statistique, théorie et gestion de portefeuille - Docs at ISFA

1. The Pareto distribution:
79
Fu(x) = 1 − (u/x) b , (22)
which corresponds to the set of parameters (b > 0,c = 0) with A(b,c,d,u) = b·u b . Several works have
attempted to derive or justified the existence of a power tail of the distribution of returns from agentbased
models (Challet and Marsili 2002), from optimal trading of large funds with sizes distributed
according to the Zipf law (Gabaix et al. 2002) or from stochastic processes (Biham et al 1998, 2002).
2. The Weibull distribution:
Fu(x) = 1 − exp −
x
c +
d
u
c , (23)
d
with parameter set (b = −c,c > 0,d > 0) and normalization constant A(b,c,d,u) = c
dc exp
u c
d .
This distribution is said to be a “Stretched-Exponential” distribution when the exponent c is smaller
than 1, namely when the distribution decays more slowly than an exponential distribution.
3. The exponential distribution:
Fu(x) = 1 − exp − x
d
u
+ , (24)
d
with parameter set (b = −1, c = 1, d > 0) and normalization constant A(b,c,d,u) = 1
d exp− u
d .
The exponential family can for instance derive from a simple model where stock price dynamics is
governed by a geometrical (multiplicative) Brownian motion with stochastic variance. Dragulescu
and Yakovenko (2002) have found an excellent fit of this model with the Dow-Jones index for time
lags from 1 to 250 trading days, within an asymptotic exponential tail of the distribution of log-returns
with a time-dependent exponent.
4. The incomplete Gamma distribution:
Fu(x) = 1 − Γ(−b,x/d)
Γ(−b,u/d)
with parameter set (b, c = 1, d > 0) and normalization A(b,c,d,u) =
d b
Γ(−b,u/d) .
Thus, the Pareto distribution (PD) and exponential distribution (ED) are one-parameter families, whereas
the stretched exponential (SE) and the incomplete Gamma distribution (IG) are two-parameter families. The
comprehensive distribution (CD) given by equation (20) contains three unknown parameters.
Interesting links between these different models reveal themselves under specific asymptotic conditions.
For instance, in the limit b → +∞, the Pareto model becomes the Exponential model (Bouchaud and Potters
2000). Indeed, provided that the scale parameter u of the power law is simultaneously scaled as u b = (b/α) b ,
we can write the tail of the cumulative distribution function of the PD as u b /(u + x) b which is indeed of the
form u b /x b for large x. Then, u b /(u + x) b = (1 + αx/b) −b → exp(−αx) for b → +∞. This shows that the
Exponential model can be approximated with any desired accuracy on an arbitrary interval (u > 0,U) by
the (PD) model with parameters (β,u) satisfying u b = (b/α) b . Although the value b → +∞ does not give
strickly speaking a Exponential distribution, the limit b → +∞ provides any desired approximation to the
Exponential distribution, uniformly on any finite interval (u,U).
More interesting for our present study is the behavior of the (SE) model when c → 0. In this limit, and
provided that
u
c c · → β,
d
as c → 0 . (26)
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(25)