Modeling drawdowns and drawups in financial markets - Risk.net

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Beatriz Vaz de Melo Mendes and Vinicius Ratton Brandi
FIGURE 2 Densities of the MGPD with ξ = 0.6 and γ = 0.5, 1.0, 1.5, and the
stretched exponential for γ = 0.8, 1.0, 1.2, respectively.
0.0 0.1 0.2 0.3 0.4
MGPD
0.5
1.0
1.5
0 2 4 6 8 10 12 14
Both distributions possess scale equal to 3.
0 2 4 6 8 10 12 14
for y ≥ 0, where ψ is the scale parameter and γ > 0 is a tuning constant. When
γ =1 we are back to the unit exponential distribution. The class of sub-exponential
distributions is generated by letting γ < 1. Distributions with γ > 1 are called
super-exponentials. The right-hand side of Figure 2 shows the stretched exponential
densities for γ = 0.8, 1, 1.2, and scale ψ = 3.
Johansen and Sornette (2001) fitted model (1) to several data sets, including
indexes, commodities and currencies. They found that, typically, this distribution
gives a good fit to the bulk of the data but underestimates between one and
10 extreme observations. Their conclusion raises the question of whether these
extreme observations should actually be considered outliers, or whether a better
fit could be found for all observations. In fact, the authors considered this issue
and commented that the success in using the stretched exponential for fitting
drawdowns does not mean that a better parametrization does not exist. Thus, it
seems natural to look for a flexible distribution from the extreme value theory
suitable for modeling long-tailed data.
In the 1990s we saw a large amount of research applying techniques from
the extreme value theory in risk management. Examples include Longin (1996),
Danielsson and de Vries (1997), McNeil and Frey (2002), Embrechts, Resnick,
and Samorodnitsky (1998), among others. Extreme value theory is mostly concerned
with the study of the behavior of extremes from a strictly stationary
stochastic process {X i }, i = 1, 2,…, where all X i has a common distribution F X . A
main result is the Fisher–Tippett theorem (Fisher and Tippett, 1928), which gives
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0.0 0.1 0.2 0.3 0.4
Stretched exponential
0.8
1.0
1.2