Modelling the Implied Probability of Stock Market Declines

In this paper we study risk-neutral densities (RNDs) for the German stock market. The use of option prices allows us to quantify the risk-neutral probabilities of various levels of the DAX index. For the period from December 1995 to November 2001, we implement the mixture of log-normals and a volatility-smoothing method. We discuss the time series behaviour of the moments of the two models and we examine linkages of the moments to macroeconomic variables, the US stock markets and credit risk. We find that the risk-neutral densities exhibit pronounced negative skewness. The spline and mixture models produce a similar shape for the entire density, but their pricing performance differs. Finally, we observe a significant volatility spillover effect, as the implied volatility and kurtosis of the DAX RND are mostly driven by the volatility of US stock prices.