Realized Quantiles^*

Abstract

This article proposes a simple approach to estimate quantiles of daily financial returns directly from highfrequency data. We denote the resulting estimator as realized quantile (RQ) and use it to forecast tail risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES). The RQ estimator is built on the assumption that financial logarithm prices are subordinated self-similar processes in intrinsic time. The intrinsic time dimension stochastically transforms the clock time in order to capture the real "heartbeat" of financial markets in accordance with their trading activity and/or riskiness. The self-similarity assumption allows to compute daily quantiles by simply scaling up their intraday counterparts, while the subordination technique can easily accommodate numerous empirical features of financial returns, such as volatility persistence and fat-tailedness. Our method, which is built on a flexible assumption, is simple to implement and exploits the rich information content of high-frequency data from another time perspective than the classical clock time. In a comprehensive empirical exercise, we show that our forecasts of VaR and ES are more accurate than the ones from a large set of up-to-date comparative models, for both, stocks and foreign exchange rates.

Conditional quantile analysis for realized GARCH models

“…To incorporate the realized quantile (Dimitriadis and Halbleib, 2021), we assume the self-similarity condition, which is often violated in the real data analysis. Thus, it is interesting to develop an estimation procedure of the realized quantile, which is robust to the self-similarity condition.…”

“…Recently, under the self‐similarity condition, Dimitriadis and Halbleib (2021) suggested the realized quantile. For example, for some H ∈ (0, 1), the stochastic process X t satisfies where denotes equality in distribution.…”

“…For example, the self‐similarity condition (2.3) with H = 0.5 is satisfied, the log‐returns are ‐mixing, and their distribution is absolutely continuous with strictly positive and continuous density. Details can be found in Theorem 2.1 (Dimitriadis and Halbleib, 2021). Assumption 3(c) is required to establish the asymptotic normality.…”

“…Recently, under the self-similarity condition, Dimitriadis and Halbleib (2021) suggested the realized quantile. For example, for some H ∈ (0, 1), the stochastic process X t satisfies…”

“…On the other hand, we impose some self‐similarity condition on the high‐frequency data to utilize the realized quantile information. The proposed quantile regression model has the open‐to‐close realized quantile (Dimitriadis and Halbleib, 2021) as the innovation instead of the realized volatility. We call this model the realized quantile and realized GARCH quantile regression (real‐realized GARCH quantile regression) model.…”