Parametric estimation of stochastic differential equations (SDEs) has been a subject of intense studies already for several decades. The Heston model for instance is driven by two coupled SDEs and is often used in financial mathematics for the dynamics of the asset prices and their volatility. Calibrating it to real data would be very useful in many practical scenarios. It is very challenging however, since the volatility is not directly observable. In this paper, a complete estimation procedure of the Heston model without and with jumps in the asset prices is presented. Bayesian regression combined with the particle filtering method is used as the estimation framework. Within the framework, we propose a novel approach to handle jumps in order to neutralise their negative impact on the estimates of the key parameters of the model. An improvement of the sampling in the particle filtering method is discussed as well. Our analysis is supported by numerical simulations of the Heston model to investigate the performance of the estimators. And a practical follow-along recipe is given to allow for finding adequate estimates from any given data.
翻译:几十年来,对随机差分方程(SDEs)的参数估计一直是一项密集研究的主题。例如,赫斯顿模型是由两个相联的SDEs驱动的,常常用于金融数学,以了解资产价格的动态及其波动性。将它校准为真实数据在许多实际情景中非常有用。然而,由于波动性不是直接可见的,因此它非常具有挑战性。本文介绍了赫斯顿模型的完整估计程序,不考虑资产价格的跳跃。贝耶斯回归与粒子过滤法结合作为估计框架。在这个框架内,我们提出了处理跳跃的新办法,以抵消其对模型关键参数估计的消极影响。讨论了改进粒子过滤方法的取样工作。我们的分析得到赫斯顿模型数字模拟的支持,以调查估量器的性能。我们提供了一种实用的后续配方,以便从任何给定的数据中找到适当的估计。