Realizing stock market crashes: stochastic cusp catastrophe model of returns under time-varying volatility

This paper develops a two-step estimation methodology that allows us to apply catastrophe theory to stock market returns with time-varying volatility and to model stock market crashes. In the first step, we utilize high-frequency data to estimate daily realized volatility from returns. Then, we use stochastic cusp catastrophe on data normalized by the estimated volatility in the second step to study possible discontinuities in the markets. We support our methodology through simulations in which we discuss the importance of stochastic noise and volatility in a deterministic cusp catastrophe model. The methodology is empirically tested on nearly 27 years of U.S. stock market returns covering several important recessions and crisis periods. While we find that the stock markets showed signs of bifurcation in the first half of the period, catastrophe theory was not able to confirm this behavior in the second half. Translating the results, we find that the U.S. stock market's downturns were more likely to be driven by the endogenous market forces during the first half of the studied period, while during the second half of the period, the exogenous forces seem to be driving the market's instability. The results suggest that the proposed methodology provides an important shift in the application of catastrophe theory to stock markets.