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

2.3. RELATIVE ENTROPY IN THE LITERATURE 73
Theorem 2.7. Let P be a Lévy process with characteristic triplet (A, ν, γ). If there exists a
constant β ∈ R such that
∫
{x>1}
γ +
e x e β(ex −1) ν(dx) < ∞, (2.14)
( 1
2 + β )
A +
∫
|x|≤1
{
} ∫
(e x − 1)e β(ex−1) − x ν(dx) + (e x − 1)e β(ex−1) ν(dx) = 0,
|x|>1
then there exists a minimal entropy martingale measure Q ∗ with the following properties:
1. The measure Q ∗ corresponds to a Lévy process: Q ∗ ∈ L with characteristic triplet
A ∗ = A,
ν ∗ (dx) = e β(ex−1) ν(dx),
∫
γ ∗ = γ + βA + x(e β(ex−1) − 1)ν(dx).
|x|≤1
2. The measure Q ∗ is an equivalent martingale measure: Q ∗ ∼ P .
3. The minimal relative entropy is given by
{ ∫ β ∞
}
I(Q ∗ |P ) = −T
2 (1 + β)A + βγ + {e β(ex−1) − 1 − βx1 |x|≤1 }ν(dx) . (2.15)
−∞
Remark 2.1. It is easy to show, along the lines of the proof of Proposition 1.8, that condition
(2.14) is satisfied, in particular, if P ∈ L NA ∩ L + B
for some B > 0, which corresponds to a stock
price process with jumps bounded from above in a market without arbitrage opportunity.
Since large positive jumps do not happen very often in real markets, (2.14) turns out to be
much less restrictive (and easier to check) than the general hypotheses in [42]. This shows that
the notion of MEMM is especially useful and convenient in the context of exponential Lévy
models.
In addition to its computational tractability, the interest of the minimal entropy martingale
measure is due to its economic interpretation as the pricing measure that corresponds to the
limit of utility indifference price for the exponential utility function when the risk aversion
coefficient tends to zero. Consider an investor with initial endowment c, whose utility function
is given by
U α (x) := 1 − e −αx , (2.16)