Consistent time‐homogeneous modeling of SPX and VIX derivatives
Consistent time‐homogeneous modeling of SPX and VIX derivatives
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DOI：
10.1111/mafi.12348
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发表时间：
2018-12
影响因子：
1.6
通讯作者：
A. Papanicolaou
中科院分区：
文献类型：
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作者：
A. Papanicolaou
本文展示了如何从 VIX 期货期限结构的市场模型中恢复随机波动率模型 (SVM)。市场模型比 SVM 具有更大的曲线拟合灵活性，因此更适合对 VIX 期货和 VIX 衍生品进行定价。但 VIX 本身是 S&P500 (SPX) 的衍生品，使用 SVM 为 SPX 衍生品定价是常见的做法。因此，SPX 和 VIX 的一致建模应该涉及可以通过反转市场模型获得的 SVM。本文的主要成果是一种通过求解反问题来恢复随机波动率函数的方法，其中输入是市场模型给出的 VIX 函数。分析将显示出该反问题的唯一解所需的条件。如果恢复的波动率函数是非负的，则模型是一致的。给出的例子是为了说明理论，强调解决方案中的消极性问题，并展示非马尔可夫环境中不一致的可能性。
This paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Therefore, consistent modeling for both SPX and VIX should involve an SVM that can be obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function by solving an inverse problem where the input is the VIX function given by a market model. Analysis will show conditions necessary for there to be a unique solution to this inverse problem. The models are consistent if the recovered volatility function is non‐negative. Examples are presented to illustrate the theory, to highlight the issue of negativity in solutions, and to show the potential for inconsistency in non‐Markov settings.