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One of the stylized facts of financial time series seems to be a fundamental asymmetry between smooth upward movements over longer periods of time followed by abrupt declines over relatively shorter time frames ("crashes").

Unfortunately I haven't found much literature on this yet and I am trying to answer two questions as a starting point:

  1. How can you test empirical time series for the degree of this kind of pattern?
  2. With which stochastic generating process can you simulate this kind of behavior?

Further research would go into the direction of how to exploit these patterns if they can really be corroborated empirically.

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3 Answers

up vote 8 down vote accepted

"Treshold Garch" or T-Garch models are designed to capture this asymmetry. See this exposition by U. Chicago's Ruey Tsay who has a terrific text on time-series models in "Analysis of Financial Time Series".

You can use the structure of the T-Garch models to simulate data with this property.

There is a package called fGarch that creates APARCH models. A T-GARCH model is a special case of an APARCH model where delta = 1. See Ruey's Lecture 5 and associated R-code for using the fGarch library.

Also, there is some outstanding theoretical research by Capital Fund Management using the statistical physics approach on how to explain the negative-skewness and other features. I include a link and abstract to their research below:

More stylized facts of financial markets: leverage effect and downside correlations

We discuss two more universal features of stock markets: the so-called leverage effect (a negative correlation between past returns and future volatility), and the increased downside correlations. For individual stocks, the leverage correlation can be rationalized in terms of a new `retarded' model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect. As for the increase of correlations in highly volatile periods, we investigate how much of this effect can be explained within a simple non-Gaussian one-factor description with time independent correlations. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations, which reflects the fat-tailed and negatively skewed distribution of market returns.

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Thank you, Quant Guy - Implementations in R are highly appreciated! –  vonjd Dec 26 '11 at 17:40
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These patterns are of course well-known enough to have been "priced in" to the financial markets. Jump diffusions are a classic way to capture the phenomenon, and often have closed-form option pricing formulas associated with them. The implied option skew, for example, gets a lot flatter when you use a JD model.

Jump diffusions are often combined with some kind of stochastic volatility model (e.g. as described by Jim Gatheral) to obtain what many practictioners consider the most descriptive tractable modeling framework. I happen to think this family of models is easily superior to a specific "regime" time series model like some of those described above, not least because I think GARCH is of only academic interest.

The key difficulty in this type of work is not finding a plausible stochastic description, but rather achieving tractability and avoiding an explosion of parameters beyond what your data supports.

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May you assess the goodness of some GEV distribution fit against a gaussian? You could also use this asymmetric curve to generate an iid sequence.

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I think it is not so simple because the form of the distribution is not the same as the unfolding of the stochastic process in time. –  vonjd Dec 26 '11 at 16:09
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