论文标题
高斯随机波动率模型的特征功能：一个分析表达
The characteristic function of Gaussian stochastic volatility models: an analytic expression
论文作者
论文摘要
基于高斯流程的随机波动率模型，例如分数布朗运动，能够重现金融市场的重要风格化事实，例如丰富的自相关结构，持久性和样本路径的粗糙度。由于在高斯过程的协方差函数中所引入的灵活性，这是可能的。要付费的代价是，通常，这样的模型不再是马尔可夫人，也不再是半明星，这限制了其实际使用。我们以两种不同的方式得出了对数值的联合特征函数及其在一般高斯随机波动率模型中的综合方差的明确分析表达。这种分析表达可以通过封闭形式矩阵表达式近似。这为使用傅立叶反转技术快速近似的关节密度和定价的衍生物的定价打开了大门。在粗糙的波动性建模的背景下，我们的结果适用于（粗糙的）分数Stein-Stein模型，并为迄今已知的选项定价提供了第一个分析公式，从而推广了Stein-Stein-Stein，Sch {Ö} Bel-Zhu和Heston的特殊情况。
Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylized facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that, in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. Such analytic expression can be approximated by closed form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realized variance using Fourier inversion techniques. In the context of rough volatility modeling, our results apply to the (rough) fractional Stein--Stein model and provide the first analytic formulae for option pricing known to date, generalizing that of Stein--Stein, Sch{ö}bel-Zhu and a special case of Heston.
