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- Handbook of Modeling High-Frequency Data in Finance.
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- Risk Estimation on High Frequency Financial Data?
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### Empirical Analysis of the DAX 30

Jiang, G. Bouchaud, J. Fundamental Problems in Statistical Physics. Schwab, M. Peng, R. Jasny, B. Cowan Research L. Cowan Research, L.

## Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data

White, S. Chen, C. Financial Management, 63—69 Google Scholar. Frankfurter, G. It is not permitted to disclose, copy or distribute the document except by written permission of Olsen Ltd. Contents 1 Introduction 2 2 Realized volatility 3 2. In particular, due to the increasingly central role played by the Value-at-Risk VaR approach in risk assessment, it is becoming progressively more important to have a good definition, measure and forecast of short term volatility.

## Paris December 2018

But despite its importance, volatility is still an ambiguous term for which there is no unique, universally accepted precise definition. The standard approaches currently compute volatilities either by fitting econometric models such as GARCH, or by studies volatilities implied from specific option pricing models such as Black-Scholes, or by studying historical indicators computed from daily squared or absolute returns.

But, especially for short term volatility forecasts, all of those approaches suffer from noticeable drawbacks. GARCH models have aggregation, scaling and long memory properties which does not corresponds to that widely found in the real data. In Black-Scholes models implied volatility presents the so called smile and can yet have a too slowly changing dynamical behaviour. Such excessive high sensitivity is primarily due to the stochastic noise introduced by the arbitrary choice of prices, which makes the results dependent on only one observation which may not be representative of the full day dynamics.

Using only one price per day may potentially lead to loose all the other information contained in the whole process of prices formation during the day. On the other hand Taylor and Xu, and Dacorogna et al.

## Risk Estimation on High Frequency Financial Data | SpringerLink

This approach, as claimed by Andersen et al. Treating the realized volatility as an observable object has the far reaching consequence of allowing to directly fit forecasting models rather than using much more complicated ARCH-type econometric models required when volatility is viewed as a latent variable. In practice however, in this study we found that for return intervals less than few hours, such definition is affected by a considerable systematic error. In other words, the expectation of the volatility computed with high frequency returns is not equal to the one obtained with daily returns.

Since our target measure is the daily volatility we shall consider the expectation of the volatility from 1-day returns the suitable one; hence we term the difference between the two measures the bias of the high frequency volatility. We believe that, once the practical subtle pitfalls contained in the definition of high frequency realized volatility will be overcame, it will be extremely advantageous to integrate this better measurement of the target function i. First illustrate how using high frequency data in the measurement of the realized volatility significantly improve the standard forecasting models currently used in the conventional approaches to VaR.

Second presents evidence that a wide spectrum of frequency ranging from few minutes to months allows to compute better forecasts of the realized volatility. The paper is organized as follows. In section 2 a proper target function for volatility forecast, realized volatility and its bias are analyzed, together with a description of a bias correction procedure. In section 3, our volatility forecasting model based on a combination of time series operator with different horizons EMA-HAR is introduced, while the methodology and data employed described in section 4.

In section 5 the empirical results of forecasting performances are summarized and section 6 concludes. The idea of employing high frequency data in the computation of volatility trace back to the seminal intuition of Merton according to which higher frequency are not useful for the mean but essential for the variance.

Yet only recently such idea has been exploited with intraday data: Dacorogna et al. Furthermore Andersen et al.

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Hence following with only slightly differences Dacorogna et al. In the computation of this quantity we will use business time scale and previous tick interpolation. From this point of view, the volatility computed with daily returns is a very loose measure since it relies only on one observation per day, taken at a certain daytime; all the other information on the price process of the day is thrown away.

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For instance, as shown by Andersen and Bollerslev, the variance of the measurement error of the volatility measured with daily squared returns, is often twenty time the unconditional variance of the squared volatility. This contrasts sharply with the theoretical result reached by Andersen et al.

In fact, when the return interval shrinks market microstructures effects arise. Departures from an i. We justify the use of this term on the basis of the simple observation that for almost the totality of the agents currently operating on financial markets, the relevant variable of interest is the daily volatility or a longer one and not the volatility observed at a say, 5-minute level. The fact that we use high frequency return to compute the daily volatility is merely a measurement issue. We employ short term return because we want to reduce the estimation error of our measure, not because we are interested in evaluating the risk existing at such fine time frame.

Not considering the bias would imply to contaminate the measure of daily risk with risk components that are present only at very short time scale, and that would never be perceived by an operator having a 1-day horizon. Taking into account the existence of this bias, leads to a trade-off between two opposite types of limitations which precludes the possibility to have an infinitively precise, easy measure of the realized volatility.

In fact, if on the one hand, statistical consideration would impose to use a very high number of return observations to reduce the stochastic error of the measurement, on the other hand, after a certain threshold market microstructure frictions come into play introducing a bias that usually increases as the sampling frequency increases. A significant bias of the volatility can only be explained by a significant departure from the i.

This bias represent nothing but the deviation of the scaling law of the realized volatility from that of an i. Then the actual amount of bias for any given interval return can be analyzed by looking at the scaling behaviour of the average normalized volatility computed at different frequencies. Using normalized volatility 4 the scaling behaviour of a standard i.

In other words, using our notation, if the price followed an i. In this case the vertical distance between the constant line passing through the volatility computed with 1-day returns and the empirical value of the volatility obtained with any other return interval, directly gives us the amplitude of the bias at any given time scale.

This leads to a negative serial correlation in returns that is positively related to the size of the spread. The dynamical behaviour of the bias as for the other financial instruments analyzed remain quite constant over time. This means that the booms and bursts of volatility result in quasi-parallel shifts of the curve depicted in figure 2. This quite general empirical property of the bias will play a central role in the bias correction method we will propose in the next paragraph. Both these features point to the direction of increasing the size of the price movements observed at very high frequency and the presence and impact of the gaps.