Influence of periodic volatility on the stability of financial market

Abstract

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.