It is known (see [25, Theorem 4.7] or [27, Remark 10.3]) that all exponential L´evy models with the exception of the Black-Scholes model and the model driven by the compensated Pois- son 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 in- vestigated 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
Q∈MI(Q|P ). (2.13)
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 ex- ponential L´evy models. The following result is proven in Miyahara and Fujiwara [73].
Theorem 2.7. Let P be a L´evy process with characteristic triplet (A, ν, γ). If there exists a constant β ∈ R such that
Z {x>1} exeβ(ex−1)ν(dx) < ∞, (2.14) γ + 1 2 + β A + Z |x|≤1 n (ex− 1)eβ(ex−1)− xoν(dx) + Z |x|>1 (ex− 1)eβ(ex−1)ν(dx) = 0, then there exists a minimal entropy martingale measure Q∗ with the following properties:
1. The measure Q∗ corresponds to a L´evy process: Q∗ ∈ L with characteristic triplet A∗ = A, ν∗(dx) = eβ(ex−1)ν(dx), γ∗= γ + βA + Z |x|≤1 x(eβ(ex−1)− 1)ν(dx).
2. The measure Q∗ is an equivalent martingale measure: Q∗∼ P . 3. The minimal relative entropy is given by
I(Q∗|P ) = −T β 2(1 + β)A + βγ + Z ∞ −∞{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 ∈ LN A∩ 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 L´evy 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
where α is a risk-aversion coefficient. Given some set of admissible trading strategies Θ, the utility indifference price pα(c, H) of a claim H for this investor is defined as the solution of the
following equation: sup θ∈Θ E[Uα(c + pα(c, H) + Z (0,T ] θudSu− H)] = sup θ∈Θ E[Uα(c + Z (0,T ] θudSu)].
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
pα(c, H) = pα(0, H) := pα(H).
Using the results in [33], Miyahara and Fujiwara [73] established the following properties of utility indifference price in exponential L´evy 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 L´evy process such that P ∈ L+B∩ LN A, and let Q∗ be the
MEMM defined by (2.13). Let St:= eXt and let Θ include all predictable S-integrable processes
θ such that R(0,t]θudSu is a martingale for each local martingale measure Q, with I(Q|P ) < ∞.
Then the corresponding utility indifference price pα(H) of a bounded claim H has the following
properties: 1. pα(H) ≥ EQ ∗ [H] for any α > 0. 2. If 0 < α < β then pα(H) ≤ pβ(H). 3. limα↓0pα(H) = EQ ∗ [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
martingale measure. The neutral pricing measure Q∗ also corresponds to the least favorable
market completion from the point of view of the investor in the following sense. Let V (c, Q) := sup{E[U(c + X − EQ[X])] : X is FT-measurable}
be the maximum possible expected utility that an investor with initial endowment c and utility function U can get by trading in a market where the price of every contingent claim X is equal to EQ[X]. Then
V (c, Q∗) = inf
Q∈EMM(P )V (c, Q),
that is, the neutral pricing measure minimizes the maximum possible expected utility that an investor can get in a completed market, over all possible arbitrage-free completions. The notion of neutral pricing measure thus coincides with the minimax measures studied in [16], [49] and other papers.