Portfolio Optimization for Exponential L´ evy Processes
3.3 Convergence to the Exponential Utility Prob- Prob-lem
In the continuous case, at first we showed L1-convergence of the terminal valuesX0(p)toX0(exp)in a quite general setting using the convergence of the q-optimal measures to the minimal entropy martingale measure. If in addition the tradeoff process is deterministic, the MEMM coincides with all qMMM.
In this case, we proved convergence of the portfolio in Lv-supremums norm (v ≥ 1). Note, under this additional stronger assumption, it would not have been necessary to show convergence of the terminal values, separately.
In the L´evy case, as long as the Structure is satisfied, the tradeoff process is deterministic. However, for L´evy processes q-optimal measures do not coincide with the minimal entropy measure. Further, we do not necessar-ily assume that the tradeoff process actually exists. Even though, we can prove convergence of the portfolios which again implies Lv-convergence of the optimal terminal values. The proof of the convergence of the portfolios is a lot different from the continuous case, whereas the proof regarding the terminal values follows almost the same steps as above with some different technical details. These details have been proven in the preceding section, we summarize:
3.3.1 Convergence of the Terminal Values and the Value Functions
Assume Cq for all q ∈(1, q0) for an q0 >1 and condition C implying that the (equivalent) qMMMs and the MEMM exist. We need to prove the three items proposed in Section 1.2.6. To establish item 1a (Lemma 1.2.18), we only need uniform integrability of (Z(q)1+˜ǫ)q, ˜ǫ > 0, see Lemma 3.2.7. For item 1b (Lemma 1.2.19) and item 2, it remains to show that E(logZq) → E(logZmin) andZq L1
→Zmin, see Lemma 3.2.8. It should be noted, that item 1b uses convergence of the Girsanov parameters which was firstly shown in Jeanblanc et al. [69], but under different assumptions. Recall, the proof of Item 3 does not rely on the continuity assumption, we therefore do not have to prove this item again, i.e. we apply Theorem 1.2.21. Finally:
Theorem 3.3.1. In a model with an exponential L´evy process S = S0 + M+A given, let one of the following sets of assumptions be satisfied:
1. N = 1. Condition C and Assumption 3.2.2 hold. If (3.16) holds, assume that jumps are bounded from above.
2. Assumptions C, Cq for all q∈(1, q0]for an q0 >1 and 3.2.4.
3. Xˇ is a piecewise constant real valued L´evy process with jumps lower than a constantL <∞ plus a Brownian motion withσ 6= 0.
Then, the solution of thep-th problem converges in L1 to the solution of the exponential problem, i.e.: X0(p)(˜x)−→L1 X0(exp)(˜x).
The values of the dual and primal problems converge.
Under the third assumption the second is satisfied, see also the example inJeanblanc et al. [69].
3.3.2 Convergence to the Optimal Portfolio
Next we clarify when the optimal portfoliosϑ(p) of the p-th problems con-verge to the optimal portfolioϑ(exp) of the exponential problem. We solve the pricing equation of the claimX0(p)(˜x), x < p˜ :
dYt(p),˜x = (ϑ(p)t )′dAt+ (ϑ(p)t )′dMt+dMˇ(p), (3.31) Y(p),˜x(T) = X0(p)(˜x)
where ˇM(p)is e.g. the orthogonal term appearing in the F¨ollmer-Schweizer-decomposition. Under the assumptions of this chapters, we are able to find a portfolio such that ˇM(p) is zero, i.e. X0(p,˜x), the optimal solution of the static problem, is hedgeable. Next, we give a short motivation for the solution (Y(p),˜x, ϑ(p)) of (3.31). A reader not interested in this motivation can skip the next paragraph. (Y(p),˜x, ϑ(p)) is given in Lemma 3.3.2:
Motivation
As in the continuous case a possible candidate for the solution is
Y˜t(p),˜x := Zˆt−1E( ˆZTX0(p)(˜x)|Ft) (3.32)
= Zˆt−1E( ˆZTp(1−Z
1
qp−1(1−x˜ p)(E(Z
p
qp−1))−1)|Ft),
where with eg∗(x) =µ′∗(ex−1) + 1,g∗(x) = log(eg∗(x)) (if well defined) and f2=µ′∗σ
Zˆt = eµ∗σWt−12µ′∗σσ′µ′∗t+
Rt 0
R
kxk>1g∗(x)NXˇ(dx,ds)+Rt 0
R
kxk≤1g∗(x) ˜NXˇ(dx,ds)
×e−tµ′∗R(ex−1−g∗(x)1kxk≤1)ν(dx).
However, ˆZ is the density of the minimal and the variance optimal martin-gale measure (under our conditions both are equal andg2(x) =g∗(x)), only if its density is positive. Otherwiseg∗(x) is not defined. That would restrict the set of processes enormously, see Example 3.3.5 and Example 4.2.6 in the next chapter. In fact, we do not have to assume that the variance optimal measure is not signed nor that it actually exists. The Structure Condition is
not necessary! This is because of the following argumentation. When heuris-tically calculating ˜Yt(p),˜x, all terms containing µ∗, g∗(x),eg∗(x) cancel out.
The version obtained from this calculation, denoted by Yt(p),˜x see Lemma 3.3.2, satisfies YT(p),˜x =X0(p)(˜x).Under Assumption 3.1.1 all integrals exist.
Itˆo’s formula and a coefficient comparison then yield that this transformed processY(p),˜x is in fact equal to a price process, that reaches X0(p),˜x. If the variance optimal measure exists and is equivalent toP: ˜Yt(p),˜x =Yt(p),˜x.
We start to derive E(Zqq),as before we get (e.g. see the proof of Lemma 3.2.7 setting κ=q):
E(Zqq) = exp(1
2q(q−1)fqfq′T+ Z
(eqgq(x)−1−q(egq(x)−1))ν(dx)T), whereZqq=Z
p
qp−1,withq= p−p1.Next, we need to deal with E( ˆZTZqq−1) : E( ˆZTZqq−1|Ft) = E( ˜MT(q)|Ft) ˜AqT (3.33) where
M˜t(q) = e
Rt 0
R
kxk>1g∗(x)+(q−1)gq(x)NXˇ(dx,ds)+Rt 0
R
kxk≤1g∗(x)+(q−1)gq(x) ˜NXˇ(dx,ds)
×e(f2+fq(q−1))Wt−12t(f2+(q−1)fq)(f2+(q−1)fq)′
×e−tR(eg∗(x)e(q−1)gq(x)−1−(g∗(x)+(q−1)gq(x))1kxk≤1)ν(dx) and
A˜(q)T = e−TR((eg∗(x)−1)+(q−1)(egq(x)−1)−(eg∗(x)e(q−1)gq(x)−1))ν(dx)
×exp(f2fq(q−1)T +1
2((q−1)fq)((q−1)fq)′T−1
2(q−1)fqfqT).
M˜(q) is a martingale, because R
kxk>1|(eg∗(x)e(q−1)gq(x)−1)|ν(dx) < ∞ by (3.13) and (3.35) below and similar arguments as in the proof of Lemma 3.2.7. With (3.33), we have
E( ˆZTZqq−1|Ft)
E(Zqq) = M˜t(q)A˜(q)T (E(Zqq))−1 = ˜Mt(q)A(q)T , (3.34) where
A(q)T = e−TR((eg∗(x)−1)+(q−1)(egq(x)−1)−(eg∗(x)e(q−1)gq(x)−1))ν(dx)
×ef2fq′(q−1)T−(q−1)fqfq′Te−R(eqgq(x)−1−q(egq(x)−1))ν(dx)T
= e−TR((eg∗(x)−1)−(egq(x)−1)−(eg∗(x)e(q−1)gq(x)−1)+(eqgq(x)−1))ν(dx)
×exp(f2fq′(q−1)T− 1
2q(q−1)fqfqT).
Further from (3.13)
e(q−1)gq(x)eg∗(x)−1 = (q−1)θ′q(ex−1)(ex−1)′µ∗+(q−1)θq′(ex−1)+µ′∗(ex−1) (3.35) we obtain
t Z
(eg∗(x)−eg∗(x)e(q−1)gq(x)+e(q−1)gq(x)−1)ν(dx)−f2fq′(q−1)t
= t Z
(−(q−1)θ′q(ex−1)(ex−1)′µ∗)ν(dx)−t(q−1)θ′qσσ′µ∗
= (q−1)θ′q Z t
0
S−u−1dhMiuλˆu= (q−1)θq′ Z t
0
S−u−1dAu. Finally, we have
Zˆt−1E( ˆZTZqq−1|Ft) E(Zqq)
= M˜t(q)A(q)T e−R0tg∗(x)NXˇ(dx,ds)+tR(eg∗(x)−1)ν(dx)−f2Wt+12f2f2′t
= Mt(q)A(q)T βt(q), where
βt(q) = etR((eg∗(x)−1)+(e(q−1)gq(x)−1)−(eg∗(x)e(q−1)gq(x)−1))ν(dx)e−f2fq′(q−1)t
= exp{(q−1)θq′ Z t
0
S−u−1dAu} and
Mt(q) = e
Rt 0
R
kxk<1(q−1)gq(x)NXˇ(dx,ds)+Rt 0
R
kxk≤1(q−1)gq(x) ˜NXˇ(dx,ds)
×efq(q−1)Wt−12fqfq′(q−1)2t−tR(e(q−1)gq(x)−1−(q−1)gq(x)1kxk≤1)ν(dx), which is again a martingale by (3.13) and an analogous argument as above.
Note,M(q)andβ(q)do not depend on the existence ofµ∗,i.e. we do not have to prove necessary conditions for the Structure Condition to hold. Further-more,A(q)T = 1 using (3.34) because EQ(X0(p,˜x)) coincides for allQ∈ Mqa,if X0(p) can be hedged which is shown next.
Optimal portfolio of the p-th Problem and its Convergence The next lemma provides the solution (Y(p),˜x, ϑ(p)) of BSDE (3.31):
Lemma 3.3.2. Under condition Cq, for all q ≥ q1 for an q1 > 1 and Assumption 3.2.4 or 3.2.2 in the one-dimensional case, there exists aq0 ≥q1
such that for all q ≥ q0 and x˜ ≤ p the optimal wealth process of the p-th
Note again, in the one-dimensional case as condition C and Cq are sat-isfied, jumps have to be either bounded from above or (3.16) cannot hold.
The assumptions of Lemma 3.3.2 are equivalent to the assumptions made in Lemma 3.2.5. Before we start with the proof. We like to give a comparison to the continuous case:
Remark 3.3.3. (i) If the L´evy process only contains a Brownian component and a drift. We get for all q >1: −θq= (σσ′)−1(µ−r) =λ′St.Hence, That is the result from the continuous Brownian case, in the case N =d.
If ˜x > p and consumption is allowed (or if we are allowed to start with less initial wealth), then we do not invest in the stock and consume ˜x−p. This obviously leads to the optimal solution. For every ˜x there is a p such that
˜
x < p,so for our convergence result this case can be ignored.
Proof of Lemma 3.3.2. Suppose ˜x ≤ p. The condition YT(p),˜x = X0(p)(˜x) is
As the supremum ofMu(q) to the power pis integrable by Doob’s inequality and the fact that (q−1)p=q, we thus get kϑ(p)kLp(A)<∞.Further,
It suffices to consider the jump part ofRT
0 (ϑ(p)t )′d[M]tϑ(p)t (denoted by the
with a constant Kq>0. So, with a constant ˜Kq >0 : E((∗)p)
≤ K˜qE( sup
t∈[0,T]|Mtq|p| Z t
0
Z
θq′(ex−1)(ex−1)′θqNXˇ(dx, du)|p2) (3.36) In the one-dimensional case, by Lemma 3.2.5 there exists anm0such that for allm≥m0, θqm →θeand an arbitrary sequenceqm ↓1.Ifθe <0,thenθqm<
0 for allm≥m1 ≥m0.Jumps have to be bounded from above otherwiseCq fails, see Chapter 4. In view of the first assertion of the proof and because jumps are bounded from above, we have kϑ(p)kLp(M) <∞.Ifθe>0,under Assumption 3.2.2, all moments of|Rt
0
R θ′q(ex−1)(ex−1)′θqNXˇ(dx, du)|exist andE(supt∈[0,T]|Mtq|p(1+ǫ)) is finite by Assumption 3.2.2, see Lemma 3.2.7.
The assertion follows by H¨older’s inequality. In the multidimensional case, Assumption 3.2.4 yields sufficient integrability of the elements in (3.36).
Next, we turn to the convergence of the solutions of the p-level BSDEs to the BSDE of the exp-problem. We establish for all v≥1:
E(sup
t |Yt(p),˜x−Yt(˜x)|v]→0, p → ∞, (3.37) where Yt(˜x) := ˜x+Rt
0−θ′qS−1s−dS. As we already know the explicit form of Zmin, it is easily seen that
YT(˜x)=X0(exp)(˜x).
Hence the optimal portfolio that reachesX0(exp) is equal to:
ϑ(exp)=−S−·−1θe ∈ Aexp. Finally, we obtain −S−·−1θe∈ Aexp and
YT(p),˜x=X0(p)(˜x)→Lv X0(exp)(˜x), v≥1.
We obtain the following theorem:
Theorem 3.3.4. Under the assumptions of Theorem 3.3.1, (−S−−t1θe,0) ∈ Aexp×C is the optimal portfolio of the problem
Vexp(˜x) = max
(ϑ,C)∈Aexp×CE(1−e−(˜x+R0Tϑ′udSu−CT)), (3.38) where C is an arbitrary class of adapted, right-continuous, increasing pro-cesses with C0≥0. Further,
E(sup
t |Yt(p),˜x−(˜x+ Z t
0 −θ′eS−s−1dSs)|v]→0, p→ ∞, v≥1 (3.39)
where Y·(p),˜x is the optimal wealth process of
Proof of Theorem 3.3.4. We proceed similar as in the proof of Lemma 3.2.8:
by definingg(p):= pp−−x1˜θqM(q)β(q), q=p/(p−1) by H¨older’s inequality as in the proof of Lemma 3.3.2 and the following argu-ment: recall,S=E( ˘L). Split up ˘Lin a martingale part M( ˘L)and a process of finite variation. E(supt≤T kMt( ˘L)kv) is finite, by Doob’s inequality and by Prop. 3.13 in Cont and Tankov [32] because R
kxkˇv(ex−1)ν(dx)<∞ with ˇv=v(1 +ǫ) by Assumption 3.2.2/3.2.4. The part of finite variation is deterministic and independent ofq, see (3.2). HenceE(supkL˘tkˇv) is finite.
Further, we know that if a sequence of a stochastic process (yt(n))nconverges in Sˇv to yt (supremums norm in Lvˇ) and a deterministic sequence (x(n)t )n converges in supremums norm to xt, then the product converges in Svˇ to xtyt. The deterministic part β(q) converges to 1 since it is continuous in t and (q−1)gq(x)−→q↓1 0.Fubini’s Theorem can be applied as in Lemma 3.2.7 and 3.2.8, especially see (3.27). Finally, we need to consider the random partM(q). We showE(supt|Mt(q)−1|ˇv)→0.By Doob’s inequality we have
We need to prove that the last part converges to zero for every fixed ˇv. We have
ifE((MT(q))u)→1 for u= 0,1, ...,2l, which remains to show:
The Brownian and the jump part of M(q) are independent, by Yor’s formula, we can split up Mt(q) in a continuous and discontinuous part Mt(q) = Et(−
Z
−(q−1)θq′σdWs)Et(− Z Z
(q−1)θ′q(ex−1) ˜NXˇ(dx, ds)).
Both parts can be treated separately because of the independence. The continuous part can be established as above (see Chapter 2) and by observing that θq(i)∈(θe(i)−ǫ, θ(i)e +ǫ). The jump part is equal to (Mtd,(q))u =euXt(q), whereX(q) is a L´evy process with characteristic triplet (Aq, νq, bq):
(0, νq,− Z
(e(q−1)gq(x)−1−(q−1)gq(x)1kxk≤1)ν(dx)), whereνq(A) =R
1(q−1)gq(x)∈Aν(dx), A∈B(RN).We have Z
kxk>1
ev(qˇ −1)gq(x)ν(dx)≤ν({x:kxk ≥1}) + Z
kxk>1
eqgq(x)ν(dx) (3.41) for allq∈(1, q(ˇv)], whereq(ˇv) denotes the conjugate of ˇv. The last integral is finite for allq∈(1, q0]. Applying Prop. 3.14 inCont and Tankov [32], we obtain:
E(euXt(q)) =etψq(−iu)
whereψqdenotes the characteristic exponent ofX(q). By the L´evy- Khinchin representation, we get:
ψq(−iu) = −u Z
(e(q−1)gq(x)−1−(q−1)gq(x)1kxk≤1)ν(dx) +
Z
(eu(q−1)gq(x)−1−u(q−1)gq(x)1kxk≤1)ν(dx) which is equal to zero for u = 0,1 and converges to zero for u = 2, ...,2l, because limit and integration can be changed as in (3.27). So for every ˇ
v=v(1 +ǫ) there exists an l(v)∈N, v(1 +ǫ)≤2l(v) such that E(sup
t |Mt(q)−1|ˇv)1/ˇv ≤(E(|MT(q)−1|2l))2l1 →0.
Apart from convergence in Sv of the price processes, we have uniform con-vergence of the integrands. This further yields uniform convergence in probability of the integrals and therefore of the price processes. Finally,
−θe′S−1·− ∈ Aexp,as in the proof of Lemma 3.3.2.
We finally apply the results of this section to a Brownian-Poisson case:
Example 3.3.5 (Brownian Motion + Compound Poisson). Let γ be the intensity andf the jump size distribution of a compound Poisson process.
σ 6= 0 is the volatility, W is a Brownian motion and β is defined as before β = −(µ−r) = −(b− 12σ2 −r). All assumptions of Theorem 3.3.1 are satisfied. We getϑ(exp)t =−St−1−θe where θe is the solution of (3.9):
θeσ2+ Z
((ex−1)eθe(ex−1)−1|x|≤1x)γf(dx) =β
We present an example when also the variance optimal measure is equiva-lent to P. In the case of a truncated normal distribution N(−∞,L)(µL, σ2L) (jump size is bounded by an L ∈ R), we have to ensure that µ∗ ∈ ((1−eL)−1,1), L > 0 or µ∗ ∈ (−∞,1), L < 0. The upper bound is always satisfied. µL, σ2L and γ have to be chosen appropriately such that µ∗ ∈((1−eL)−1,1), where
µ∗ =
β− NµL,σLγ (L)(µLN,1−1−R1
−1xnµL,σL(x)dx) (σ2+γµLN,2−2γµLN+γ)
and µLN,n denotes the n-th moment of x =ey with y ∼ N(−∞,L)(µL, σL2).
Choose e.g.L= 0.75, γ= 0.001, µ=b+12σ2 = 0.09,σ2 = 0.101,σ2L= 0.07, orσ = 0.3178, σL = 0.2646. A slight increase of some parameters and the condition fails! The coefficients above imply that ϑ(exp)S = −θe = 0.8923.
This can be compared to the continuous portfolio, here we obtainϑ(exp)c Sc = µ/σ2 = 0.8911. Hence, in this case we invest more in the stock if jumps are considered, because the jump measure is defined onxand not ony=ex−1.
The jump measure of ˇX is symmetric. As we consider its exponential large jumps obtain a larger weight. The expected value increases and we invest more in the processes possessing jumps.
3.4 Concluding Remarks
In Chapter 2, we provide a framework to solve the dynamic utility maxi-mization problem for an exponential utility function via an approximation approach. Optimal terminal values as well as its optimal portfolios are con-sidered. However, this is done in a continuous semimartingale setting. We are able to expand these results to exponential L´evy processes in this chap-ter. In particular, we prove convergence of the static and dynamic problems (i.e. the optimal terminal wealth and the explicit portfolios attaining those for the p-th problem to the exponential one). Results from the continuous setting on the sequence−|1−xp|p (p-th problem) are almost taken over to solve the static problem, whereas proofs of our convergence results become
quite different. To obtain convergence several additional results onq-optimal equivalent and the minimal entropy martingale measure are proven. Finally, our convergence result yields a construction method for an explicit portfolio in the exponential case with jumps. Remarkably, it looks very similar to the portfolio in the continuous case. Nevertheless, the numerical difference can be quite huge. Even the signs of the two portfolios can differ, see Chapter 5, in particular Figure 5.1.
This chapter is a slightly generalized and extended version of the article Niethammer[106] (there for the casep= 2m). In the next chapter, we will consider theq-optimal signed or absolutely continuous martingale measures further, in particular its explicit form and the convergence to the minimal entropy measure. The problem of portfolio optimization can be solved also for unbounded jump size. This is particularly interesting as typical models, like the variance gamma model (if θe exists, it is always negative), have unbounded jumps. That means that condition Cq proposed in [69] always fails. The dual solution (theq-optimal signed martingale measure or the q-optimal absolutely continuous martingale measure) can become signed resp.
zero in these cases. In some odd cases, we can even derive an optimal portfolio (and optimal consumption) for thep-th problem although there is arbitrage, see Section 4.5. We refrained from a more detailed presentation in the last chapters because of the questionable interpretation of signed or absolutely continuous measures for pricing issues. In the next chapter however, we will observe that signed and absolutely continuous measures can make perfect sense.
On q -Optimal Martingale Measures
This chapter is devoted to portfolio optimization in the presence of un-bounded jumps. In this case, still the q-optimal martingale measure is the optimal solution of the dual problem to thep-th problem. However, if jumps are unbounded the correspondingq-optimal martingale measure might fail to be equivalent. We therefore discuss theq-optimal signed martingale mea-sure (qSMM) and theq-optimal absolutely continuous martingale measure (qAMM) in this chapter and derive some consequences for portfolio opti-mization. In particular, we explain when the qSMM and when the qAMM is appropriate. As signed martingale measures cannot necessarily be de-scribed by a single stochastic exponential, techniques usually used to find equivalent versions cannot be applied. In fact, by applying a novel verifica-tion procedure we derive an explicit form of theq-optimal signed martingale measure using the primal side of thep-th problem and an analogous hedging argument as derived in Section 3.3. To find the q-optimal absolutely con-tinuous martingale measure, we need to widen the class of strategies. This is because the set of claims allowed in the static primal problem becomes larger as budget constraints are taken with respect to the smaller class Mqa
instead ofMqs.Allowing for consumption will be sufficient, see also Remark 1.2.15. A similar verification procedure then yields the explicit form of the qAMM. In both cases, we relax assumption Cq, in particular the required positivity of (q−1)θq(ex−1) + 1. Moreover, if the qSMM is strictly signed, i.e. P(Zq <0) > 0, then an equivalentq-optimal martingale measure does not exist. Finally, we show convergence of the q-optimal signed/absolutely continuous martingale measures to the minimal entropy measure in both cases. Note that there are simple examples of sequences of the qSMMs where all elements are strictly signed and yet the sequence converges to the MEMM.
The chapter is organized as follows: after an extended discussion of con-95
ditionCq in Section 4.1, we suggest some new assumptions for the existence of an explicit form of theq-optimal signed martingale measure, qSMM. Sec-tions 4.2.2 is devoted to the verification of theq-optimal signed martingale measure. Afterwards a similar procedure is presented for the q-optimal absolutely continuous martingale measure. Section 4.4 then presents the convergence of theq-optimal signed/absolutely continuous martingale mea-sure to the minimal entropy one. Examples are given for all cases. Section 4.5 collects some consequences for thep-th and the exponential utility max-imization problem. We close the chapter by discussing the existence of an equivalent qMMM if the q-optimal signed martingale measure is strictly signed.