Working Paper: CEPR ID: DP2711
Authors: Thorsten Lehnert; Christian C. P. Wolff
Abstract: Returns in financial assets show consistent excess kurtosis, indicating the presence of large fluctuations not predicted by Gaussian models. Mandelbrot (1963) first proposed the idea that price changes distributed according to a Lévy stable law. The unique feature of Lévy-stable distributions in general is that they are stable under addition. However, these distributions have power law tails that decay too slowly from the point of view of financial modelling. In recent studies the truncated Lévy Flight has been shown to eliminate this problem and to be very promising for the modelling of financial dynamics. An exponential decay in the tails ensures that all moments are finite and the distribution is fat-tailed for short time scales and converges in a Gaussian process for increasing time scales, a feature observed in financial data. We propose a model with time varying scale parameter (GARCH process) with error terms that are truncated Lévy distributed. We determine the appropriate GARCH specification for each data set by conducting a specification test based on a generalization of the augmented GARCH process of Duan (1997). The Lévy flight includes a method for scaling up a single-day volatility to a multi-day volatility, precisely a ?-root-of-time rule, where ? is the characteristic parameter of the process. We use this rule to forecast future volatility and as a result estimate Value-at-Risk (VaR) several days ahead and compare it to the RiskMetricsTM (1996) approach, which is a special case of our method. We compare the models in in-sample- and out-of-sample analyses for a sample of stock index returns.
Keywords: Augmented GARCH process; In and out-of-sample analysis; Scale consistency; Truncated Lévy flight; Value-at-risk
JEL Codes: C22; C52; G10
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| Traditional Gaussian model (C29) | Underestimates the tails of return distributions (G17) |
| Truncated Lévy distribution (C46) | Captures the dynamics of financial returns (G17) |
| Truncated Lévy distribution (C46) | Allows for finite moments and converges to Gaussian model (C58) |
| GARCH specifications (C58) | Effectively model volatility clustering in financial data (C58) |
| Truncated Lévy distribution (C46) | Outperforms traditional methods in estimating value-at-risk (VaR) (C58) |
| Incorporating asymmetry in volatility process and skewed distributions (C46) | Leads to more accurate representation of financial returns (G19) |
| Model's ability to capture conditional tail fatness and skewness (C46) | Provides significant improvement over existing methodologies (C52) |