High Watermarks of Market Risk

Abstract

The volatility has long been used as an auxiliary variable in the processes explaining the returns on risky assets. In this traditional framework, the observable were the returns and the volatility remained a latent variable, whose value or possible values were a by-product of the estimation. Recently, the focus has changed and many studies have been devoted to empirical estimates of the volatility itself, without specifying necessarily any model for the prices themselves. This has been made possible by the increased availability of high-frequency data, and the theoretical works of Barndorff-Nielsen and Shephard (2002) showing convergence between an empirical measure of volatility and its theoretical expression. The empirical measure of volatility has been progressively refined, from a simple sum of squared returns to more sophisticated measures taking into account microstructure biases (see for instance Oomen, 2005). In parallel, some theoretical developments have put back into focus the role of jumps. There are now procedures to disentangle the jump part of the empirical volatility from its regular fluctuations. Taking the volatility as a random variable in itself means studying its characteristics. It is well known that volatility dynamics are autoregressive but also that obviously its process is stationary. Given that, it is natural to look for the best fit for the distribution of the volatility, given that the theory yields several possible candidates. Of special interest is the estimation of the likelihood of the volatility peaks, which relies on Extreme Value Theory. In this article, we first present several estimates of measures of risk, using both high frequency data and lower frequency data. The aim here is also to show what lower frequency measures can be substitute to the high precision measures when transaction data is unavailable. The second part is devoted to the studies of the distribution of the volatility, using general forms of common distribution functions. Finally, we focus on the slope of the tail of the various risk measure distribution, in order to estimate the frequency of extreme events and define the high watermarks for market risks. Using several techniques of estimation, we finally do not find evidence for the need of a specification with heavier tails than the ones of the traditional log-normal. The tail estimates additionally yield return times for the extreme market events, as another reality check.