Existence of the q -Optimal Equivalent Martin- Martin-gale Measure in the Presence of Unbounded

Jumps

In this section, we ask whether theq-optimal equivalent martingale measure qEMM exists if theq-optimal signed martingale measure qSMM is strictly signed, i.e. P(Z_q<0)>0.

From convex analysis, we know that

Q∈Mmin^qa

E

dQ dP

q

has a solution, qAMM, ifM^qa6=∅. (M^qa resp. Da^q is closed, the functional is convex and sufficiently smooth.) If the solution is an equivalent martingale measure, it coincides with the qEMM.D^qeis not closed so it may happen that the qEMM does not exist. This is the case, if the qAMM is not equivalent as De^q is dense in D^qa. The above results already show that under C_q^A, the qAMM is not equivalent when C_q⁺ fails (see Corollary 4.3.4). We give an alternative, maybe more intuitive proof in this section. We show that under some minor assumptions the qEMM does not exist if the qSMM is already signed, i.e. the density has negative points with positive probability:

Theorem 4.6.1. Suppose σ has full rank and for some q >1 condition C_q⁻ is satisfied, but C_q⁺ is not. Then the equivalent q-optimal measure qEMM does not exist, but the qSMM does.

The last assertion is implied by Theorem 4.2.2. From Jeanblanc et al. [69, Theorem 2.6.3], we know if the qEMM exists, it has deterministic Girsanov parameters, see Section 3.2.2. Hence, we just need to look for an equivalent version within the class of densities with deterministic Girsanov parameters. To prove Theorem 4.6.1, we need to show:

For every deterministic Girsanov parameters f and h(·) such that E¯(f, h− 1)∈ D^eq there exist f˜andh(˜ ·) such that E¯( ˜f ,˜h−1)∈ D^qe and

E( ¯E( ˜f ,˜h−1)^q)< E( ¯E(f, h−1)^q).

Intuitively, the proof works as follows: we choose ˜f and ˜h(·)−1 as a convex combination of (f, h−1) and the optimal Girsanov parameters of the qSMM, (θ_q^′σ,eg_q−1). Every convex combination not equal to (f, h−1) then leads to a strictly smallerq-moment than the suggested pair (f, h−1). We finally need to push ˜f and ˜h(·)−1 close enough tof andh−1 such that ¯E( ˜f ,h˜−1)>0.

Some technicalities are necessary to show that this is actually possible, i.e.

that we do not suddenly jump out of the set equivalent martingale measures.

For the rest of the proof, we setf^(q) =θ^′_qσ. We establish a lemma before we give a proof of Theorem 4.6.1:

Lemma 4.6.2. We have

1. E¯(f, h−1)∈ D^qe, if and only if h >0, ν-a.s., σf+

Z

R^N\{0}

((e^x−1)h−x1_kxk≤1)ν(dx) =β, (4.46) and

Z

RN\{0}|h^q(x)−1−q(h(x)−1)|ν(dx)<∞. (4.47) 2. Let (f1, h1) and (f0, h0) two pairs satisfying (4.46) and (4.47) then for an arbitrary convex combination f_η := ηf₁+ (1−η)f₀ and h_η :=

ηh₁+ (1−η)h₀, η∈(0,1), the pair(f_η, h_η) satisfies (4.46) and (4.47).

3. If E(|E¯(f1, h1 −1)|^q) < E(|E¯(f0, h0 −1)|^q) then also E(|E¯(fη, hη − 1)|^q)< E(|E¯(f₀, h₀−1)|^q) for η ∈(0,1) andf_η, h_η as defined in (2).

Proof. For item 1 see [69, Theorem 2.7].

We continue with item 2: Firstly, we need to check the martingale condition:

σ(ηf₁+ (1−η)f₀) + Z

RN\{0}

((e^x−1)(ηh₁+ (1−η)h₀)−x1_kxk≤1)ν(dx)

= η(σf₁+ Z

RN\{0}

((e^x−1)h₁−x1_kxk≤1)ν(dx)) +(1−η)(σf₀+

Z

RN\{0}

((e^x−1)h₀−x1_kxk≤1)ν(dx))

= ηβ+ (1−η)β =β.

As we have built a convex combination, we obtain sufficient integrability:

Z

R^N\{0}

((ηh₁+ (1−η)h₀)^q−1−q(ηh₁+ (1−η)h₀−1))ν(dx)

≤ Z

R^N\{0}

((ηh^q₁+ (1−η)h^q₀)−1−q(ηh₁+ (1−η)h₀−1))ν(dx)

= η Z

R^N\{0}

((h^q₁−1)−q(h₁−1))ν(dx) +(1−η)

Z

R^N\{0}

((h^q₀−1)−q(h₀−1))ν(dx)

< ∞ (4.48)

Lemma 4.2.1 together with (4.48) implies item 3:

E[|E¯(f_η, h_η−1)|^q] = e^{Tq(q−1)2 fη′fη}e^T

R

RN\{0}(|hη(x)|^q−1−q(hη(x)−1))ν(dx)

≤ E[|E¯(f₀, h₀−1)|^q]^1−η(e^{Tq(q−1)2 f1′f1})^η

×(e^T

R

RN\{0}(|h1(x)|^q−1−q(h1(x)−1))ν(dx)

)^η

= E[|E¯(f₀, h₀−1)|^q]^1−ηE[|E¯(f₁, h₁−1)|^q]^η

< E(|E¯(f₀, h₀)|^q).

Idea: The proof of Theorem 4.6.1 is divided into several steps: firstly, we choose an arbitraryf andh(·) such that ¯E(f, h−1)∈ De^q. We then setf₀ = f, and h₀ =h and look for functionsf₁ and h₁ such that E(|E¯(f₁, h₁)|^q)<

E(|E¯(f0, h0)|^q). We build the convex combination fη, hη for an η ∈ (0,1).

By Lemma 4.6.2, ¯E(f_η, h_η −1) then also possesses a smaller q-th moment then ¯E(f, h−1). Its Girsanov parameters satisfy condition (4.46) and (4.47).

It remains to choose f₁,h₁, andη such thath_η >0 : so suppose for the moment thath > ǫ for anǫ >0. We then set

h₁=h⁽ⁿ⁾₁ = 1_kxk≤neg_q andf₁=f₁⁽ⁿ⁾=ζ_n^′σ with

ζ_n= (σσ^′)⁻¹(β− Z

RN\{0}

((e^x−1)h⁽ⁿ⁾₁ −x1_kxk≤1)ν(dx))

for large enoughnsuch thatE( ¯E|f₁⁽ⁿ⁾, h⁽ⁿ⁾₁ −1)|^q)< E(|E¯(f , h−1)|^q). This is possible ifE(|E¯(f₁⁽ⁿ⁾, h⁽ⁿ⁾₁ −1)|^q)→E(|E¯(f^(q),eg_q−1)|^q),as ¯E(f^(q),eg_q−1)) is optimal. Finally, ash⁽ⁿ⁾₁ is a continuous function on a compact set andh

cannot come arbitrarily close to zero on non-negligible sets we can chooseη close enough to zero such that ˜h:=ηh⁽ⁿ⁾₁ + (1−η)h >0 and ˜f = ˜ζ^′σ with

ζ˜= (σσ^′)⁻¹(β− Z

R^N\{0}

((e^x−1)˜h−x1_kxk≤1)ν(dx)).

By Lemma 4.6.2.4, we have found our pair ˜f ,˜h that dominates f, h.

If we do not find an ǫas in the last step, we define J_ǫ ={x:h(x)≤ǫ}. (Note,h >0 a.s. henceν(T

J¹

n) = 0). We then choose hn = 1x∈J^c₁

n

1_kxk≤neg_q (4.49)

fn = ζ_n^′σ, ζn= (σσ^′)⁻¹(β− Z

RN\{0}

((e^x−1)hn−x1_kxk≤1)ν(dx)).(4.50) The result follows as in the previous step, if

E(|E¯(f₁⁽ⁿ⁾, h⁽ⁿ⁾₁ )|^q)→E(|E¯(θ^′_qσ,eg_q−1)|^q).

To make these argument rigorous, we have to prove that all functions exist and the last convergence result holds. In particular, we have to show that integral and limitation are interchangeable. We start with a lemma on the set J_n:

Lemma 4.6.3. For sufficiently large n, it holds that ν(J_n)<∞ and

nlim→∞ν(J_n) = 0.

Proof. Note, |h| = h. Choosen₀ (independent of q) such that |y^q−qy| ≤

|1/n^q₀−q/n₀| ≤(q−1)/2 for all 0< y ≤1/n₀. Then, forn≥n₀, q−1

2 ν(J_n) ≤ Z

Jn

(|q−1| − |h^q(x)−qh(x)|)ν(dx)

≤ Z

RN\{0}|h^q(x)−1−q(h(x)−1)|ν(dx)<∞, asR

RN\{0}|h^q(x)−1−q(h(x)−1)|ν(dx) <∞. Since the sequence (Jn) of sets is decreasing, we obtain

nlim→∞ν(J_n) =ν(\

n∈N

J_n).

However, the latter expression is zero, since h >0 ν-a.s..

Hence, we can fix an n₀ such that ν(J_n0) < ∞ for all n ≥ n₀, the definition of the pair (f_n, h_n) given in (4.49) and (4.50) makes sense. The integral inf_n exists forn≥n₀ as it can be rewritten as

Z

RN\{0}

((e^x−1)h_n(x)−x1_kxk≤1)ν(dx)

= Z

{1<kxk≤n}∩Jn

(e^x−1)eg_q(x)ν(dx)− Z

{kxk≤1}∩Jn

(e^x−1)eg_q(x)ν(dx) +

Z

kxk≤1

((e^x−1)eg_q(x)−x1_kxk≤1)ν(dx)

andν(Jn)<∞. Moreover, applying dominated convergence, Lemma 4.6.3, and the martingale condition (4.46), we obtain

n→∞lim ζ_n=θ_q ⇒ lim

n→∞f_n=f^(q) and lim

n→∞h_n= eg_q. (4.51) To complete the proof of Theorem 4.6.1, we need to show the following lemma:

Lemma 4.6.4. Supposen≥n₀ is sufficiently large andη >0(depending on n) is sufficiently small. Choose fn and hn as defined in (4.50) and (4.49).

Then,

Z^η,n:= ¯E(ηf_n+ (1−η)f, ηh_n+ (1−η)h−1)∈ D^qe

and

E[|Z_T^η,n|^q]< E[|E¯(f, h−1)|^q].

Proof. We denote the convex combinations in the definition of Z^η,n by fη,n

and h_η,n, respectively. We first show that Z^η,n ∈ De^q for all n ≥ n₀ and sufficiently small η depending on n by verifying the conditions in Lemma 4.6.2.1. As in the proof of Lemma 4.6.2 by convexity,

Z

R^N\{0}||h_η,n(x)|^q−1−q(h_η,n(x)−1)|ν(dx)

≤ η Z

R^N\{0}||h_n(x)|^q−1−q(h_n(x)−1)|ν(dx) +(1−η)

Z

R^N\{0}||h(x)|^q−1−q(h(x)−1)|ν(dx). (4.52) The second integral is finite by assumption, while the first one can be de-composed into

Z

Jn^c∩{kxk≤n}||eg_q(x)|^q−1−q(eg_q(x)−1)|ν(dx) + (q−1)ν(J_n∪ {kxk> n}).

The first term is finite by assumption, and so is the second one, because ν(Jn) < ∞ for n ≥n0. The integrability condition (4.47) is satisfied and

the martingale condition follows by construction of (f_n, h_n). Hence, by Lemma 4.6.2 the convex combination satisfies both conditions as well. It suffices to prove the positivity ofh_η,n for smallη. We have already observed that hn is bounded from below by some constant −an, an ≥ 0, since the continuous function eg_q is bounded on the compact set {kxk ≤ n}. We defineη_n:= (a_n+ 1/n)⁻¹/n≤1 and have on J_n^c

h_η,n≥ −ηa_n+ (1−η)/n >0

for all 0 ≤ η < η_n, as by construction of h_n, on J_n, h_η,n = (1−η)h > 0 ν-a.s.. Hence, for all η∈[0, ηn), we have ¯E(fη,n, hη,n−1)∈ De^q.

It remains to show the second assertion

E(|E¯(f_n, h_n|^q)→E(|E¯(f^(q),eg_q−1)|^q), which is equivalent to

exp(1

2q(q−1)(f^(q))^′f^(q)T+ Z

(|eg_q(x)|^q−1−q(eg_q(x)−1))ν(dx)T)

= lim

n→∞exp(1

2q(q−1)f_n^′f_nT+ Z

(|h_n(x)|^q−1−q(h_n(x)−1))ν(dx)T).

Finally, we need to check the conditions of dominated convergence: the decomposition of the first integral on the right-hand side of (4.52) yields

n→∞lim Z

R^N\{0}

(|h_n(x)|^q−1−q(h_n(x)−1))ν(dx)

= Z

R^N\{0}

(|eg_q(x)|^q−1−q(eg_q(x)−1))ν(dx), (4.53) as lim_n→∞h_n = eg_q, by the dominated convergence theorem, and Lemma 4.6.3. The proof is completed by Lemma 4.6.2.3.

4.7 Concluding Remarks

In the presence of jumps theq-optimal signed martingale measure and the q-optimal absolutely continuous martingale measure may fail to be equivalent, but belong to the larger class of signed martingale measures or absolutely continuous martingale measures, respectively. In the class of signed mea-sures however, an analogous representation for the densities of equivalent martingale measures as stochastic exponentials is not available. Thus tech-niques for the equivalent case cannot be generalized. One might think that a solution to the portfolio optimization problem is now hard to find. This concern is not confirmed and in fact thanks to the portfolio optimization problem, we can establish an explicit form of the q-optimal signed martin-gale measure and the q-optimal absolutely continuous martingale measure:

In detail, we show that the optimal terminal value with respect to the utility function−|1−^x_p|^p (case i=s) can be hedged by almost the same strategy as in Chapter 3, Lemma 3.3.2 (now depending on the potential candidate of the qSMM). In the reverse direction of the duality procedure, we can then verify that the candidate is actually the qSMM as the optimal claim can be hedged. Analogously for the absolutely continuous case, here we have to find a superhedging strategy to find the correct measure. This is be-cause we allow for a larger class of claims (we have less budget constraints asM^qa ⊂ M^qs). Consequently, we have to enlarge the class of strategies to reach this claim: a consumption process has to be included to make sure that the optimal claim is met, i.e. if we overshoot the maximum utility we consume the difference to the maximum.

Summarizing, a verification procedure based on a hedging or superhedg-ing problem yields an explicit representation of theq-optimal signed mar-tingale measure or theq-optimal absolutely continuous martingale measure while significantly relaxing the assumptions in [69]. In fact, if those assump-tions fail (C_q⁺), the qSMM becomes strictly signed, i.e. P(Z_q <0) > 0. In this case we prove that the qEMM does not exist. Nevertheless, convergence of the qAMM to the minimal entropy martingale measure is shown if the MEMM exists in the typical case in finance ((4.12) is valid), although po-tentially the qAMM will never be equivalent. Under a further integrability assumption the convergence of the qSMM to the minimal entropy measure is established as well, see also Table 4.3. Finally, we discuss some consequences for portfolio optimization also valid in typical models like the Merton, Kou, or variance gamma model.

We close these concluding remarks by presenting the results and condi-tions of this chapter within three overview schemes. In all tables we consider a single stock, whereν(−∞,0)6= 06=ν(0,∞)∨σ >0:

In Table 4.1, we describe when C_q^A,C_q⁻,C_q, andC_q⁻∧C_q⁰ are satisfied.

The matrix describes the additional conditions necessary for C_q^A, C_q⁻, C_q, and C_q⁻∧C_q⁰ under the two possible distinct sets of the model parameters (σ, ν, b): (4.12) is valid or not. All posted conditions for C_q and C_q⁻∧C_q⁰ hold forq >1 small enough only. In Table 4.2, we summarize under which conditions the different versions of theq-optimal martingale measure exist, have an explicit representation, and when the versions coincide. In Table 4.3, we assume that the MEMM exists with parameterθ_e as defined in As-sumption 3.2.1 (condition C ⇔ ∃˙ MEMM ⇒ R

x≥1e^θeexν(dx) < ∞). We then collect sufficient assumptions for our convergence result again by split-ting up into the two cases possible forθe.Recall, a negativeθecorresponds to a positive optimal exponential portfolio and therefore describes the typical case.

C^j_i holds if C^A_q C⁻_q ∃q>1 : Cq ∃q>1: C⁻_q ∧C⁰_q C, θe<0 no ass. (4.11) ∃L ν([L,∞)) = 0 (4.12) no ass. (4.11) ∃L ν([L,∞)) = 0

C, θe>0 no ass.

¬(4.12) (4.11)

Cq= (C^Aq ∧Cq^E)∨(Cq⁻∧Cq⁺), q >1, no ass. = no further assumptions

(4.12) :R

RN\{0}((e^x−1)−x1|x|≤1)ν(dx) + (b+¹₂σ²)>0, (C,θe<0⇒(4.12)) (4.11):R

x>1e^{q−1q x}ν(dx)<∞,(C,θe>0⇒(4.11)∧¬(4.12))

Table 4.1: Conditions leading to theq-optimal martingale measure

C^A_q ∧ ¬C^E_q C⁻_q ∧ ¬C⁺_q C^A_q ∧C^E_q C⁻_q ∧C⁰_q C⁻_q ∧C⁺_q

qAMM ∃˙ + EF ∃˙ + EF =qEMM =qSMM =qEMM

qSMM ∃˙ ∃˙ + EF =qEMM =qAMM =qEMM

qEMM ∄ ∄ =qAMM ? =qSMM

EF = explicit formula, see Theorem 4.3.2 (qAMM), 4.2.2 (qSMM), 4.1.3 (qEMM).

Table 4.2: Versions of the q-optimal martingale measure (existence and conditions)

θ_e<0 θ_e>0

qAMM →MEMM no ass. (4.35)

qSMM →MEMM (4.36) (4.35)

qEMM →MEMM ∃L: ν([L,∞)) = 0 (4.35)

no ass. = no further assumptions

(4.35): ∃δ >0 : R

x≥1e^(θe+δ)exν(dx)<∞ (4.36): ∃δ >0 : R

x≥1e^{(max{θe,−0.28θe}+δ)ex}ν(dx)<∞

Table 4.3: Convergence to the minimal entropy martingale measure - differ-ent conditions for differdiffer-ent versions of theq-optimal martingale measure.

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