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

5.3. TWO-DIMENSIONAL VARIANCE GAMMA MODEL 183
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.2 −0.1 0 0.1 0.2
−0.2 −0.1 0 0.1 0.2
Figure 5.5: Scatter plots of returns in a 2-dimensional variance gamma model with correlation
ρ = 50% and different tail dependence. Left: strong tail dependence (η = 0.75 and θ = 10).
Right: weak tail dependence (η = 0.99 and θ = 0.61).
and the discounted payoff of the basket option by V T = e −rT H T . Then the Monte Carlo
estimate of basket option price is given by
Ê[V T ] = ¯V T + a 1 (E[V 1 T ] − ¯V 1 T ) + a 2 (E[V 2 T ] − ¯V 2 T ),
where a bar over a random variable denotes the sample mean over N i.i.d. realizations of this
variable, that is, ¯V T = 1 ∑ N
N i=1 V (i)
(i)
T
, where V
T
are independent and have the same law as V T .
The coefficients a 1 and a 2 should be chosen in order to minimize the variance of Ê[V T ]. It is easy
to see that this variance is minimal if a = Σa 0 , where Σ ij = Cov(V i T , V j T ) and a0 i = Cov(V T , V i T ).
In practice these covariances are replaced by their in-sample estimates; this may introduce a
bias into the estimator Ê[V T ], but for sufficiently large samples this bias is small compared to
the Monte Carlo error [45].
To illustrate the option pricing procedure, we fixed the following parameters of the marginal
distributions of the two assets: c 1 = c 2 = 25, λ 1 + = 28.9, λ 1 − = 21.45, λ 2 + = 31.66 and
λ 2 − = 25.26. In the parametrization (1.18) this corresponds to θ 1 = θ 2 = −0.2, κ 1 = κ 2 = 0.04,
σ 1 = 0.3 and σ 2 = 0.25. To emphasize the importance of tail dependence for pricing multi-asset
options, we used two sets of dependence parameters, which correspond both to a correlation
of 50% (the correlation is computed numerically) but lead to returns with very different tail
dependence structures: