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

72 CHAPTER 2. THE CALIBRATION PROBLEM
solution Q ∗ of problem (2.11), if it exists, satisfies I(Q ∗ |P ) ≤ C. This entails that Q ∗ ≪ P ,
which means by Proposition 1.5 that Q ∗ ∈ L + B . Therefore, Q∗ belongs to the set
L + B ∩ {Q ∈ M ∩ L : ‖CQ − C M ‖ = ‖C Q+ − C M ‖} ∩ {Q ∈ L : I(Q|P ) ≤ C}. (2.12)
Lemma 2.10 and the Prohorov’s theorem entail that the level set {Q ∈ L : I(Q|P ) ≤ C}
is relatively weakly compact. On the other hand, by Corollary 2.1, I(Q|P ) is weakly lower
semicontinuous with respect to Q for fixed P . Therefore, the set {Q ∈ P(Ω) : I(Q|P ) ≤ C} is
weakly closed and since by Lemma 2.4, M ∩ L + B is also weakly closed, the set M ∩ L+ B ∩ {Q ∈
L : I(Q|P ) ≤ C} is weakly compact. Lemma 2.2 then implies that the set (2.12) is also weakly
compact. Since I(Q|P ) is weakly lower semicontinuous, it reaches its minimum on this set.
2.3 The use of relative entropy for pricing and calibration: review
of literature
2.3.1 Pricing with minimal entropy martingale measure
It is known (see [25, Theorem 4.7] or [27, Remark 10.3]) that all exponential Lévy models
with the exception of the Black-Scholes model and the model driven by the compensated Poisson
process correspond to incomplete markets meaning that there exists no unique equivalent
martingale measure.
In the incomplete market setting many authors including [24, 41, 42, 49, 72, 73, 89] have investigated
the minimal entropy martingale measure, that is, the pricing measure that minimizes
the relative entropy with respect to the historical probability P . A probability measure Q ∗ ∈ M
is called minimal entropy martingale measure if
I(Q ∗ |P ) = min I(Q|P ). (2.13)
Q∈M
Frittelli [42] proves the existence of a minimal entropy martingale measure provided that the
stock price process is bounded and that there exists a martingale measure with finite relative
entropy with respect to P .
He further shows that if there exists an equivalent martingale
measure with finite relative entropy with respect to P , the MEMM is also equivalent to P .
Several authors including [24, 49, 73] discuss minimal entropy martingale measures for exponential
Lévy models. The following result is proven in Miyahara and Fujiwara [73].