtue of the deterministic change of measure, produces flat VIX smiles, in contrast to the upward sloping smiles observed in the market. We use the exact solution for a certain inhomogeneous fractional Ornstein-Uhlenbeck equation to build a regime switching stochastic change of measure for the rBergomi model that both yields upward slopping VIX smiles and is equipped with an efficient semi-analytic Monte Carlo method to price VIX options. The model also allows an approximation of the VIX, which leads to a significant reduction of the computational cost of pricing VIX options and futures. A variance reduction technique based on the underlying continuous time Markov chain allows us to further reduce the computational cost. We verify the capabilities of our model by calibrating it to observed market smiles and discuss the results.
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