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

74 CHAPTER 2. THE CALIBRATION PROBLEM
where α is a risk-aversion coefficient. Given some set of admissible trading strategies Θ, the
utility indifference price p α (c, H) of a claim H for this investor is defined as the solution of the
following equation:
∫
sup E[U α (c + p α (c, H) +
θ∈Θ
(0,T ]
∫
θ u dS u − H)] = sup E[U α (c + θ u dS u )].
θ∈Θ
(0,T ]
Due to the special form (2.16) of the utility function, the initial endowment c cancels out of the
above equation and we see that
p α (c, H) = p α (0, H) := p α (H).
Using the results in [33], Miyahara and Fujiwara [73] established the following properties of
utility indifference price in exponential Lévy models. Similar results have been obtained by El
Karoui and Rouge [39] in the setting of continuous processes.
Proposition 2.8. Let (X, P ) be a Lévy process such that P ∈ L + B ∩ L NA, and let Q ∗ be the
MEMM defined by (2.13). Let S t := e Xt and let Θ include all predictable S-integrable processes
θ such that ∫ (0,t] θ udS u is a martingale for each local martingale measure Q, with I(Q|P ) < ∞.
Then the corresponding utility indifference price p α (H) of a bounded claim H has the following
properties:
1. p α (H) ≥ E Q∗ [H] for any α > 0.
2. If 0 < α < β then p α (H) ≤ p β (H).
3. lim α↓0 p α (H) = E Q∗ [H].
The price of a claim H computed under the MEMM thus turns out to be the highest price
at which all investors with exponential utility function will be willing to buy this claim.
Kallsen [58] defines neutral derivative prices for which the optimal trading strategy consists
in having no contingent claim in one’s portfolio. This approach to valuation in incomplete
markets corresponds to the notion of fair price introduced by Davis [31]. Kallsen further shows
that such prices are unique and correspond to a linear arbitrage-free pricing system, defined
by an equivalent martingale measure (neutral pricing measure). If the utility function of an
investor has the form (2.16), the neutral pricing measure coincides with the minimal entropy