周期波动性对金融市场稳定性的影响

Influence of periodic volatility on the stability of financial market

利用平均逃逸率和逃逸时间分别研究了周期性波动对股票价格稳定性在金融常态和金融危机下的影响.基于Heston模型、引入单稳势函数和周期函数,构建了描述股票价格处于稳定状态和崩盘的逃逸状态的动力学模型.通过数值模拟和实际数据结合,发现:1)利用道琼斯指数成分股的实际金融数据对模型参数进行估计,对模型和实际金融数据的概率密度函数做了比较,发现模型和实际情况较为符合;2)从金融常态到金融危机逃逸率的研究中,发现较强的经济增长率、较小的周期波动强度、较小的长期波动值和较弱的波动的振幅都会增强股票价格处于稳定状态的机率;3)通过研究金融危机周期性波动对价格平均逃逸时间的影响,发现存在一个最佳的周期波动振幅能最大化股票价格稳定性,某个最佳的波动均值回归速度、变弱的周期波动频率、变强的噪声关联强度和增加的经济增长率会进一步加强该最佳周期波动振幅从而进一步促进稳定性.

Various stochastic volatility models have been designed to model the variance of the asset price. Among these various models, the Heston model, as one-factor stochastic volatility mode, is the most popular and easiest to implement. Unfortunately, recent findings indicate that existing Heston modelis not able to characterize all aspects of asset returns, such as persistence, mean reverting, and clustering. Therefore, a modified Heston model is proposed in this paper. Compared with the original Heston model, the mean-reverting Cox Ingersoll and Ross process is modified to include a cosine term with the intention of capturing the periodicity of volatility. The phenomenon that high-volatile period is interchanged with low-volatile periods can thus be better described by adding such a period term to the volatility process. In addition, the geometric Brownian motion is replaced by a random walk in the presence of a cubic nonlinearity proposed by Bonanno et al. By doing so, a financial market with two different dynamical regimes （normal activity and extreme days） can be modeled. Closed-form solution for the modified Heston model is not derived in this paper. Instead, Monte-Carlo simulation is used to generate the probability density function of log-return which is then compared with the historical probability density function of stock return. Parameters are adjusted and estimated so that the square errors can be minimized. Daily returns of all the component stocks of Dow-Jones industrial index for the period from 3 September 2007 to 31 December 2008 are used to test the proposed model, and the experimental results demonstrate that the proposed model works very well. The mean escape time and mean periodic escape rate of the proposed modified Heston model with periodic stochastic volatility are studied in this paper with two different dynamical regimes like financial markets in normal activity and extreme days. Also the theoretical results of mean escape time and mean periodic escape rate can be calculated by numerical simulation. The experimental results demonstrate that 1） larger value of rate of return, smaller long run average of variance and smaller magnitude of periodic volatility will reduce the mean periodic escape rate, and thus stabilize the market; 2） by analyzing the mean escape time, an optimal value can be identified for the magnitude of periodic volatility which will maximize the mean escape time and again stabilize the market. In addition, an optimal rate of relaxation to long-time variance, smaller frequency of the periodic volatility, larger rate of return, and stronger correlation between noises will furtherreduce the mean escape time and enhance the market stability.