ON q-OPTIMAL MARTINGALE MEASURES IN EXPONENTIAL L ´ EVY MODELS CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER TU Braunschweig and University of Konstanz Abstract. We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential L´ evy models. As a consequence we find that, in the one- dimensional case, the q-optimal equivalent martingale measures may ex- ist only, if the tails for upward jumps are extraordinarily light. Moreover, we derive convergence of the q-optimal signed, resp. absolutely continu- ous, martingale measures to the entropy minimal martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed. Keywords: stochastic duality, q-optimal martingale measure, minimal entropy martingale measure, L´ evy processes 2000 AMS subject classification: 91B28, 60H10, 60G51, 60J75 JEL classification: G11, C61 1. Introduction Recently, the characterization of q-optimal equivalent martingale mea- sures in market models with jumps has been studied in several papers. Jeanblanc et al. (2007) consider exponential L´ evy processes. By a point- wise minimization procedure and formal application of the Kuhn-Tucker theorem they give sufficient conditions to identify the q-optimal equivalent martingale measures by finding a root of a deterministic equation, provided this root satisfies a positivity condition related to the L´ evy measure. Their procedure generalizes and simplifies earlier work by Fujiwara and Miyahara (2003) and Esche and Schweizer (2005) for the minimal entropy martingale measure. As demonstrated by Choulli et al. (2007) the same type of root and positivity condition can be used to characterize minimal Hellinger mar- tingale measures of order q in a general semimartingale framework. In a parallel development Kohlmann and Xiong (2007) derive a semimartingale backward equation with jumps to identify q-optimal equivalent martingale Date : January 4, 2008. 2000 Mathematics Subject Classification. 91B28, 60H10, 60G51, 60J75 . Key words and phrases. stochastic duality, q-optimal martingale measure, minimal entropy martingale measure, L´ evy processes. 1
Transcript
ON q-OPTIMAL MARTINGALE MEASURES INEXPONENTIAL LEVY MODELS
CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
TU Braunschweig and University of Konstanz
Abstract. We give a sufficient condition to identify the q-optimalsigned and the q-optimal absolutely continuous martingale measures inexponential Levy models. As a consequence we find that, in the one-dimensional case, the q-optimal equivalent martingale measures may ex-ist only, if the tails for upward jumps are extraordinarily light. Moreover,we derive convergence of the q-optimal signed, resp. absolutely continu-ous, martingale measures to the entropy minimal martingale measure asq approaches one. Finally, some implications for portfolio optimizationare discussed.
Recently, the characterization of q-optimal equivalent martingale mea-sures in market models with jumps has been studied in several papers.Jeanblanc et al. (2007) consider exponential Levy processes. By a point-wise minimization procedure and formal application of the Kuhn-Tuckertheorem they give sufficient conditions to identify the q-optimal equivalentmartingale measures by finding a root of a deterministic equation, providedthis root satisfies a positivity condition related to the Levy measure. Theirprocedure generalizes and simplifies earlier work by Fujiwara and Miyahara(2003) and Esche and Schweizer (2005) for the minimal entropy martingalemeasure. As demonstrated by Choulli et al. (2007) the same type of rootand positivity condition can be used to characterize minimal Hellinger mar-tingale measures of order q in a general semimartingale framework. In aparallel development Kohlmann and Xiong (2007) derive a semimartingalebackward equation with jumps to identify q-optimal equivalent martingale
Date: January 4, 2008.2000 Mathematics Subject Classification. 91B28, 60H10, 60G51, 60J75 .Key words and phrases. stochastic duality, q-optimal martingale measure, minimal
entropy martingale measure, Levy processes.
1
2 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
measures extending methods developed by Mania et al. (2002) for modelswith continuous trajectories.
It is well known, however, that, in the presence of jumps, q-optimal mar-tingale measures may fail to be equivalent, but belong to the larger classof signed martingale measures. As all mentioned papers exploit represen-tation results for densities of equivalent martingale measures as stochasticexponentials, these techniques cannot be generalized to study characteri-zation results for q-optimal signed martingale measures. As a main resultof this paper we demonstrate how the root condition in Jeanblanc et al.(2007) can be modified in two ways such that the positivity condition canbe dropped. Depending on the modification of the root condition, the ex-istence of the respective root turns out to be sufficient to identify the q-optimal signed martingale measure and the q-optimal absolutely continuousmartingale measure in exponential Levy models. Our proof is based on averification procedure in terms of a hedging problem, which holds true in ageneral semimartingale setting and is based on duality for convex optimiza-tion. In the one-dimensional case we give necessary and sufficient conditionsfor the existence of the roots in terms of the Levy measure. In particular, ifthe stock model has a pth moment, where p is the conjugate exponent to q,
• we explicitly characterize the q-optimal signed martingale measure;• we explicitly characterize the q-optimal absolutely continuous mar-
tingale measure (here the condition on the pth moment can bedropped in many models);
• we show that the q-optimal equivalent martingale measure exists ifand only if the q-optimal signed martingale measure is equivalent,in which case all the three measures coincide
We also demonstrate that the positivity assumption in Jeanblanc et al.(2007), which guarantees that the q-optimal signed martingale measure isequivalent, typically requires the tails of the upward jumps to be extraordi-narily light. (Indeed, it typically amounts to the existence of some exponen-tial moments of the stock model). Therefore, in many practically relevantmodels such as generalized hyperbolic models or the Merton model, theq-optimal equivalent martingale measure does not exist.
As a second contribution we study the convergence of the q-optimalsigned, respectively absolutely continuous, martingale measures to the mini-mal entropy equivalent martingale measure for one-dimensional exponentialLevy processes. This convergence problem was treated by Grandits andRheinlander (2002) in semimartingale models with continuous trajectories.Our results significantly generalize the corresponding results in Jeanblanc etal. (2007), even in situations where the q-optimal measures are equivalent.Theorem 2.7 below entails, for instance, convergence under the assumptionof bounded upward jump heights without any assumptions on the activity ofthe jumps. When the positivity condition fails and the q-optimal measuresare absolutely continuous, resp. signed, Theorem 2.7 provides, to the best of
q-OPTIMAL MARTINGALE MEASURES 3
our knowledge, the first convergence result to the minimal entropy measurefor exponential Levy processes. As already noted by Grandits (1999) in hisstudy in discrete time, the convergence to the minimal entropy measuresadds importance to the q-optimal martingale measures for pricing purposes:Given a bounded contingent claim, the (discounted) expectation with re-spect to a q-optimal martingale measures will lie within the no arbitrage-pricing interval for this claim when q is sufficiently close to one, even if theq-optimal martingale measure is not equivalent.
Finally, the convergence results for the measures are applied to establishconvergence of an approximating sequence of optimal investment problems(with and without consumption) to the exponential utility maximizationproblem, complementing earlier work by Kohlmann and Niethammer (2007)and Niethammer (2008). The corresponding sequence of utility functionsrewards high odd moments and penalizes even moments, whose order tendsto infinity. In this way the risk of large downwards jumps in the exponentialLevy model can be taken into account more flexibly than in the mean-variance case.
The paper is organized as follows: After an extended discussion of theresults in Section 2, Sections 3 and 4 are devoted to the verification ofthe q-optimal martingale measures. In Section 5 we prove the convergenceto the minimal entropy one. Section 6 collects some consequences for theexponential utility maximization problem. Some proofs are postponed to anAppendix.
2. Discussion of the results
We recall the standard setting for exponential Levy models. Let (Ω,F , P )be a probability space, T ∈ (0,∞) a finite time horizon, and F = (Ft)t∈[0,T ] afiltration satisfying the usual conditions, i.e. right-continuity and complete-ness. X is supposed to be an Rn-valued Levy process with characteristictriplet (σσ′, ν, b) on (Ω,F , P ). By the Levy-Ito-decomposition, (see e.g.Cont and Tankov, 2004), X has the following form:
Xt = bt + σWt +∫ t
0
∫
‖x‖>1xN(dx, ds) +
∫ t
0
∫
‖x‖≤1xN(dx, ds) (1)
where W is a n-dimensional Brownian motion, N is a Poisson random mea-sure with intensity measure ν(dx)dt, and N(dx, dt) = N(dx, dt) − ν(dx)dt.Here, the Levy measure ν is defined on Rn
0 := Rn \ 0.We suppose that a discounted market with n assets is given by
St = diag(S(1)0 , . . . , S
(n)0 )eXt , t ∈ [0, T ], S
(i)0 > 0. (2)
The following standing assumptions are in force throughout the paper:
Standing Assumption:
(i) The filtration F coincides with FX , the completion of the filtration gen-erated by the Levy process X.
4 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
(ii) E[ ‖St‖ ] < ∞ for all t ∈ [0, T ].
The second assumption guarantees that S is a special semimartingale withdecomposition St = S0 + Mt + At, where
dMt = St−(σdWt +∫
Rn0
(ex − 1)N(dx, dt))
anddAt = St−(−β +
∫
Rn0
(ex − 1− x1‖x‖≤1)ν(dx))dt.
Here, β = −(b+ 12
∑j σ2
·,j) and S = diag(S(1), ..., S(n)). 1 denotes the vectorin Rn having all entries equal to one, and expressions such as ex are to beinterpreted componentwise, i.e. ex = (ex1 , ..., exn)′ .
To properly explain the problems to be discussed in this paper, for q > 1we first introduce the set of q-integrable density processes of signed localmartingale measures,
Dqs = Z ∈ Uq|E(ZT ) = 1, SZ is a local P -martingale,
where Uq denotes the set of R-valued Lq(Ω, P )-uniformly integrable mar-tingales. Clearly Z ∈ Dq
s can be identified with a signed measure settingdQZ = ZT dP . We will first study the following problem:
(Mins,q) Find Z(q) ∈ Dqs such that
E[|Z(q)T |q] = inf
Z∈Dqs
E[|ZT |q].
dQ(s,q) = Z(q)T dP is called the q-optimal signed martingale measure Q(s,q)
(qSMM).
A closely related problem is the minimization over the subset of density pro-cesses corresponding to q-integrable absolutely continuous local martingalemeasures
Dqa = Z ∈ Dq
s | ZT ≥ 0 P -a.s.,only. Hence:
(Mina,q) Find Z(q) ∈ Dqa such that
E[|Z(q)T |q] = inf
Z∈Dqa
E[|ZT |q].
dQ(a,q) = Z(q)T dP is called the q-optimal absolutely continuous martingale
measure Q(a,q) (qAMM). If Q(a,q) is equivalent to P , then it is called theq-optimal equivalent martingale measure (qEMM) and is denoted by Q(e,q).
Apart from characterizing the q-optimal signed and absolutely continuousmartingale measures, we study their behaviour as q tends to 1. For mod-els with continuous trajectories it was shown in Grandits and Rheinlander(2002) that, under some technical assumptions, the q-optimal equivalent
q-OPTIMAL MARTINGALE MEASURES 5
martingale measures converge to the minimal entropy martingale measure(as q → 1). The latter one is defined to be the equivalent local martingalemeasure, which minimizes the entropy relative to P over the set of all equiv-alent local martingale measures with finite relative entropy:
(Mine,log) Find Zmin ∈ Dloge such that
E[ZminT log Zmin
T ] = infZ∈Dlog
e
E[ZT log ZT ].
where correspondingly
Dloge = Z ∈ D1
a| ZT > 0 P -a.s. and E(ZT log ZT ) < ∞.dQmin = Zmin
T dP is called the minimal entropy martingale measure Qmin
(MEMM).
Jeanblanc et al. (2007) discussed, for the first time, the convergence of theqEMMs to the MEMM in the framework of exponential Levy models undersome technical conditions. To this end, they first propose the followingsufficient condition to identify the qEMM.
Condition Cq:There exists a θq ∈ Rn such that
(q − 1)θ′q(ex − 1) + 1 > 0 ν-a.s., (3)
σσ′θq +∫
Rn0
((ex − 1)((q − 1)θ′q(e
x − 1) + 1)1
q−1 − x1‖x‖≤1
)ν(dx) = β
and ∫
Rn0
∣∣((q − 1)θ′q(ex − 1) + 1)
qq−1
−1− (q − 1)(((q − 1)θ′q(ex − 1) + 1)
1q−1 − 1)
∣∣ν(dx) < ∞.
Theorem 2.1 (Jeanblanc et al. (2007), Theorem 2.9). Suppose Cq holds.Then the qEMM exists and is given by
E(θ′qσ, ((q − 1)θ′q(ex − 1) + 1)
1q−1 − 1),
where E(f, g) denotes the stochastic exponential with Girsanov parametersf, g, i.e.
Et(f, g) = exp∫ t
0f(s)dWs − 1
2
∫ t
0‖f(s)‖2ds +
∫ t
0
∫
Rn0
g(s, x)N(dx, ds)
×∏
s≤t
(1 + g(s,∆X(s)))e−g(s,∆X(s)).
In the one-dimensional case, we now argue that condition Cq is ratherrestrictive in ‘reasonable’ models, i.e. we shall assume:
(H) n = 1, [σ 6= 0 or ν((−∞, 0)) 6= 0 6= ν((−∞, 0))]
6 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Condition (H) means that upwards and downwards movements of the stockare subject to random influences. Note, that the positivity assumption (3)rules out negative values for θq, if the possible heights of upward jumps areunbounded. This fact can be employed to derive the following necessarycondition for Cq. Its proof is postponed to the appendix.
Proposition 2.2. Suppose (H) and P is not a martingale measure. Then:(i) If Cq holds for some q > 1, then∫
x≥1eθex
ν(dx) < ∞ (4)
for some θ > 0 or the MEMM does not exist.(ii) If Cq holds for some q > 1, then∫
R0
(ex − 1)− x1|x|≤1ν(dx) + (b +12σ2) < 0 (5)
or upward jumps heights are bounded, i.e. ν([L,∞)) = 0 for some L > 0.
Remark 2.1. Suppose n = 1.Note that most of the concrete models discussed in the literature, such as thevariance gamma model, tempered stable models or the popular jump-diffusionmodels by Merton or Kou satisfy∫
x≥1eθex
ν(dx) = ∞
for all θ > 0. Hence, Cq and the existence of the MEMM cannot holdsimultaneously for these models.
The positivity condition (3) guarantees that the (q − 1)th root of (q −1)θ′q(ex − 1) + 1 defines a real valued function. We shall now present twonatural extensions of Cq which allow to drop the positivity assumption:Condition Cs
q :There exists a θq ∈ Rn such that
egq(x) := |(q − 1)θ′q(ex − 1) + 1| 1
q−1 sgn((q − 1)θ′q(ex − 1) + 1)
satisfies
σσ′θq +∫
Rn0
((ex − 1)egq(x)− x1‖x‖≤1
)ν(dx) = β (6)
and ∫
Rn0
∣∣ |egq(x)|q − 1− q(egq(x)− 1)∣∣ ν(dx) < ∞. (7)
The second modification of Cq truncates (q − 1)θ′q(ex − 1) + 1 at zero:Condition Ca
q :There exists a ξq ∈ Rn such that
egq(x) := ((q − 1)ξ′q(e
x − 1) + 1)1
q−1
+
q-OPTIMAL MARTINGALE MEASURES 7
satisfies
σσ′ξq +∫
Rn0
((ex − 1)eg
q(x)− x1‖x‖≤1
)ν(dx) = β (8)
and ∫
Rn0
| egqq(x)− 1− q(eg
q(x)− 1)|ν(dx) < ∞ (9)
If we assume Csq , resp. Ca
q , only, then E(θ′qσ, egq−1), resp. E(θ′qσ, egq−1),
still defines a martingale owing to the fact that the stochastic exponentialof a Levy martingale is always a martingale (see Cont and Tankov, 2004,Prop. 8.23). However, in general they define a signed, resp. an absolutelycontinuous measure only. It is a good guess that they are q-optimal in therespective class.
Theorem 2.3. Suppose q > 1.(i) If Cs
q holds, then
Z(q) = E(θ′qσ, egq − 1)
is the density process of qSMM.(ii) If Ca
q holds, then
Z(q) = E(θ′qσ, egq− 1)
is the density process of qAMM.
The proof will be given in Sections 3 and 4. Obviously, if egq(x) > 0 ν-almost surely or eg
q(x) > 0 ν-almost surely, the densities Z(q) and Z(q)
coincide and induce the qEMM.
Remark 2.2. The proof of Theorem 2.9 in Jeanblanc et al. (2007) is veryintuitive: Exploiting that all equivalent martingale measures can be repre-sented by stochastic exponentials, they first show that passing from randomto appropriate deterministic and time-independent Girsanov parameters, theLq-norm can always be reduced. Then Cq is derived by a formal applicationof the Kuhn-Tucker theorem to the problem of finding optimal deterministicand time-independent Girsanov parameters. This line of arguments cannotbe applied to the problem of finding the q-optimal signed martingale mea-sure, as signed measures cannot be represented by stochastic exponentials ingeneral. Instead we will prove Theorem 2.3 (i) by a verification procedurethat makes use of a hedging argument and duality. An analogous verificationprocedure will be applied for the absolutely continuous case.
The following proposition gives a complete characterization of conditionsCs
q and Caq under assumption (H). Its proof will also be postponed to the
appendix.
8 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Proposition 2.4. Suppose (H), q > 1, and P is not a martingale measure.Then,
Cq ⇒ Csq ⇔
∫
x≥1epxν(dx) < ∞
⇒ Caq ⇔
(∫
x≥1epxν(dx) < ∞
)or
(∫
R0
(ex − 1)− x1|x|≤1ν(dx) + (b +12σ2) > 0
)
where p = q/(q − 1) is the conjugate exponent to q.
Example 2.1. Suppose (H) holds and P is not a martingale measure. Weconsider models in which ν(dx) behaves (up to a slowly varying function) ase−λ+xdx for x → ∞. This tail behavior of the Levy measure is inherent inthe Kou model, the variance gamma model, tempered stable models, and intypical generalized hyperbolic models. We shall also assume that∫
R0
(ex − 1)− x1|x|≤1ν(dx) + (b +12σ2) > 0,
which is equivalent to the existence of the minimal entropy martingale mea-sure in this situation, see Fujiwara and Miyahara (2003) and Theorem 2.6below. Then,
• Cq fails for all q > 1.• Cs
q holds for q such that p := q/(q − 1) < λ+ and fails for q suchthat p > λ+.
• Caq holds for all q > 1.
In fact, in this situation ξq from condition Caq is negative. Therefore, the
qEMM does not exist, and the qAMM and the qSMM are different in therange of q, where Cs
q is satisfied.
The specific form of the q-optimal martingale measures derived in The-orem 2.3 can also be exploited to state results on the non-existence of theq-optimal equivalent martingale measure. This is exemplified by the follow-ing corollary:
Corollary 2.5. Suppose that (H) holds. If, for some q > 1, Csq holds and
the qSMM is not equivalent, then the qEMM does not exist.
Proof. By Proposition 2.4, Csq implies Ca
q . Let us suppose that the qEMMexists. Then the qAMM coincides with the qEMM and consequently eg
q(x) >
0 ν-almost surely. This implies that the qAMM and the qSMM are identical,and therefore the qSMM is equivalent, since the qAMM is. ¤
Our study on the convergence of the q-optimal signed martingale measuresto the minimal entropy martingale measure for exponential Levy models willmake use of the following characterization of the minimal entropy measurebased on Condition C.
q-OPTIMAL MARTINGALE MEASURES 9
Condition C:There exists a vector θe ∈ Rn satisfying∫
Rn0
‖(ex − 1)eθ′e(ex−1) − x1‖x‖≤1‖ν(dx) < ∞ (10)
and
σσ′θe +∫
Rn0
(ex − 1)eθ′e(ex−1) − x1‖x‖≤1ν(dx) = β. (11)
Theorem 2.6. (i) If condition C is satisfied, then the minimal entropymartingale measure is given by
E(θ′eσ, eθ′e(ex−1) − 1).
(ii) If n = 1 and there is no θe satisfying C, then the minimal entropymartingale measure does not exist.
Item (i) is due to Fujiwara and Miyahara (2003) for n = 1 and Escheand Schweizer (2005) for the multidimensional case. Item (ii) was provedby Hubalek and Sgarra (2006).
In the one-dimensional case, we derive the following result concerningthe convergence of the q-optimal martingale measures for exponential Levymodels, which will be proved in Section 5:
Theorem 2.7. Suppose (H) holds and the minimal entropy martingale mea-sure exists.(i) If θe specified in condition C satisfies θe > 0 and∫
x≥1e(θe+δ)ex
ν(dx) < ∞ (12)
for some δ > 0, then Cq is satisfied for sufficiently small q > 1 and theq-optimal equivalent martingale measures converge to the minimal entropymartingale measure in Lr(P ), for some r > 1, as q ↓ 1 (in the sense thatthe densities converge). Moreover, convergence holds in entropy.(ii) If θe specified in condition C satisfies θe < 0, then:
a. Caq is satisfied for all q > 1 and the q-optimal absolutely continu-
ous martingale measures converge to the minimal entropy martingalemeasure in Lr(P ), for some r > 1, as q ↓ 1.
b. If additionally ∫
x≥1e−0.28θeex
ν(dx) < ∞, (13)
then Csq is satisfied for all q > 1 and the analogous statement of
convergence holds for the q-optimal signed martingale measures.c. If additionally the upward jumps heights are bounded, then Cq is sat-
isfied for sufficiently small q, the q-optimal signed martingale mea-sures are equivalent and convergence holds in entropy as well.
10 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Remark 2.3. Suppose (H) holds.(i) If θe = 0, then P itself is the qEMM for all q > 1 and the MEMM. Thus,convergence is trivial.(ii) Item (i) and (iic) of Theorem 2.7 significantly generalize an example inJeanblanc et al. (2007), which requires that jumps are of finite activity andupward jumps heights are bounded.(iii) If in item (iib) of Theorem 2.7 the upward jump heights are not bounded,then the q-optimal signed martingale measures are not absolutely continuouswith respect to P and nonetheless convergence in Lr(P ) holds. Clearly,convergence in entropy is not a meaningful concept in this situation.(iv) If the density of the MEMM has a (1 + ε)th moment for some ε > 0,then θe < 0 or (12) is satisfied.(v) The constant 0.28 in condition (13) can be replaced by any constanta > 0 such that y ≤ e(a−δ)y + 1 for all y ∈ R and some δ > 0.(vi) If in the models of Example 2.1 the MEMM exists, then θe < 0 (or P isa martingale measure) and therefore item (iia) of Theorem 2.7 is in force.
3. Verification of the q-optimal signed solution
This section is devoted to the proof of Theorem 2.3 (i). As a first stepwe derive a verification theorem for the q-optimal signed martingale mea-sure. This theorem does not rely on the Levy setting, but holds for generalsemimartingale models.
Theorem 3.1. Suppose Z ∈ Dqs , q > 1, p = q/(q− 1) and, for some x < p,
the contingent claim
X(p)(Z) := p− sgn(ZT ) |ZT |qp
(p− x
E(|ZT |q)
)(14)
is replicable with a predictable strategy ϑ (the number of shares held), suchthat
‖ϑ‖Lp(M) := ‖(∫ T
0ϑd[M ]tϑ′)
12 ‖Lp(Ω,P ) < ∞, (15)
‖ϑ‖Lp(A) := ‖∫ T
0|ϑdAt|‖Lp(Ω,P ) < ∞. (16)
Then Z is the density process of the q-optimal signed martingale measure.
From now on, A(p) denotes the set of predictable process ϑ such that(15)–(16) holds.
Remark 3.1. Suppose q > 1, p = q/(q − 1), ϑ ∈ A(p), x ∈ R, and considerthe wealth process
Vt(ϑ, x) = x +∫ t
0ϑudSu.
q-OPTIMAL MARTINGALE MEASURES 11
Then, by the Burkholder-Davis-Gundy inequality, ZtVt(ϑ,X) is a martingalefor every Z ∈ Dq
s. In particular,
E[ZT VT (ϑ, x)] = x.
Therefore, a direct calculation shows, that the initial capital required to repli-cate X(p)(Z) in (14) with a strategy of class A(p) is x (provided replicationis possible).
Proof of Theorem 3.1. We consider the following maximization problemswith utility function up(x) = −|1− x
p |p:Max1 : X(1) := argmaxE(up(X)); X ∈ Lp s.t. E(ZT X) ≤ xMax2 : X(2) := argmaxE(up(X)); X ∈ Lp s.t. ∀Z ∈ Dq
s : E(ZT X) ≤ xMax3 : X(3) := argmaxE(up(X)); X ∈ Θ(p),xwhere
A straightforward calculation shows that the convex dual of up is given by
up(y) = (p− 1)|y|q − py. (18)
Therefore standard duality theory can be applied to verify that X(p)(Z) isthe maximizer of problem Max1. The details are carried out in Kohlmannand Niethammer (2007) for q = 2m/(2m − 1). In particular, we observethat all inequalities turn into identities in formula (17).
In a next step we exploit that the problem Max2 is dominated by its dualminimization problem (see e.g. Luenberger, 1969, p. 225), and thus, by (18),
E(up(X(Z))) = E(up(X(2))) ≤ infZ∈Dq
s , y≥0(E(up(y · ZT )) + xy)
= infy≥0
((p− 1)yq
(inf
Z∈Dqs
E[|ZT |q])− (p− x)y
)(19)
Define
yp =(
(p− x)(pE
(|ZT |q
))−1)p−1
.
AsX(p)(Z) = p− p sgn(ZT )|ypZT |
qp = (u′p)
−1(ypZT )and
up(y) = up((u′p)−1(y))− (u′p)
−1(y)y,
we have, due to Remark 3.1,
E(up(X(Z))) = E(up(yp · ZT )) + xyp.
12 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Consequently, the pair (Z, yp) is the minimizer of the problem on the righthand side of (19). This immediately implies that Z is the density process ofthe q-optimal signed martingale measure. ¤
We now apply the verification theorem to the exponential Levy modelsintroduced in Section 2. Suppose that q > 1 and that Cs
q holds. Define
Z = E(θ′qσ, egq − 1),
which is the candidate density for the q-optimal signed martingale measure.One can easily verify that Z belongs to Dq
s . Indeed, equation (6) guaranteesthat SZ is a martingale, see Kunita (2004) where this type of conditionis referred to as the ‘martingale condition’. Moreover, a direct calculationshows that, thanks to (7), for all 1 ≤ κ ≤ q
|Et(θ′qσ, egq − 1)|κ = Et(κθ′qσ, |egq|κ − 1)
× et κ2−κ
2θ′qσσ′θq+t
∫Rn0|egq(x)|κ−1−κ(egq(x)−1)ν(dx)
(20)
making use of the identity sgn(Et(θ′qσ, egq−1)) =∏
s≤t sgn(egq(∆Xs)). Tak-ing expectation in the previous formula with κ = q yields the q-integrabilityof Z, whence Z ∈ Dq
s .Therefore, in order to prove Theorem 2.3, it remains to show that X(p)(Z)
is replicable within the class A(p) for some x < p. We will now derive areplicating strategy constructively. To this end we fix some x < p.
If X(p)(Z) is replicable, then its conditional expectation under every mar-tingale measure is equal to the price process of X(p)(Z). We define thecandidate Yt for the price process via the relation
Since p(q − 1) = q, (20) yields that Et((q − 1)θ′qσ, (q − 1)θ′q(e· − 1)) is ap-integrable martingale under Cs
q . By Doob’s inequality, we immediatelyobserve that ϑ(p) satisfies (16). To prove (15), we first notice that, for aconstant Kq > 0,
E
[(∫ T
0ϑ
(p)t d[M ]t(ϑ
(p)t )′
) p2
]
≤ KqE
[[∫ ·
0Et−((q − 1)θ′qσ, (q − 1)θ′q(e
· − 1))
×d((q − 1)θ′qσWt + (q − 1)θ′q
∫ t
0
∫
Rn0
(ex − 1)N(dx, ds))] p
2
T
]
= KqE[[E((q − 1)θ′qσ, (q − 1)θ′q(e
· − 1))− 1] p
2
T
],
applying the Doleans-Dade formula for the last identity. By the Burkholder-Davis-Gundy inequality and since E((q−1)θ′qσ, (q−1)θ′q(e·−1)) is p-integrable,the expression on the right hand side is finite. ¤
4. Verification of the q-optimal absolutely continuoussolution
The verification of the q-optimal absolutely continuous martingale mea-sure follows similar lines. However, the replication argument is to be re-placed by a super-replication argument. We first introduce the class ofadmissible strategies which we apply throughout this section. We say a pair
q-OPTIMAL MARTINGALE MEASURES 15
(ϑ,C) belongs to the class A(p,c), if ϑ is a predictable, S-integrable process,C is an RCLL, non-decreasing and adapted process, and
Vt(ϑ,C, x) := x +∫ t
0ϑudSu − Ct
satisfies: VT (ϑ,C, 0) ∈ Lp(Ω, P ) and ZtVt(ϑ,C, 0) is a P -supermartingalefor every Z ∈ Dq
a, q = p/(p− 1). This class of strategies is closely related tothe class H2 introduced by Schachermayer (2003).
Then, a contingent claim ξ is super-replicable with initial capital x withinthe class of strategies A(p,c), if ξ = VT (ϑ,C, x) for (ϑ,C) ∈ A(p,c).
Clearly, if ξ is super-replicable with initial capital x within the class ofstrategies A(p,c), then for all Z ∈ Dq
a
E[ZT ξ] = x + E[ZT VT (ϑ,C, 0)] ≤ x.
Theorem 4.1. Suppose Z ∈ Dqa, q > 1, p = q/(q− 1) and, for some x < p,
the contingent claim
X(p)(Z) := p− |ZT |qp
(p− x
E(|ZT |q)
)
is super-replicable with initial capital x within the class A(p,c). Then Z is thedensity process of the q-optimal absolutely continuous martingale measure.
The proof is more or less the same as that of Theorem 3.1 by consideringthe modified maximization problems
Maxa1 : X(1) := arg maxE(up(X)); X ∈ Lp s.t. E(ZT X) ≤ x
3 : X(3) := arg maxE(up(X)); X is super-replicable with initial
capital x within the class of strategies A(p,c).In order to prove Theorem 2.3 (ii), we assume Ca
q and consider the candidatedensity
Z = E(θ′qσ, egq− 1).
The martingale condition (8) and the integrability condition (9) ensure thatZ ∈ Dq
a. The construction of the superhedge is similar to the constructionof the hedge in the previous section. We now define
Y at := p− (p− x)Ma
t Aat ;
Mat := Et((q − 1)ξ′qσ, ((q − 1)ξ′q(e
x − 1) + 1)+ − 1)
Aat := exp
(q − 1)ξ′q
∫ t
0S−1
u−dAu + t
∫
Rn0
((q − 1)ξ′q(ex − 1) + 1)− ν(dx)
.
16 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Straightforward modifications to the arguments in the previous section showthat
Y at = x +
∫ t
0ϑ(p,a)
u dSu − C(p,a)t , Y a
T = X(p)(Z),
where
ϑ(p,a)t := −p− x
p− 1Et−((q − 1)ξ′qσ, ((q − 1)ξ′q(e
· − 1) + 1)+ − 1)
× exp
t
∫
Rn0
((q − 1)ξ′q(ex − 1) + 1)− ν(dx)
× exp
(q − 1)ξ′q
∫ t
0S−1
u−dAu
ξ′qS
−1t− (21)
C(p,a)t := (p− x)
∫ t
0
∫
Rn0
Mau−Aa
u((q − 1)ξ′q(ex − 1) + 1)−N(dx, du).(22)
Note, that the consumption term C(p,a)t can be rewritten as
C(p,a)t = 1t≥τMa
τ−Aaτ ((q − 1)ξ′q(e
∆Xτ − 1) + 1)
where the stopping time τ is defined as
τ := inft ≥ 0; (q − 1)ξ′q(e∆Xt − 1) + 1 ≤ 0.
This expression for C(p,a) shows that the consumption term is right-continuous.Moreover, it yields an intuitive interpretation. If, due to a large jump in thestock price, the wealth of the investor exceeds the level p, then she consumesthe surplus over p, stops investing into the stock and therefore has wealth pin the whole remaining period [τ, T ].
The proof of Theorem 2.3 (ii) will be completed by observing that thesuper-replicating strategy belongs to the classA(p,c) for p = q/(q−1). To thisend, it remains to show that ZtY
at is a supermartingale for all Z ∈ Dq
a. Wefix Z ∈ Dq
a and define dQ = ZT dP . Then,∫ t0 ϑ
(p,a)u dSu is a local martingale
under Q. We denote by τn a localizing sequence of stopping times. Then, thestopped processes Y a
t∧τnare supermartingales under Q, whence, for s ≤ t,
by Bayes formula
E[Zt∧τnY at∧τn
|Fs] = Zs∧τnEQ[Y at∧τn
|Fs] ≤ Zs∧τnY as∧τn
→ ZsYas , n →∞.
Finally, E[Zt∧τnY at∧τn
|Fs] → E[ZtYat |Fs] by dominated convergence, since
|Zt∧τnY at∧τn
| ≤ sup0≤u≤T
|Zu| sup0≤u≤T
|Y au |
and
E[ sup0≤u≤T
|Zu| sup0≤u≤T
|Y au |] ≤ E[ sup
0≤u≤T|Zu|q]1/qE[ sup
0≤u≤T|Y a
u |p]1/p.
q-OPTIMAL MARTINGALE MEASURES 17
The first factor is finite, since Z is a q-integrable martingale. For the sec-ond one, we observe that Aa
t is deterministic and bounded and Mat is a p-
integrable martingale due to (9), (following the same argument as in Lemma3.3). Therefore, the second factor is finite as well.
5. Convergence to the minimal entropy measure
In this section we will prove Theorem 2.7, hence convergence of the q-optimal signed, respectively absolutely continuous, martingale measures tothe minimal entropy martingale measure. Throughout the section we shallassume that the exponential Levy process is one-dimensional. The mainingredient of the proof is the convergence of the roots θq, resp. ξq to θe,which we provide in this section. Theorem 2.7 will then follow by somerather technical, but standard arguments, which are given in Appendix B.
We will first study some properties of the following mappings which aregeneralizations of those introduced in Jeanblanc et al. (2007):
Φ(s)(q, θ) = −β + σ2θ +∫
R0
((ex − 1)|(q − 1)θ(ex − 1) + 1)| 1
q−1
×sgn((q − 1)θ(ex − 1) + 1))− x1|x|≤1
)ν(dx)
and
Φ(a)(q, θ) = −β + σ2θ
+∫
R0
((ex − 1)((q − 1)θ(ex − 1) + 1))
1q−1
+ − x1|x|≤1
)ν(dx)
which are, for fixed q > 1, defined on the domains
Dom(s)q := θ;
∫
x≥1(ex − 1)|(q − 1)θ(ex − 1) + 1| 1
q−1 ν(dx) < ∞,
respectively
Dom(a)q := θ;
∫
x≥1(ex − 1)((q − 1)θ(ex − 1) + 1)
1q−1
+ ν(dx) < ∞.
Moreover, we consider
Φe(θ) := Φ(1, θ) := −β + σ2θ +∫
R(ex − 1)eθ(ex−1) − x1|x|≤1ν(dx)
defined on
Dome := θ;∫
x≥1(ex − 1)eθ(ex−1)ν(dx) < ∞.
Note that
(−∞, 0] ⊂ Dome ⊂ Dom(a)q , Dom(s)
q ⊂ Dom(a)q . (23)
18 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Lemma 5.1. (i) Assume (H). Then the mappings Φ(i)(q, ·), i = s, a, andΦe are continuous and strictly increasing on their respective domains.(ii) For all q > 1, 0 ∈ Dom(i)
q ∩Dome and Φ(i)(q, 0) = Φe(0), i = s, a.(iii) Suppose q > 1, p = q/(q − 1) and∫
x≥1epxν(dx) < ∞.
Then, Dom(i)q = R for i = s, a.
Proof. (i) follows directly from the monotone convergence theorem.(ii) is trivial.(iii) Since 1
q−1 = p− 1, there is a constant Kp such that for x > 1 and θ ∈ R
(ex − 1)|(q − 1)θ(ex − 1) + 1| 1q−1 ≤ Kp
p− 1|θ|(ex − 1)p + Kp(ex − 1),
which is ν-integrable over (1,∞) by assumption. Hence, R ⊂ Dom(s)q ⊂
Dom(a)q . ¤
Lemma 5.2. Suppose θ ∈ Dome.(i) If θ ≥ 0, then θ ∈ Dom(i)
q for all q > 1 and
limq↓1
Φ(i)(q, θ) = Φe(θ), i = s, a.
(ii) Suppose θ < 0, and additionally in the case i = s∫
x≥1e0.28 |θ|ex
ν(dx) < ∞.
Then, for i = a, s, θ ∈ Dom(i)q for all q > 1 and
limq↓1
Φ(i)(q, θ) = Φe(θ).
Proof. (i) The trivial case θ = 0 is included in Lemma 5.1 (ii). Henceforthwe will suppose θ > 0. Since θ ∈ Dome we may conclude from Lemma 5.1(iii), that θ ∈ Dom(i)
q for all q > 1. Moreover, there is a q0 such that for all1 < q < q0, (q−1)θ(ex−1)+1 > 0 for all x ∈ R. Hence, Φ(s)(q, θ) = Φ(a)(q, θ)for all 1 < q < q0. Notice that ((q − 1)θ(ex − 1) + 1)
1q−1 is monotonically
decreasing in q < q0. Hence , by monotone convergence, we get as q ↓ 1,∫
x>0(ex − 1)((q − 1)θ(ex − 1) + 1)
1q−1 − x1|x|≤1ν(dx)
→∫
x>0(ex − 1)eθ(ex−1) − x1|x|≤1ν(dx),
∫
x<0(ex − 1)((q − 1)θ(ex − 1) + 1)
1q−1 − x1|x|≤1ν(dx)
→∫
x<0(ex − 1)eθ(ex−1) − x1|x|≤1ν(dx).
q-OPTIMAL MARTINGALE MEASURES 19
Since θ ∈ Dome, both limits on the right hand side are finite and, hence,can be combined to get
limq↓1
Φ(s)(q, θ) = limq↓1
Φ(a)(q, θ) = Φe(θ).
(ii) Now suppose θ < 0. By (23), θ ∈ Dom(a)q . For the signed case, the
additional integrability ensures that θ ∈ Dom(s)q for all q > 1 due to Lemma
5.1 (iii). Notice that for sufficiently small q, (q − 1)θ(ex − 1) + 1 > 0 for allx ≤ 1. Hence, we obtain, analogously to the case θ > 0,∫
x∈R0; x≤1(ex − 1)((q − 1)θ(ex − 1) + 1)
1q−1 − x1|x|≤1ν(dx)
→∫
x∈R0; x≤1(ex − 1)eθ(ex−1) − x1|x|≤1ν(dx) (24)
as q ↓ 1. We shall first treat the case i = s. In view of (24), it remains toshow ∫
x>1(ex − 1)|(q − 1)θ(ex − 1) + 1| 1
q−1 sgn((q − 1)θ(ex − 1) + 1)ν(dx)
→∫
x>1(ex − 1)eθ(ex−1)ν(dx)
as q tends to one. This follows from the dominated convergence theorem,since on the set x ≥ 1; (q − 1)θ(ex − 1) ≥ −2
|((q − 1)θ(ex − 1) + 1)| 1q−1 ≤ 1
and on the set x ≥ 1; (q − 1)θ(ex − 1) < −2|(q − 1)θ(ex − 1) + 1| 1
q−1 ≤ ((q − 1)|θ|(ex − 1)− 1)1
q−1 ≤ e0.279 |θ|(ex−1),
since y ≤ e0.279 y + 1 for all y ∈ R.In the case i = a, one has∫
x>1(ex − 1)((q − 1)θ(ex − 1) + 1)
1q−1
+ ν(dx) →∫
x>1(ex − 1)eθ(ex−1)ν(dx)
by dominated convergence, since ((q − 1)θ(ex − 1) + 1)+ ≤ 1 for x > 1. ¤Remark 5.1. We shall argue that the estimates used to justify dominatedconvergence in the above proof for i = s are rather tight. A direct calculationshows that
maxq>1
((q − 1)|θ|(ex − 1)− 1)1
q−1 = ec
1−c|θ|(ex−1)
where c solves the equation
e1/(1−c) + 1 = 1/c.
Note that c/(1− c) ≈ 0.2785.
With this lemma at hand, we can prove the convergence of the roots θq
(resp. ξq) to θe:
20 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Theorem 5.3. (i) Under the assumptions of Theorem 2.7 (i), (iib) (resp.(iia)), Cs
q (resp. Caq ) is satisfied for all q > 1 and θq → θe (resp. ξq → θe)
holds as q ↓ 1.(ii) Under the assumptions of Theorem 2.7 (i), (iic), Cq is satisfied forsufficiently small q > 1.
Proof. (i) We first give the proof for the signed case. Assumption (12)-(13)and Proposition 2.4 guarantee that Cs
q is satisfied for all q > 1. Moreover,due to (23) under (iib), resp. assumption (12) under (i), we have θe− ε, θe +ε ∈ Dome for all sufficiently small 0 < ε < δ. Fix some 0 < ε < δ. Then, bythe strict monotonicity of Φe on its domain we derive
Φe(θe − ε) < 0 = Φe(θe) < Φe(θe + ε).
By the previous lemma, there is a q(ε) > 1 such that for all 1 < q < q(ε)
Φ(s)(q, θe − ε) < 0 < Φ(s)(q, θe + ε).
By continuity and monotonicity of Φ(s)(q, ·), the root of Φ(s)(q, ·), θq, isunique and satisfies θe − ε < θq < θe + ε. Hence |θq − θe| < ε for all1 < q < q(ε).
The argumentation for absolutely continuous case is almost identical. Wemerely note that, for θe < 0, by the monotonicity of Φe∫
R0
(ex − 1)− x1|x|≤1ν(dx) + (b +12σ2) = Φe(0) > 0
holds. Therefore, Caq is fulfilled for all q > 1 by Proposition 2.4.
(ii) We need to verify (3) for θq and sufficiently small q. If θe > 0 orν((K,∞)) = 0 for some K > 0, then, by (i), for every sufficiently smallε > 0 there is a q0 > 1 such that for all 1 < q < q0 and ν-almost every x,
0 < miny∈[θe−ε,θe+ε]
(q − 1)y(ex − 1) + 1 ≤ (q − 1)θq(ex − 1) + 1
¤
The convergence of the roots θq (resp. ξq) to θe is the main ingredient forthe convergence of the measures. Indeed, with this result at hand, Theorem2.7 can be proved along the lines presented in Niethammer (2008). Nonethe-less we give a complete proof in Appendix B that takes the additional diffi-culties stemming from possibly non-positive densities into account.
6. Some consequences for portfolio optimization
In this section we collect some consequences for the portfolio selectionproblem with utility function
up,α(x) := −∣∣∣∣1−
αx
p
∣∣∣∣p
, 1 < p < ∞, α > 0.
q-OPTIMAL MARTINGALE MEASURES 21
These utility functions are particularly interesting, when p = 2m for somem ∈ N. E.g., for p = 2,
u2,α(x) = αx− α2
4x2 − 1.
Depending on the risk aversion parameter α, one maximizes the expectedterminal wealth while penalizing potential risk from a large second moment.This is intimately related to the Markowitz problem of identifying efficientportfolios, see e.g. Markowitz (1959), and to mean variance hedging, seee.g. Duffie and Richardson (1991).
In the presence of jumps the second moment may be insufficient to mea-sure risk, but higher moments can contain useful information about the riskof large downwards jumps. Similarly, to the case p = 2, for p = 2m theutility function u2m,α rewards large odd moments up to order 2m − 1 andpenalizes the even moments up to order 2m. For instance,
u4,α(x) = αx +α3
16x3 − 3α2
8x2 − α4
256x4 − 1
additionally rewards a non-centralized version of the skewness and penalizesa non-centralized version of the kurtosis.
We therefore consider the portfolio selection problems (with and withoutconsumption), for x < p/α,
arg maxE[up,α(x +∫ T
0ϑudSu)]; ϑ ∈ A(p), (25)
arg maxE[up,α(x +∫ T
0ϑudSu − CT )]; (ϑ,C) ∈ A(p,c), (26)
where the classes of strategies A(p) and A(p,c) where introduced in Sections3 and 4, respectively. Our findings complement earlier results by Kohlmannand Niethammer (2007) and Niethammer (2008):
(1) Suppose q = p/(p− 1) is the conjugate exponent to p. If Csq holds, then
the optimal portfolio for problem (25) with α = 1 was identified throughoutthe verification procedure in Section 3. A straightforward generalization toα > 0 yields
ϑ(p,α)t = −
pα − x
p− 1Et((q − 1)θ′qσ, (q − 1)θ′q(e
· − 1))e(q−1)θ′q∫ t0 S−1
u−dAuθ′qS−1t−
as the optimal portfolio.
22 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Analogously, applying the results of Section 4, under Caq , an optimal pair
of investment and consumption processes for the problem (26) is given by
ϑ(p,α,a)t := −
pα − x
p− 1Et−((q − 1)ξ′qσ, ((q − 1)ξ′q(e
· − 1) + 1)+ − 1)
×et∫Rn0((q−1)ξ′q(ex−1)+1)− ν(dx)+(q−1)ξ′q
∫ t0 S−1
u−dAuξ′qS
−1t−
C(p,α,a)t :=
( p
α− x
) ∫ t
0
∫
Rn0
Mau−Aa
u((q − 1)ξ′q(ex − 1) + 1)−N(dx, du).
In the case ν(egq(x) < 0) > 0, these optimal strategies for problems(25) and (26) differ. This is due to the fact that, employing the strategyϑ(p,α), the corresponding wealth process may exceed the optimal level p/αof the utility function up,α. Overshooting this level is penalized by theutility function. Contrarily, if consumption is allowed as in problem (26),at the first time, when the level p/α is exceeded, the surplus over p/α isconsumed and investment is stopped thereafter. In this way the optimallevel p/α is maintained for the remaining time. Therefore, the disadvantageof the utility function up,α of not being everywhere increasing is resolved forthe investment/consumption problem. The optimal wealth process neverreaches the region in which up,α is decreasing, and the level p/α can beinterpreted as the amount of money at which an investor is saturated anddoes not gain additional utility from higher wealth.
(2) It is straightforward to check that, under the assumptions of Theorem2.7, ϑ(p,α) and ϑ(p,α,a) converge to
ϑ(∞,α)t = −θe
αS−1
t−
uniformly on [0, T ] in probability, as p tends to infinity, owing to Theorem5.3. Moreover, C(p,α,a) converges to zero almost surely uniformly on [0, T ].Therefore, by the continuity of the stochastic integral, the wealth processescorresponding to the optimal strategies of both problems (25) and (26) con-verge to
x +∫ t
0ϑ(∞,α)
u dSu
uniformly in probability as p goes to infinity. A direct calculation, makinguse of the semimartingale decomposition of S and Theorem 2.6, yields
∫ T
0ϑ(∞,α)
u dSu = − 1α
(log E(θeσ, eθe(e·−1) − 1) + const.
).
By martingale considerations under the minimal entropy measure, the con-stant must coincide with the relative entropy of the minimal martingalemeasure with respect to P and consequently x +
∫ T0 ϑ
(∞,α)u dSu is the op-
timal payoff of the exponential utility problem with risk aversion α andinitial endowment x. Note that the optimal portfolio ϑ(∞,α) for the expo-nential problem can already be found in Kallsen (2000). Different choices of
q-OPTIMAL MARTINGALE MEASURES 23
portfolio classes for the exponential problem are discussed in Schachermayer(2003) and Biagini and Fritelli (2007).
We have, thus, in generalization of results in Niethammer (2008), derivedconvergence of the portfolio selection problem (with and without consump-tion) for the sequence of utility functions up,α to the exponential utilityproblem even in situations where the q-optimal martingale measures neednot be equivalent.
Appendix A. Proofs of Propositions 2.2 and 2.4
Throughout Appendix A we assume that (H) holds and P is not a mar-tingale measure.
We start with the proof of Proposition 2.2.
Proof of Proposition 2.2. (i) Assume that Cq holds for some q > 1, andhence the q-optimal equivalent martingale measure exists. Moreover, sup-pose that, for all θ > 0,
∫
x≥1eθex
ν(dx) = ∞ (27)
and the MEMM exists. Then, due to Theorem 2.6 (ii), Φe has a zeroθe on its domain Dome. Since, by (27), Dome = (−∞, 0], and P is nota martingale measure, we get θe < 0. Hence, by Lemma 5.1, we haveΦ(s)(q, 0) = Φe(0) > 0, and consequently the zero θq of Φ(s)(q, ·) is negative.As the upward jump heights are not bounded, thanks to (27), we have acontradiction to (3).
(ii) Suppose that Cq holds for some q, (5) fails, and upward jump heightsare unbounded. Then we can again conclude from Lemma 5.1 that θq isnegative contradicting (3) as in (i). ¤
It remains to give the proof of Proposition 2.4.
Proof of Proposition 2.4. The implication ‘Cq ⇒ Csq ’ is obvious. Suppose
now that Csq holds for some q > 1. Then, by definition,
are non-decreasing. For x < 0 and θ → −∞ both mappings coincide andconverge to −∞. For x > 0 and θ → +∞ both mappings coincide andconverge to +∞. This implies (30) due to (H).
If we do not assume (28), the same argument still shows that
limθ↓−∞
Φ(a)(q, θ) = −∞,
because (−∞, 0] ⊂ Dom(a)q by (23). If we now assume (29), i.e. Φ(a)(q, 0) >
0, then there is a root ξq < 0 of Φ(a)(q, ·) and thus Caq is satisfied. ¤
Appendix B. Proof of Theorem 2.7: the technical details
In this appendix we show how to derive the convergence of the q-optimalsigned martingale measures to the minimal entropy martingale measure fromthe convergence of the corresponding roots. The latter convergence wasproved in Theorem 5.3 (i). The argumentation for the absolutely continuousis analogous and even a little bit easier. We therefore omit it.
Theorem B.1. Under the conditions of Theorem 2.7 (i), (iib), the densitiesof the qSMM converge to that of the MEMM in probability as q ↓ 1.
Proof. Recall that θq → θe and therefore pointwise egq(x) → eθe(ex−1), asq ↓ 1. By Yor’s formula we can decompose
ET (θqσ, egq − 1) = ET (θqσ, 0) ET (0, 1|·|≤1(egq − 1)) ET (0, 1|·|>1(egq − 1)).
q-OPTIMAL MARTINGALE MEASURES 25
Obviously, we have P -almost surely,
ET (θqσ, 0) → ET (θeσ, 0). (31)
Moreover,
ET (0, 1|·|>1(egq − 1)) = e−T
∫|x|>1[egq(x)−1] ν(dx)
∏
s≤T
1|∆Xs|>1egq(∆Xs).
The product consists pathwise of finitely many factors only, whence∏
s≤T
1|∆Xs|>1egq(∆Xs) →∏
s≤T
1|∆Xs|>1eθe(eXs−1), P -a.s.
Moreover, |egq(x)| is dominated on x, |x| > 1 by a ν-integrable function.Indeed, if θe < 0, this follows as in the proof of Lemma 5.2. For θe > 0 andq sufficiently small, we obtain, by (12),
|egq(x)| ≤ eθq(ex−1) ≤ e(θe+δ)(ex−1).
Hence, we get P -almost surely
ET (0, 1|·|>1(egq − 1)) → ET (0, 1|·|>1(eθ(e·−1) − 1)). (32)
To treat the remaining factor, note that there is an ε > 0 depending on θe
such that egq(x) > ε for all x, |x| ≤ 1, provided q is sufficiently small. Thus,
ET (0, 1|·|≤1(egq − 1)) = exp
∫ T
0
∫
|x|≤1log(egq(x))N(dx, ds)
× exp
−T
∫
|x|≤1[egq(x)− 1− log(egq(x))] ν(dx)
=: e(I)+(II)
To justify dominated convergence for the term (II), note that by Taylor’stheorem
log(egq(x)) = (egq(x)− 1)− 12ξ(x)2
(egq(x)− 1)2,
egq(x) = 1 +((q − 1)θq(eη(x) − 1) + 1
)1/(q−1)θq(ex − 1)
with intermediate points ξ(x), η(x) between 1 and egq(x) and between 0and x, respectively. Consequently, for all x, |x| ≤ 1, and sufficiently small q(independent of x),
|egq(x)− 1− log(egq(x))| ≤ 12ε2
|θe + δ|2e4|θe+δ|(ex − 1)2,
which is ν-integrable over x, |x| ≤ 1. Thus,∫
|x|≤1[egq(x)−1− log(egq(x))] ν(dx) →
∫
|x|≤1[eθ(ex−1)−1− θ(ex−1)] ν(dx).
26 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
Concerning (I), we obtain from the isometry property of the stochastic in-tegral,
E
∣∣∣∣∣∫ T
0
∫
|x|≤1[log(egq(x))− θe(ex − 1)] N(dx, ds)
∣∣∣∣∣2
= T
∫
|x|≤1| log(egq(x))− θe(ex − 1)|2 ν(dx).
The right hand side converges to zero by dominated convergence, since theintegrand can be estimated for sufficiently small q by 2(|θe| + δ)2(ex − 1)2.We, thus, obtain, in probability,
ET (0, 1|·|≤1(egq − 1)) → ET (0, 1|·|≤1(eθ(e·−1) − 1)). (33)
Combining (31)–(33) yields the assertion thanks to Yor’s formula. ¤
Finally, we generalize from convergence in probability to Lr(Ω, P )-conver-gence for some r > 1, and to convergence in entropy in the appropriate cases.
Proof of Theorem 2.7. We first prove that the convergence stated in Theo-rem B.1 also holds in Lr(Ω, P ) for some r > 1. In view of Theorem B.1 itsuffices to show that there are r, q0 > 1 such that the family
(∣∣∣ET (θqσ, egq − 1)− ET (θeσ, eθe(ex−1))∣∣∣r)
1<q≤q0
is uniformly integrable.Choosing κ > 1 and 1 < q ≤ q0 sufficiently small, we get
∫
R0
||egq(x)|κ − 1− κ(egq(x)− 1)| ν(dx) ≤ K(κ) (34)
for some constant K(κ) independent of q. Indeed, as in the previous proof1|x|>1|egq(x)|κ is dominated by an ν-integrable function for sufficiently smallκ and 1 < q ≤ q0. A straightforward application of Taylor’s theorem showsthat 1|x|≤1||egq(x)|κ − 1 − κ(egq(x) − 1)| is dominated by an ν-integrablefunction as well. Hence, (34) follows and implies that
|ET (θqσ, egq − 1)|κ = ET (κθqσ, |egq|κ − 1)
× exp
T
(κ2 − κ
2θ2qσ
2 +∫
R0
|egq(x)|κ − 1− κ(egq(x)− 1) ν(dx))
.
As the stochastic exponential of a Levy martingale is a martingale, we havefor all 1 < q ≤ q0
E[∣∣ET (θqσ, egq − 1)
∣∣κ]
≤ exp
T
(κ2 − κ
2θ2qσ
2 +∫
R0
| |egq(x)|κ − 1− κ(egq(x)− 1)| ν(dx))
≤ K(κ). (35)
q-OPTIMAL MARTINGALE MEASURES 27
In view of either (12) or the condition θe < 0, we can assume without lossof generality that κ is small enough such that
E[∣∣∣ET (θeσ, eθe(ex−1))
∣∣∣κ]
= exp
T
(κ2 − κ
2θ2eσ
2 +∫
R0
|eκθe(ex−1) − 1− κ(eθe(ex−1) − 1)| ν(dx))
< ∞.
Combining these two estimates, the de la Valle-Poussin criterion yields theuniform integrability for all 1 < r < κ. This proves (iib) and, in view ofTheorem 5.3 (ii), the Lr-convergence stated in item (i).The proof of item (iia) is completely analogous to that of item (iib).
It remains to show the convergence in entropy stated in (i) and (iic).We assume that q is sufficiently small such that Cq is satisfied thanks toTheorem 5.3 (ii). We shall prove that the relative entropy
Taking the definition of egq and the convergence of θq to θe into account,the terms in the second and third line converge to zero in L2m(Ω, P ) for allm ∈ N. (Of course, the term in the third line requires a, by now standard,application of the dominated convergence theorem). To treat the remainingterm, we note that
X(q)t :=
∫ t
0
∫
R0
[log(egq(x))− θe(ex − 1)] N(dx, ds)
is a zero expectation Levy process with Levy measure
νq(A) =∫
R0
1(log(egq(x))−θe(ex−1))∈A ν(dx).
28 CHRISTIAN BENDER AND CHRISTINA R. NIETHAMMER
The (2m)th moment of X(q)T can be calculated by differentiating its char-
acteristic function. Performing this differentiation explicitly shows thatthe (2m)th moment of X
(q)T tends to zero for q ↓ 1, if and only if for all
k = 2, 3, . . . , 2m∫
R0
xk νq(dx) =∫
R0
(log(egq(x))− θe(ex − 1))k ν(dx) → 0 q ↓ 1.
This follows by a routine application of the dominated convergence theorem.¤
Acknowledgement
The paper benefited from the helpful comments by Walter Schachermayer,Michael Kohlmann, Yoshio Miyahara, and an anonymous referee.
CN gratefully acknowledges financial support by UniCredit, Markets andInvestment Banking. However, this paper does not reflect the opinion ofUniCredit, Markets and Investment Banking, it is the personal view of theauthors.
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Christian Bender, Institute for Mathematical Stochastics, TU Braunschweig,Pockelsstr. 14, D-38106 Braunschweig, Germany.Email: [email protected]
Christina R. Niethammer, Department of Mathematics and Statistics, Uni-versity of Konstanz, D-78464 Konstanz, Germany.Email: [email protected]
