Dynamic behaviors and measurements of financial market crash rate

Abstract

This paper proposes the conditional crash rate (CCR) to study the degree of financial market crisis and evaluate the risk of financial market crash, based on maximum drawdown and escape rate. It is found that the CCR of stock price in financial market is equivalent to the escape rate of the drawdown time series corresponding to price series. Based on the mean escape rate, the analytical solution of CCR of financial market of stock price described by Black–Scholes model is obtained. For the CCR, empirical researches show that the theoretical results of the Heston model are in good agreement with the results of the real financial data of the S&P 500. The parameters of Heston model are estimated based on the least square estimation of the mean square error of probability density of return between the proposed model and S&P 500 data set. Then, we discuss the dynamic behaviors of CCR and the results indicate that (i) the monotonic and non-monotonic features are observed in the behaviors of CCR as a function of system parameters; (ii) we can find critical phenomena, resonance and inverse resonance behaviors in the function of CCR vs. mean reversion of volatility; (ii) the parameter variation induces the appearance of noise enhancement stability in the function of CCR vs. amplitude of volatility fluctuation.