Processus de Lévy en Finance - Laboratoire de Probabilités et ...

4.1. INTRODUCTION 135
as jump times of a standard Poisson process {N t } t≥0 . This leads us to the following model for
the log-price processes of d assets:
∑N t
Xt i = µ i t + Bt i + Yj i , i = 1 . . . d,
j=1
where (B t ) is a d-dimensional Brownian motion with covariance matrix Σ, and {Y j } ∞ j=1 are
i.i.d.
d-dimensional random vectors which determine the sizes of jumps in individual assets
during a market crash. This model contains only one driving Poisson shock because we only
account for jump risk of one type (global market crash affecting all assets). To construct a
parametric model, we need to specify the distribution of jumps in individual assets during a
crash (distribution of Y i
∗
a simplifying assumption that {Y i
j }d i=1
for all i) and the dependence between jumps in assets. If we make
are Gaussian random vectors, then we need to specify
their covariance matrix Σ ′ and the mean vector m, thus obtaining a multivariate version of
Merton’s model [71].
If the jumps are not Gaussian, we must specify the distribution of jumps in each component
and the copula 1 describing their dependence.
The model is thus completely specified by a
covariance matrix Σ, d jump size distributions, a d-dimensional copula C and a jump intensity
parameter λ. However, sometimes it is necessary to have several independent shocks to account
for events that affect individual companies or individual sectors rather than the entire market.
In this case we need to introduce several driving Poisson processes into the model, which now
takes the following form:
where N 1 t , . . . , N M t
X i t = µ i t + B i t +
N M∑ ∑t
k Yj,k i , i = 1 . . . d,
k=1 j=1
are Poisson processes driving M independent shocks and Y i
j,k
is the size of
jump in i-th component after j-th shock of type k. The vectors {Yj,k i }d i=1 for different j and/or k
are independent. To define a parametric model completely, one must specify a one-dimensional
distribution for each component for each shock type — because different shocks influence the
same stock in different ways — and one d-dimensional copula for each shock type. This adds up
to M × d one-dimensional distributions and M one-dimensional copulas. How many different
1 For an introduction to copulas see [76] — this monograph treats mostly the bivariate case — and [56] for
the multivariate case.