Modeling drawdowns and drawups in financial markets - Risk.net

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Beatriz Vaz de Melo Mendes and Vinicius Ratton Brandi
tions. For the IBOVESPA, the strings of consecutive gains are longer than those
of consecutive losses and may result in extreme cumulative gains.
The longest duration observed correspond to 19 consecutive gains of the
Nasdaq. Nasdaq also showed the smallest frequency of gains of duration one day,
and the highest frequency of consecutive losses and gains lasting more than 15
days.
In Table 3 we present results from the MGPD and the stretched exponential fits.
This table gives the parameters maximum likelihood estimates and the log-likelihood
value for each model. For the sake of brevity, we do not report the results
from the statistical tests. The estimates of ξ and γ suggest that, for most indexes,
drawdowns possess heavier tails than drawups. 7 An exception is the IBOVESPA,
whose drawups possess the heavier tail among all indexes, as indicated by the
estimate of the shape parameter, ˆ ξ = 0.35.
As noted by Johansen and Sornette (2001), extreme (larger than 10%) drawdowns
and drawups from the three US indexes, showed a strong departure from
the stretched exponential. The full MGPD model provided a better fit for them and
the statistical tests strongly rejected the stretched exponential.
Figure 3 shows QQ-plots based on the Nasdaq drawdowns and drawups fits. In
this figure the reference distribution is the MGPD for the two upper plots, and the
stretched exponential for the two lower plots. At the left-hand side, we can see that
the five large drawdowns outliers observed on the stretched exponential fit do not
seem to be so atypical under the MGPD model. The MGPD fitted the moderately
large drawups well and pointed out a single outlier (right-hand side). We note that
this largest observation had not shown up as an outlier under the stretched exponential
model (bottom right). This phenomenon is well known in robustness as the
masking effect (Rousseeuw and Leroy, 1987).
For the drawdowns from the Dow Jones, the stretched exponential fit detected
one large and five moderately large outliers. The MGPD also found the large
outlier, but fitted the remaining data points well. For the drawups from the Dow
Jones the stretched exponential provided a nice fit only for drawups with size up
to 6%, whereas the MGPD gave a good fit to all data, spotting only the largest
observation.
In the case of the S&P500, the stretched exponential provided a poor fit for
drawdowns smaller than –8% and for drawups greater than 8%. The MGPD
found one extreme and one large drawdown outlier and gave a good fit to drawups
up to 14%, detecting just two moderately large outliers.
Johansen and Sornette (2001) found that drawdowns from CAC40 are almost
perfect exponential. Our results shown in Table 3 confirm this statement. The
QQ-plots indicated two drawups outliers.
The Hang Seng index is another nice example that improvements may be
achieved by using the MGPD distribution. The stretched exponential fit on the
7 It seems there might be a connection between the value of the modifying parameter γ and
the duration of a drawdown. This requires a more comprehensive investigation.
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