arXiv 1706.045662017-06q-fin.CPmath.PR

Heston随机微分方程的已实现波动率及其参数估计

Realized volatility and parametric estimation of Heston SDEs

Robert Azencott, Peng Ren, Ilya Timofeyev

摘要：

我们提出了一个详细的分析\emph{observable}矩为基础的参数估计的赫斯顿SDES共同驱动的回报率$R_t$和平方波动率$V_t$。由于波动率是不可直接观测的，我们的参数估计是从经验矩的已实现波动率$Y_t$，这当然是可观测的。在$\varepsilon$大小的滑动窗口上计算已已实现波动率，并将其划分为$J(\varepsilon)$区间。我们建立标准的联合选择的$J(\varepsilon)$和子采样频率的回报率数据。我们得到了$L^q$已实现波动率率收敛到真实波动率的速度的显式界为$\varepsilon \to 0$。反过来，这些界限也提供了$L^q$的速度收敛的赫斯顿波动率的参数，我们的可观察估计。我们的理论分析是由广泛的联合赫斯顿随机微分方程的数值模拟，以调查我们的矩为基础的参数估计的实际表现。我们的研究结果提供了实际的指导方针，充分拟合赫斯顿SDES参数观察到的股票价格序列。

英文摘要：

We present a detailed analysis of \emph{observable} moments based parameter estimators for the Heston SDEs jointly driving the rate of returns $R_t$ and the squared volatilities $V_t$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realized volatilities $Y_t$, which are of course observable. Realized volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon)$ intervals. We establish criteria for the joint selection of $J(\varepsilon)$ and of the sub-sampling frequency of return rates data. We obtain explicit bounds for the $L^q$ speed of convergence of realized volatilities to true volatilities as $\varepsilon \to 0$. In turn, these bounds provide also $L^q$ speeds of convergence of our observable estimators for the parameters of the Heston volatility SDE. Our theoretical analysis is supplemented by extensive numerical simulations of joint Heston SDEs to investigate the actual performances of our moments based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDEs parameters to observed stock prices series.

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