We present a detailed analysis of observable moment-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 realised volatilities \((Y_{t})\), which are of course observable. Realised 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 subsampling frequency of return rates data.We obtain explicit bounds for the \(L^{q}\) speed of convergence of realised 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 moment-based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDE parameters to observed stock price series.

中文翻译：

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Heston SDE的已实现波动性和参数估计

我们对Heston SDE的可观察的基于矩的参数估计量进行了详细分析，这些参数估计量共同驱动了回报率\（（R_ {t}）\）和波动率平方（{V_ {t}）\）。由于不能直接观察到波动率，因此我们的参数估计器是根据已实现的波动率\（（Y_ {t}）\）的经验矩构造的，这些波动当然是可以观察到的。在大小为\（\ varepsilon \）的滑动窗口上计算已实现的波动率，并划分为\（J（\ varepsilon）\）区间。我们为\（J（\ varepsilon）\）和返回率数据的二次采样频率的联合选择建立标准。\（L ^ {q} \）的已实现挥发度到真实挥发度的收敛速度为\（\ varepsilon \ to 0 \）。反过来，这些界限也为我们对Heston波动SDE的参数的可观测估计量的收敛速度提供了（（L ^ {q} \））。基于矩的参数估计器 我们的结果为将Heston SDE参数充分拟合到观察到的股票价格序列提供了实用指导。