Abstract
In this paper, we examine the S&P 500 index log-returns on short intraday time scales with three different ARMA-GARCH models. In order to forecast market risk, we describe the innovation process with tempered stable distributions which we compare to commonly used methods in financial modeling. Value-at-risk backtests are provided where we find that models based on the tempered stable innovation assumption significantly outperform traditional models in forecasting risk on short time-scales. In addition to value-at-risk, the idiosyncratic differences in average value-at-risk are compared between the models.
- 1
High-frequency in the context of this paper is used to address time-scales ranging from 75 s to 5 min in order to clearly distinguish it from time-series based on daily recorded data. It is not to be confused with high-frequency in the context of sub-millisecond trading. Very high-frequency time-series exhibit phenomena such as long-range dependency. Those effects cannot be captured with the ARMA-GARCH approach and as a consequence, different models, such as fractional Brownian motion or neural network approaches, have to be employed [see Sun, Rachev, and Fabozzi (2008)].
- 2
Moreover, many studies have shown that the normal distribution hypothesis for most financial assets traded is not an adequate model to describe observed return distributions. This was first reported by Mandelbrot (1963). A summary of studies is provided in Rachev, Menn, and Fabozzi (2005).
- 3
Some dates are excluded from the study where either the data were completely missing or the dataset is affected by large gaps. The exclusion criteria for 1 day are fulfilled if either more than 3% of the values are missing or if gaps occur which exceed 1% of the number of values for 1 day.
- 4
N=10 is chosen such that a time horizon of two business weeks is used to forecast the volatility. τ=0.1 has been chosen such that the 10th days till has a non-vanishing influence on the volatility.
- 5
The period has been chosen under the constraint that the dataset contains no missing days.
- 6
In order to conserve space, we restricted the analysis to only two time-series on different time-scales.
- 7
The CLR test is an enhancement of the Kupiec test [see Kupiec (1995)].
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