基于PCRLB开关滤波器的Heston模型自适应校正
Adaptive calibration of Heston Model using PCRLB based switching Filter
摘要：
自1987年股市崩盘以来，随机波动率模型一直存在于期权定价理论中，它违背了布莱克-斯科尔斯模型关于波动率恒定的假设。Heston模型就是这样一种随机波动率模型，被广泛应用于波动率估计和期权定价。本文结合文献[1]中提出的正态最大似然估计（NMLE），利用贝叶斯滤波理论和后验Cramer-Rao下界（PCRLB），设计了一种新的状态空间表示下Heston模型参数估计方法。几个贝叶斯滤波器，如扩展卡尔曼滤波器（EKF），无迹卡尔曼滤波器（UKF），粒子滤波器（PF）用于潜在的状态和参数估计。本文采用文献[2]中提出的一种切换策略，对非线性离散时间状态空间模型（SSM）（如Heston模型）进行自适应状态估计。我们使用粒子滤波器近似PCRLB [3]的性能指标来判断每个时间步的最佳滤波器。我们测试我们提出的框架，从标准普尔500指数和NSE指数的定价数据，估计潜在的波动率和参数的指数。我们提出的方法进行了比较，与波动率指数和历史波动率的指数。结果表明，一个有效的框架，估计波动自适应不断变化的市场动态。
英文摘要：
Stochastic volatility models have existed in Option pricing theory ever since the crash of 1987 which violated the Black-Scholes model assumption of constant volatility. Heston model is one such stochastic volatility model that is widely used for volatility estimation and option pricing. In this paper, we design a novel method to estimate parameters of Heston model under state-space representation using Bayesian filtering theory and Posterior Cramer-Rao Lower Bound (PCRLB), integrating it with Normal Maximum Likelihood Estimation (NMLE) proposed in [1]. Several Bayesian filters like Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Particle Filter (PF) are used for latent state and parameter estimation. We employ a switching strategy proposed in [2] for adaptive state estimation of the non-linear, discrete-time state-space model (SSM) like Heston model. We use a particle filter approximated PCRLB [3] based performance measure to judge the best filter at each time step. We test our proposed framework on pricing data from S&P 500 and NSE Index, estimating the underlying volatility and parameters from the index. Our proposed method is compared with the VIX measure and historical volatility for both the indexes. The results indicate an effective framework for estimating volatility adaptively with changing market dynamics.
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