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

182 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
5.3 A two-dimensional variance gamma model for option pricing
In this section we present a case study showing how one particular model, constructed using
Lévy copulas, can be used to price multi-asset options.
The model We suppose that under the risk-neutral probability, the prices {S 1 t } t≥0 and
{S 2 t } t≥0 of two risky assets satisfy
S 1 t = e rt+X1 t , S 2 t = e rt+X2 t , (5.18)
where (X 1 , X 2 ) is a Lévy process on R d with characteristic triplet (0, ν, b) with respect to
zero truncation function. X 1 and X 2 are supposed to be variance gamma processes, that is,
the margins ν 1 and ν 2 of ν are of the form (1.19) with parameters c 1 , λ 1 +, λ 1 − and c 2 , λ 2 +, λ 2 −.
The Lévy copula F of ν is supposed to be of the form (5.4) with parameters θ and η. The
no-arbitrage condition imposes that for i = 1, 2, λ i + > 1 and the drift coefficients satisfy
(
b i = c i log 1 − 1
λ i + 1
+ λ i − 1 )
− λ i .
+ λi −
The problem In the rest of this section, model (5.18) will be used to price two different kinds
of multi-asset options: the option on weighted average, whose payoff at expiration date T is
given by
( 2∑ +
H T = w i ST i − K)
with w 1,2 ≥ 0 and w 1 + w 2 = 1,
i=1
and the best-of or alternative option with payoff structure
H T =
( ( S
1
) ) +
N max T
S0
1 , S2 T
S0
2 − K
Option pricing by Monte Carlo Basket options, described above can be priced by Monte
Carlo method using European options on individual stocks as control variates. Denote the
discounted payoffs of European options by
V i T = e −rT (S i T − K) + for i = 1, 2.