Discussion of Condition C q

Throughout the chapter, we again assume that Assumption 1.2.2 holds, i.e.

our stocks are driven by an exponential L´evy process. Up to now, we have only solved the portfolio problem with respect to−|1−^x_p|^p under condition Cq, see Assumption 3.2.3. We have further mentioned that this condition makes perfectly sense under the assumption of upward bounded jumps only.

This first section is therefore supposed to shed some light on the exact meaning of this condition.

Before, we start recalling some important facts on the considered stocks and (signed) martingale measures:

4.1.1 Review

Assumption 1.2.2 requires that E[kS(t)k] < ∞ for all t ∈ [0, T] which is satisfied if and only if R

kxk>1e^xν(dx) <∞ (see Theorem 1.1.19). By Itˆo’s formula it thus guarantees that S is a special semimartingale with decom-positionS_t=S₀+M_t+A_t, where

dM_t=S_t−(σdW_t+ Z

R^N\{0}

(e^x−1) ˜NXˇ(dx, dt)) and

dA_t=S_t−(−β+ Z

RN\{0}

(e^x−1−x1_kxk≤1)ν(dx))dt.

Here,β=−(b+¹₂P

jσ²_·,j) andS=diag(S⁽¹⁾, ..., S^(N)). 1denotes the vector inR^N having all entries equal to one, and expressions such as e^x are to be interpreted componentwise, i.e. e^x = (e^x1, ..., e^xN)^′ .

To explain the problem to be discussed in this chapter in comparison to Chapter 2 and 3, we recall the q-optimal martingale measure in its three different variants, the qSMM, the qEMM, and the qAMM. Recall forq >1,

the set of q-integrable density processes of signed local martingale measures is given by

Ds^q={Z ∈ U^q|E(Z_T) = 1, SZ ∈M_loc},

where M_loc is the space of local martingales and U^q denotes the set of R-valued L^q(Ω, P)-uniformly integrable martingales. Again Z ∈ D^qs can be identified with a signed measure settingdQ_Z =Z_TdP. We will start study-ing the followstudy-ing problem:

(qSMM) Find Z^(q)∈ Ds^q such that E[|Z_T^(q)|^q] = inf

Z∈D^qs

E[|Z_T|^q].

dQ^(s,q)=Z_T^(q)dP is called theq-optimal signed martingale measure Q^(s,q). In the continuous case the obtained measure is an equivalent martingale measure. However, in the considered L´evy case this is not necessarily the case. So recently and applied in Chapter 3, Jeanblanc et al. [69] dis-cussed the closely related problem, where minimization is considered over the subset of density processes corresponding toq-integrable equivalent local martingale measures

D^qe={Z ∈ Ds^q|Z_T >0P-a.s.}, only. In short:

(qEMM) Find ˜Z^(q)∈ De^q such that E[|Z˜_T^(q)|^q] = inf

Z∈D^qe

E[|Z_T|^q].

If ˜Z^(q)∈ D^eq,dQ^(e,q) = ˜Z_T^(q)dP is called theq-optimal (equivalent) martingale measure Q^(e,q).

To stress the fact that it is now important to distinguish between the signed version qSMM and the equivalent version qEMM, we rather use the acronym qEMM for theq-optimal equivalent martingale measure instead of qMMM.

We discuss a third measure, the situation when the qMMM is not strictly positive, but non-negative:

(qAMM) Find ˇZ^(q) ∈ D^aq such that E[|Zˇ_T^(q)|^q] = inf

Z∈D^qa

E[|Z_T|^q].

dQ^(a,q)= ˇZ_T^(q)dP is called theq-optimal (absolutely continuous) martingale measure Q^(a,q).

D^qa denotes as before the set of q-integrable absolutely continuous local martingale measures, i.e.Z ∈ D^sqandZ_T ≥0.Apart from characterizing the q-optimal signed or absolutely continuous martingale measures, we signifi-cantly generalize the study of their behavior asq tends to 1: we show that theq-optimal signed or absolutely continuous martingale measures converge to the minimal entropy martingale measure (as q → 1). Recall, the latter one is defined to be the equivalent local martingale measure, which mini-mizes the entropy relative toP over the set of all equivalent local martingale measures with finite relative entropy:

(MEMM) FindZ^min∈ De^log such that E[Z_T^minlogZ_T^min] = inf

Z∈De^log

E[Z_T logZ_T], where

De^log={Z ∈ D¹e, E(Z_TlogZ_T)<∞}.

dQ_min=Z_T^mindP is called theminimal entropy martingale measure Q_min. Recall, we set Z_min = Z_T^(min). The qSMM, the qAMM and the MEMM always exist, if D^qs, Da^q, or De^log are nonempty, respectively. However, the qEMM might not exist ifDe^q6=∅. See Section 4.6.

4.1.2 Reformulation and Discussion of Condition C_q

We start with the mentioned reformulation of condition C_q, generalizing conditionC_q given in Assumption 3.2.3:

Assumption 4.1.1. (C_q⁻,C_q⁺, C_q⁰) C⁻_q : There exists a θq∈R^N such that

eg_q(x) :=|(q−1)θ_q^′(e^x−1) + 1|^q−11 sgn((q−1)θ^′_q(e^x−1) + 1) (4.1) satisfies

σσ^′θ_q+ Z

R^N\{0}

((e^x−1)eg_q(x)−x1_kxk≤1)ν(dx) =β (4.2)

and Z

R^N\{0}

|eg_q(x)|^q−1−q(eg_q(x)−1)

ν(dx)<∞. (4.3) C⁺_q : C_q⁺ is verified if eg^qq⁻¹(x) = (q−1)θ_q^′(e^x−1) + 1>0, ν-a.s.

C⁰_q: We say that C_q⁰ is satisfied if

(q−1)θ_q^′(e^x−1) + 1≥0, ν-a.s..

Clearly, we have

Lemma 4.1.1. IfC_q⁻ and C_q⁺are satisfied, thenC_q, as defined in Assump-tion 3.2.3, holds.

Remark 4.1.2. (i) As long asC_q⁺holds, the (q−1)-th root of (q−1)θ_q^′(e^x− 1) + 1 is well defined. Consequently, we can leave out the absolute value and the signum function. Hence,C_q⁻and C_q⁺implyC_q.Assumption 3.2.3 is therefore part of the above reformulation, i.e. ifC_q holds we have: eg = eg.

If (q−1)θ_q^′(e^x−1) + 1<0 we have to set absolute values and adjust by the signum function. In general, Z^q6=|Z|^q.

(ii) In the one-dimensional caseC_q⁰ reduces toC_q⁺, ifν possesses a Lebesgue density.

(iii) C_q⁻ guarantees that the qSMM has actually got the right integrability and moreover thatS is a local martingale under the qSMM.C₊^q (C_q⁰) implies that the resulting measure is positive (non-negative).

Recall, in a L´evy setting, Jeanblanc et al. [69] have derived the explicit form of the qEMM for the first time. Rewriting Equation 3.14 in the sense of Assumption 4.1.1, we get:

Theorem 4.1.3. Jeanblanc et al. [69, Theorem 2.9]

Suppose C_q holds. Then the qEMM exists and is given by E¯(θ^′_qσ,eg_q−1),

where E¯(f, h) denotes the stochastic exponential with Girsanov parameters f, h, i.e.

E¯t(f, h) = e

Rt

0f(s)dWs−¹₂Rt

0kf(s)k²ds+Rt 0

R

RN\{0}h(s,x) ˜NXˇ(dx,ds)

×Y

s≤t

(1 +h(s,∆ ˇX(s)))e^{−h(s,∆ ˇX(s))}. (4.4) The bar on top of E is to indicate that we now write the stochastic exponential with respect to the Girsanov parametersf andh.If ¯E(f, e^g−1) is positive, we have ¯E(f, e^g−1) =Z(f, g) as defined in (3.12).

As already mentioned in the chapters before C_q is quite restrictive, if we do not want to restrict our results to upward bounded jumps. We next summarize these findings and give a formal proof. In the one-dimensional case, we give some necessary conditions for C_q. In particular, we find that C_q only holds, if the tails for upward jumps are extraordinarily light or the MEMM does not exist.

Proposition 4.1.4. Suppose N = 1and the true measure P is not a mar-tingale measure. Then:

(i) IfC_q holds for some q >1, then Z

x≥1

e^θexν(dx)<∞ (4.5)

for some θ >0 or the MEMM does not exist.

(ii) IfC_q holds for someq >1, then Z

R\{0}

((e^x−1)−x1_|x|≤1)ν(dx) + (b+1

2σ²)<0 (4.6) or upward jumps are bounded, i.e. ν([L,∞)) = 0 for some L >0.

Before we continue with the proof, we like to comment on the obtained results:

Remark 4.1.5. SupposeN = 1.

(i) We have already seen in Section 1.1 that most of the concrete models discussed in the literature, such as generalized hyperbolic models or the popular jump-diffusion models by Merton or Kou (see Section 1.1) satisfy

Z

x≥1

e^θexν(dx) =∞

for allθ >0. Hence,C_q and the existence of the MEMM cannot hold simul-taneously for these models.

(ii) In condition (4.6) upward jumps are exponentially weighted and down-ward jumps are exponentially damped. Hence,

Z

R\{0}

((e^x−1)−x1_|x|≤1)ν(dx)

can become negative only, if the L´evy measure gives much more weight to negative jumps than to positive jumps, leading to an extreme gain-loss asymmetry in the jumps. In such situation we expect that the deterministic trend b is sufficiently large to compensate for the risk of downward jumps.

Otherwise nobody would invest in the stock, which is indicated by a positive θ_e leading to a negative optimal exponential portfolio share into the risky asset. So condition (4.6) may be rather unlikely to occur.

Proof of Proposition 4.1.4. Recall, Φ(i,·), i=q, e and Dom_e are explained in Section 3.2.

(i) Assume that Cq holds for some q > 1 implying the existence of the

q-optimal equivalent martingale measure. Moreover, suppose that, for all θ >0,

Z

x≥1

e^θexν(dx) =∞ (4.7)

and the MEMM exists. As condition C is equivalent to the existence of the MEMM in the one-dimensional case, Φ_e has a zeroθ_e on its domain Dom_e. As Dom_e = (−∞,0), by (4.7) and becauseP is not a martingale measure, we getθe<0. By Lemma 3.2.6, we thus have Φ(q,0)>0, and consequently the root of Φ(q,·),θ_q, is negative. As the upward jumps cannot be bounded due to (4.7), we have a contradiction toC_q⁺.

(ii) Suppose thatC_q holds for someq, (4.6) fails, and upward jumps are un-bounded. Then we can again conclude from Lemma 3.2.6 thatθ_qis negative contradictingC_q⁺ as in (i).

An examination of the proof shows, that the positivity assumption inC_q, C_q⁺, is particularly restrictive. As already mentioned in the previous chap-ters,C_q⁺rules out negative values forθ_q, if the upward jumps are unbounded.

Negative values ofθ_q however lead to a positive optimal exponential invest-ment in the stock and are thus the typical case. The aim of the following section is therefore to reduceC_q to C_q⁻(later toC_q^A), i.e. C_q⁺need not hold necessarily. However, assuming C_q⁻ implies an explicit form of the qSMM only, but does not guarantee the existence of the qEMM, see Section 4.6.

To establish an explicit form of the qSMM, techniques usually applied for the equivalent case, as e.g. in Jeanblanc et al. [69], do not work in the signed case, since not every Z ∈Ds^q can be represented as a stochastic exponential:

Example 4.1.6. LetS be of the following form S_t=e^bt+σWt+Ut,

where σ > 0, b ∈ R, W denotes a one-dimensional standard Brownian motion andU is a one-dimensional Poisson process with intensityγ >0 and jump height 1. W andU are supposed to be independent. We can build two different equivalent martingale measures by the following density processes:

1. Z^(W)= ¯E(f_W, h_W), where h_W = 0, f_W =σ⁻¹(β+γ).

2. Z^(U)= ¯E(f_U, h_U), where f_U = 0, h_U = _γ(e−1)^β+γ .

We define a new signed martingale measure by the density process Z^{(W U)}:= 2Z^(U)−Z^(W)⇒E(Z^{(W U)}) = 1, SZ^{(W U)}= 2SZ^(U)−SZ^(W). Sufficient integrability is obvious, hence Z^{(W U)} ∈ D^qs. Further, when there have been no jump yet Z^(U) is equal to Z_t^(U) =e^{−t(e−1)β+γ} =g(t). Moreover,

at t= 1,

P(e^fWW1−12fW2 >2g(1), U₁ = 0) =P(e^fWW1−21fW2 > g(1))P(U₁ = 0)>0.

Hence, Z₁^{(W U)} = 2g(1)−e^fWW1−12fW2 is negative and Z_t^{(W U)}, t ∈ [0,1] is continuous on the setD:={e^fWW1−12fW2 >2g(1), U₁ = 0}.Thus,

P({ω: ∃s∈[0,1)Z_s^{(W U)}(ω) = 0,∃s≥1Z_s^{(W U)}(ω)6= 0})≥P(D)>0.

But the last probability would be zero if Z^{(W U)} could be represented by a stochastic exponential. If a stochastic exponential is zero once, it will be zero in the sequel as the product always possesses a zero factor afterwards.

An alternative is to merge a family of stochastic exponentials and represent the exponential as a combination of those. Roughly speaking, if the first becomes zero, a second is added, see alsoChoulli et al.[31].

Im Dokument Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps (Seite 122-128)