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

134 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
elled as follows:
S 1 t = exp(X 1 t ), X 1 t = B 1 (Z t ) + µ 1 Z t ,
S 2 t = exp(X 2 t ), X 2 t = B 2 (Z t ) + µ 2 Z t ,
where B 1 and B 2 are two components of a planar Brownian motion, with variances σ 2 1 and
σ 2 2 and correlation coefficient ρ, and {Z t} t≥0 is the stochastic time change (an increasing Lévy
process). The correlation of returns, ρ(X 1 t , X 2 t ), can be computed by conditioning with respect
to Z t :
ρ(X 1 t , X 2 t ) =
σ 1 σ 2 ρE[Z t ] + µ 1 µ 2 Var Z t
(σ 2 1 E[Z t] + µ 2 1 Var Z t) 1/2 (σ 2 2 E[Z t] + µ 2 2 Var Z t) 1/2 .
In the completely symmetric case (µ 1 = µ 2 = 0) and in this case only ρ(X 1 t , X 2 t ) = ρ: the
correlation of returns equals the correlation of Brownian motions that are being subordinated.
However, the distributions of real stocks are skewed and in the skewed case the correlation of
returns will be different from the correlation of Brownian motions that we put into the model.
Even if the Brownian motions are independent, the covariance of returns is equal to µ 1 µ 2 Var Z t
and if the distributions of stocks are not symmetric, they are correlated.
In the symmetric case, if Brownian motions are independent, the two stocks are decorrelated
but not independent. Since the components of the Brownian motion are time changed with
the same subordinator, large jumps in the two stocks (that correspond to large jumps of the
subordinator) will tend to arrive together, which means that absolute values of returns will be
correlated. If µ 1 = µ 2 = 0 and ρ = 0 then the covariance of squares of returns is
Cov((X 1 t ) 2 , (X 2 t ) 2 ) = σ 1 σ 2 Cov((B 1 (Z t )) 2 , (B 2 (Z t )) 2 ) = σ 1 σ 2 Var Z t ,
which means that squares of returns are correlated unless the subordinator Z t is deterministic.
This phenomenon can lead to mispricing and errors in evaluation of risk measures.
In finite activity models, a more accurate modelling of dependence may be achieved by
specifying directly the dependence of individual jumps in one-dimensional compound Poisson
processes (see [65]). This approach is useful in presence of few sources of jump risk (e.g., when all
components jump at the same time) because in this case it allows to achieve a precise description
of dependence within a simple model. Suppose that we want to improve a d-dimensional Black-
Scholes model by allowing for “market crashes.” The dates of market crashes can be modelled