Modelling the Implied Probability of Stock Market Movements Modelling the Implied Probability of Stock Market Movements E U R O P E a N C E N T R a L B a N K Contents Non-technical Summary

1 This paper was presented at the ECB workshop “The Measures and Determinants of Financial Market Uncertainty”, at the 2002 meeting of the European Finance Association in Berlin, the Bank for International Settlements and the Oesterreichische Nationalbank. We are grateful to Claudio Borio, Eli Remolona, Christian Upper and an anonymous referee for their helpful comments. The opinions expressed herein are those of the authors and do not necessarily represent those of the Oesterreichische Nationalbank or the European Central Bank. Fax +49 69 1344 6000 Telex 411 144 ecb d All rights reserved. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. The views expressed in this paper do not necessarily reflect those of the European Central Bank. Abstract 4 Non-technical summary 5 1. Introduction 7 2. Abstract 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 model and a volatility-smoothing method. We discuss the time series behaviour of the implied PDFs and we examine the relations between the moments and observable factors such as macroeconomic variables, the US stock markets and credit risk. We find that the risk-neutral densities exhibit pronounced negative skewness. Our second main observation is a significant spillover of volatility, as the implied volatility and kurtosis of the DAX RND are mostly driven by the volatility of US stock prices. In Financial Economics, many researchers have studied option prices, because these derivatives contain unique information that is not available from the prices of other financial instruments. A call option gives the buyer the right to purchase in the future a certain asset at a price fixed today. The value of such an option is determined by the distance between the current stock price and the exercise price. When market participants price option contracts in the course of trading, they use forecasts of the probability of different asset prices for the period until the derivative expires. The perception of market participants about the movement of the asset price, in particular the probability density until expiry, is thus incorporated into the market price of the puts and calls through transactions made on the derivatives exchange. Therefore, the observed market prices of the options convey information about …