In light of recent empirical research on jump activity, this article study the calibration of a new class of stochastic volatility models that include both jumps in return and volatility. Specifically, we consider correlated jump sizes and both contemporaneous and independent arrival of jumps in return and volatility. Based on the specifications of this model, we derive a closed-form relationship between the VIX index and latent volatility. Also, we propose a closed-form logarithmic likelihood formula by using the link to the VIX index. By estimating alternative models, we find that the general counting processes setting lead to better capturing of return jump behaviors. That is, the part where the return and volatility jump simultaneously and the part that jump independently can both be captured. In addition, the size of the jumps in volatility is, on average, positive for both contemporaneous and independent arrivals. However, contemporaneous jumps in the return are negative, but independent return jumps are positive. The sub-period analysis further supports above insight, and we find that the jumps in return and volatility increased significantly during the two recent economic crises.

中文翻译：

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具有相关跳跃大小和独立到达的随机波动率模型

根据最近对跳跃活动的实证研究，本文研究了一类新的随机波动率模型的校准，该模型包括收益跳跃和波动率。具体来说，我们考虑相关的跳跃大小以及回报和波动性跳跃的同时和独立到达。基于该模型的规范，我们推导出了 VIX 指数与潜在波动率之间的封闭式关系。此外，我们通过使用 VIX 指数的链接提出了一个封闭形式的对数似然公式。通过估计替代模型，我们发现一般计数过程设置可以更好地捕捉返回跳跃行为。即收益和波动率同时跳跃的部分和独立跳跃的部分都可以被捕获。此外，平均而言，同时期和独立抵达者的波动性跳跃幅度为正。然而，回报中的同期跳跃是负的，但独立的回报跳跃是正的。子期分析进一步支持了上述观点，我们发现在最近两次经济危机期间，收益和波动性的跳跃显着增加。