Some problems of portfolio optimization andhedging in a Levy market via fictitious completions

Enrique Figueroa-Lopez and Jin Ma

October 2005

Abstract

The classical Merton’s problem of utility maximization was recently solvedin [2] in a market consisting of a bond with constant interest rate, a stock thatfollows a geometric Levy model, and certain “fictitious” stocks called power-jump assets. Using their previous work [3] on the completeness of such a marketand the martingale method, it was proved there that for certain utility func-tions, it is possible to choose the fictitious stocks so that they are not needed toreplicate the optimal final wealth. In this paper, we study other properties ofthese fictitious completions of the Levy market. Problems that are consideredhere include optimal portfolios that maximize state-dependent utilities, or asa particular case, that minimize the “shortfall risk” in replicating a contingentclaim. Also, we analyze conditions for replicating a contingent claim, possessinga discounted payoff of X, with an initial endowment w = supQ E

[X

], where

the supremum is only taken over equivalent martingale measures associatedto fictitious completions of the market, and not over all equivalent martingalemeasures as the fundamental theorem of supereplication establish.

1 Introduction

In a market consisting of a bond and a stock with price process modeled by a geometricLevy model, we consider two classical optimization problems: the Merton’s problemof utility maximization and the minimization of the shortfall risk in replicating acontingent claim as described by [4]. Both problems are intimately related in differentways. For instance, a well-known approach to deal with the first problem consists oftwo-steps: first, determine the optimal terminal wealth, and then, decide whether ornot this wealth can be replicated by a self-financing portfolio (this approach is usuallycalled the martingale method). On the other hand, the problem of optimal hedging ofcontingent claims is equivalent to an utility optimization problem where the utilityfunction is state-dependent (see e.g. [4]).The above problems are analyzed by a method of fictitious completion. This idea canbe traced back to Karatzas et. al. [8] in the context of finitely many stocks drivenby independent Brownian motions. In that context, the incompleteness arises when

1

the sources of randomness are more than the number assets available for investment.A natural idea is to augment the market with as many fictitious assets as necessaryto complete the market, and then, show the existence of a completion so that anagent, by optimal choice, does not invest on them at all. Recently, Nualart et. al. [2]employ the same technique for the problem of utility maximization in a market con-sisting of a bond, a stock that follows a geometric Levy model, and some “fictitious”stocks called power-jump assets. Using their previous work [3] on the completeness ofsuch a market, they derive an explicit formula for the optimal final wealth using themartingale method. Furthermore, for a class of utility functions that includes HARAand exponential utility functions, they are able to derive expressions for the optimaltrading strategy, and show that for some completions of the market, the fictitiousstocks will actually be superfluous in the sense that the terminal expected utility isnot improved by including these new assets in the market. This in turn provides thesolution to the problem of utility maximization in the real market, consisting only ofthe bond and the stock.In this paper, we study further properties of the fictitious completions of the Levymarket introduced in [2]. In particular, some results in [2] are extended to the caseof state-dependent utility functions. As it was mentioned before, this type of utilitiesare motivated by the problem of minimizing the “shortfall risk” in replicating a con-tingent claim (see [4]). Also, we analyze conditions for replicating a contingent claim,

possessing a discounted payoff of X, with an initial endowment w∗ = supQ E[X

],

where the supremum is taken over equivalent martingale measures associated to ficti-

tious completions of the market. It is know that if the supremum in w′ := supQ E[X

]

is taken over all equivalent martingale measures, then X can be attained with anyinitial endowment w ≥ w′. We essentially address conditions on X for w∗ = w′.

2 The financial model and preliminaries

The following points introduce the financial model and the terminology we use throughthe paper as well as some preliminaries results.

1. The price of the stock at time t is given by the Geometric Levy process

St = S0 + b

∫ t

0

Ss−ds +

∫ t

0

Ss−dZs

= S0 exp

Zt + bt− σ2

2t +

∑0<s≤t

[log (1 + ∆Zs)−∆ Zs]

, (1)

where Z is a Levy process on a probability space (Ω,F ,P) with Brownian partσWt and with ∆Zt > −1, a.s. Moreover, the Levy measure ν of the Levyprocess Z is assumed to satisfy the exponential-moment condition

∫

(−ε,ε)c

eλ|x|ν(dx) < ∞, (2)

2

for some λ, ε > 0. It is important to remark that in recent years differentgeometric Levy models have actively been studied, and widely acknowledged asmore accurate alternatives to the geometric Brownian motion inherent in theclassical Black-Scholes model. See for instance [18] and [1] for accounts on theirapplications in mathematical finance.

2. The value of the risk-free bond or bank account at time t is given by

dBt = rBtdtB0 = 1

=⇒ Bt = ert.

3. Let F := Ftt≥0 be the natural filtration generated by Z that satisfies theusual conditions (see e.g. [15], Theorem I.31). We assume the existence ofequivalent risk-neutral probability measures Q that preserves the structure ofthe model in the sense that, under Q, the discounted process B−1

t St0≤t≤T

is a local martingale, and Z is a Levy process with Levy measure, denotedby νQ , satisfying (2). Let us remark that B−1

t St0≤t≤T will be de facto amartingale (see Lemma 4.4. in [7]). Throughout, Q stand for a family ofstructure preserving risk-neutral measures. Section 6 completely describes thisfamily. Also, EQ denotes the expectation with respect to Q.

4. For each structure-preserving measure Q ∈ Q, one can construct a sequence ofpairwise strongly orthogonal normal martingales. We follow the exposition of[14] and [3]. Define the processes

Z(1)t := Zt, and Z

(i)t :=

∑s≤t

(∆Zs)i , i ≥ 2.

The Teugels martingales are defined by

Y(1)t := Z

(1)t − EQ

[Z

(1)t

], and Y

(i)t := Z

(i)t − EQ

[Z

(1)t

], i ≥ 2.

Letqi−1(x) = ci,ix

i−1 + · · ·+ ci,1, i ≥ 1,

be the orthonormal polynomials obtained from the Gram-Schmidt process ap-plied to the polynomials 1, x, x2, . . . with respect to the inner product

(p(x), q(x))eν =

∫ ∞

−∞p(x)q(x)x2νQ(dx) + σ2p(0)q(0). (3)

Then, the orthonormalized Teugels martingales are given by

T (i) := ci,iY(i) + · · ·+ ci,1Y

(1).

It can be proved that T (i) : i ≥ 1 are pair-wise strongly orthogonal normalmartingales (see [13]). Also, it can be verified that

∆T(i)t = pi(∆Zt), (4)

where pi(x) := xqi−1(x).

3

5. For each structure-preserving measure Q ∈ Q, let us introduce a sequence offictitious assets with prices given by the

H(i)t = BtT

(i)t , i ≥ 2,

where T (i) stands for the orthonormalized Teugel’s martingale of order i con-structed from the Levy process Z with respect to the martingale measure Q.The augmented market consisting of the bond, the stock, and the fictitiousjump-power assets is denoted MQ.

6. Generally speaking, a self-financing portfolio in the augmented market MQ

during a time horizon [0, T ] is given by predictable processes (α, β, β(1), . . . ) sothat both processes below

Gt :=

∫ t

0

αudBu +

∫ t

0

βudSu +∞∑i=2

∫ t

0

β(i)u dH(i)

u , (5)

Vt := αtBt + βtSt +∞∑i=2

β(i)t H

(i)t , (6)

are “well-defined” on [0, T ], and it holds that

Vt = V0 + Gt, (7)

for all t ∈ [0, T ]. Here, αt and βt represent the number of bonds and stock shares

held at time t, respectively. Similarly, β(i)t stands for the number of shares of

the ith jump-power asset at time t. The processes G and V are respectivelycalled the gain process and value or wealth process associated with the port-folio (α, β, β(1), . . . ). The process α can actually be determined by the initialendowment and the trading strategies for the risky assets.

Let us formalize the previous concepts. The trading strategies at hand arecharacterized by an initial endowment w and predictable processes (β, β(1), . . . )satisfying

Condition set 1.

(a) EQ[∫ T

0

∑i≥2

(β

(i)u

)2

du

]< ∞,

(b)∫ t

0βud (B−1

u Su) is well-defined.

We prefer to adopt the above rather general condition for β instead of restrictingthe trading strategy for the “real” stock. Condition 2 can be satisfied if forinstance β is locally bounded or

EQ∫ T

0

β2uS

2udu < ∞, (8)

4

(see Section 4.d in [5]). Notice that the processes

∫ t

0

β(i)u d

(B−1

u H(i)u

), i ≥ 2,

are square-integrable martingales and there exists a square-integrable martin-gale Mt0≤t≤T such that

sup0≤t≤T

∣∣∣∣∣Mt −n∑

i=2

∫ t

0

β(i)u d

(B−1

u H(i)u

)∣∣∣∣∣

n→∞−→ 0

in L2(Q) (see e.g. Section I.4a in [5]). The process Mt, denoted throughout∑∞i=2

∫ t

0β

(i)u d

(B−1

u H(i)u

), can be thought of as the gain coming from the ficti-

tious stocks. Given an initial endowment w and a trading strategy (β, β(1), . . . )satisfying the Condition Set 1, define

αt := w +

∫ t−

0

βud(B−1

u Su

)+ Mt− − βtSt−B−1

t −∞∑i=2

β(i)t H

(i)

t−B−1t ,

where the convergence of the last series is in L2(Q)1. Then, it can be provedthat taking α as above, the portfolio (α, β, β(1), . . . ) is self-financing in the sensethat the gain and value processes (5) and (6) satisfy (7). Also,

B−1t Vt = w +

∫ t

0

βud(B−1

u Su

)+

∞∑i=2

∫ t

0

β(i)u d

(B−1

u H(i)u

). (9)

Definition 1. We denote V w,πt the value process resulting from the self-financing

portfolio with initial endowment w and a trading strategy π := (β, β(1), . . . ). We saythat a value process V w,π is admissible or that π is admissible for w if

V w,πt ≥ 0, ∀ t ∈ [0, T ].

Remark 1. Since

Vt := B−1t Vt

0≤t≤T

is a local martingale, admissibility implies

that this discounted value process is a supermartingale and

EQ[B−1

T V w,πT

] ≤ w.

Remark 2. It is important to our analysis the result in [3] about the completenessof the augmented market MQ; namely, any FT -measurable Q-square-integrable ran-dom variable X can be replicated by a self-financing strategy with the initial capitalEQ

[B−1

T X].

1The convergence is guaranteed because T (i) = B−1t H

(i)t are orthogonal in L2(Q) and the as-

sumptions on (β(2), β(3), . . . ).

5

3 Formulation of the problems

Finding optimal trading strategies is one of the fundamental problems in mathemat-ical finance. A standard approach consists of ranking the trading strategies based ontheir resultant utilities, which are modeled as functions of their final wealths and/orthe consumption capability that the strategies allow during the trading period. Eventhough it is accustomed to consider utilities that depend solely on the final wealth,there are applications where state-dependent utility functions are desirable. One ofthese cases is related to the partial replication of contingent claims described below.The problem of utility maximization was formulated as early as Merton [11]-[12],who solved it in a Markovian Ito-process model. Let us remark that the problemsbelow depend on the structure-preserving measure Q because the admissible tradingstrategies are allowed to invest on the fictitious Tuegel’s stocks constructed from Q.

Minimization of shortfall risk: Given a contingent claim with payoff H at timeT , the problem is to find a trading strategy which minimizes the loss associatedwith the shortfall risk. Concretely, given a loss function l : R+ → R+, weconsider the minimization of

E[l((H − VT )+

)],

over all admissible portfolios under a constraint in the maximum allowed ini-tial endowment. More precisely, VT represents the final wealth of an arbitraryadmissible portfolio and the minimization is under the constraint

V0 ≤ c,

for some predetermined c > 0. The loss function l represents the investor’sattitude towards the shortfall, and is assumed to be convex increasing withl(0) = 0.

Maximization of utility functions: The above problem can readily be reformu-lated as a problem of utility maximization. The utility in question is state-dependent defined by

U(z, ω) = l(H(ω))− l((H(ω)− z)+).

Therefore, it is natural to consider the more general problem

uQ(c) := sup E [U(VT (ω), ω)] : V is admissible and V0 ≤ c , (10)

where U(z, ω) : R+ × Ω → R+ is a utility function for each ω. More precisely,we will assume the following:

Condition set 2.

1. U(·, ω) is nonnegative, non-decreasing, and continuous on R+,

6

2. There is a non-negative random variable H such that U(·, ω) is strictly con-cave with continuous derivative on (0, H(ω)) and constant on [H(ω),∞),

3. The mapping ω → U(·, ω) from Ω to C(0,∞) is FT -measurable.

From now on we shall focus on the more general problem of utility maximization.Using the completeness of the market MQ and the assumptions on U together withEQ [H2] < ∞, the problem in (10) is equivalent to

uQ2 (c) := sup E [U(Z(ω), ω)] : Z ∈ L2 (Ω,FT ,Q) satisfying (11)

0 ≤ Z ≤ H and EQ[B−1

T Z] ≤ c

.

Remark 3. Without assuming EQ [H2] < ∞, we simply have that uQ2 (c) ≤ uQ(c).Furthermore, defining

uQ1 (c) := sup E [U(Z(ω), ω)] : Z ∈ L1 (Ω,FT ,Q) satisfying (12)

0 ≤ Z ≤ H and EQ[B−1

T Z] ≤ c

,

we haveuQ2 (c) ≤ uQ(c) ≤ uQ1 (c).

Indeed, the second inequality follows because for every admissible portfolio with valueprocess V satisfying V0 ≤ c, the associated random variable Z := VT ∧ H is inL1 (Ω,FT ,Q), 0 ≤ Z ≤ H, and EQ

[B−1

T Z] ≤ c because of Remark 1.

Remark 4. As in [8] and [2], a standard strategy to prove the existence of an optimalportfolio made up only of bonds and the real stock is to show that for a completion(determined by Q), the fictitious stocks are actually superfluous.

Remark 5. Notice that in the case that l(x) = x, the corresponding state-dependentutility is not strictly concave on (0, H(ω)), and thus, we will need to make somemodification to the arguments below.

4 Existence and uniqueness of fictitious optimal

portfolio

Lemma 1. If E [U(H, ω)] < ∞, then uQ(c) < ∞. Moreover, the minimum value

(11) (respectively, (12)) is attained at unique Z provided that EQ [H2] < ∞ (resp.EQ [H] < ∞).

Proof. The first part is trivial. Let H be the family of random variables Z ∈L2(FT ,Q) such that EQ

[B−1

T Z] ≤ c and 0 ≤ Z ≤ H. Consider a sequence Zii≥1 ⊂

H such thatlim

n→∞E [U(Zn, ω)] = uQ2 (c).

7

By the Komlos theorem, there exists a subsequence Z ′ii≥1 of Zii≥1 such that

limn→∞

1

n

n∑i=1

Z ′i = Z, P− a.s.,

for a random variable Z ∈ L(FT ,P). Moreover, it is easy to check that Z ∈ H. Bythe monotone convergence theorem and the concavity of U(·, ω):

E[U(Z, ω)

]= lim

n→∞E

[U

(1

n

n∑i=1

Z ′i, ω

)]≥ lim

n→∞1

n

n∑i=1

E [U (Z ′i, ω)] = uQ2 (c).

For the uniqueness, suppose that Z1, Z2 ∈ H satisfy

E[U(Z1, ω)

]= E

[U(Z2, ω)

]= uQ2 (c).

Clearly, Z(ρ) = ρZ1 + (1− ρ)Z2 is in H and by the concavity of U ,

U(Z(ρ); ω

)≥ ρU

(Z1; ω

)+ (1− ρ)U

(Z2; ω

).

Moreover, since E[U

(Z(ρ); ω

)]= uQ2 (c), it follows that

U(Z(ρ); ω

)= ρU

(Z1; ω

)+ (1− ρ)U

(Z2; ω

), P− a.s.

Since U(·; ω) is strictly concave in (0, H(ω)), we have that Z1(ω) = Z2(ω), P−a.s.

5 Computation of the optimal final wealth in the

fictitious market

Inspired by the methods of duality in optimization, we can deduce an upper boundfor (11) in a standard manner. Denote

ξT :=dQT

dPT

,

where QT and PT stand for the restrictions of Q and P to FT , respectively. Now, takea square-integrable Z such that EQ

[B−1

T Z] ≤ z and 0 ≤ Z ≤ H. Then, for all y ≥ 0,

E [U (Z(ω), ω)] ≤ E [U (Z(ω), ω)]− y(EQ

[B−1

T Z]− z

)

= E[U (Z(ω), ω)− yξT B−1

T Z(ω)]+ zy

≤ E[

sup0≤u≤H(ω)

U (u, ω)− yξT B−1

T u]

+ zy

= E[U(yξT B−1

T , ω)]

+ zy,

8

where U(·; ω) is the dual function of U(·; ω) and I(·; ω) is the “inverse” of U ′(·; ω);namely,

U(y, ω) := sup0≤z≤H(ω)

U (z, ω)− yz

= U (I(y, ω) ∧H(ω), ω)− y (I(y, ω) ∧H(ω)) ,

I(y, ω) := inf z ∈ [0, H(ω)]|U ′(z, ω) < y ,

under the convention inf ∅ = ∞. Therefore, writing I(·) instead of I(·, ω),

uQ(z) ≤ E [U

(I(yξT B−1

T ) ∧H, ω)]− yE

[ξT B−1

T

(I

(yξT B−1

T

) ∧H)]

+ yz. (13)

The above bound is useful to characterize the solution to the primal problem (10).

Proposition 2. IfEQ

[H2

]< ∞ and E [U(H; ω)] < ∞, (14)

then for everyc < EQ

[B−1

T H], (15)

the problem (10) is attainable and the optimal final wealth takes the form:

V ∗T = I

(Y(c)ξT B−1T , ω

) ∧H, (16)

where Y(z) ≥ 0 is given by the equation

E[ξT B−1

T

(I

(Y(z)ξT B−1T

) ∧H)]

= z, (17)

for any 0 < z < EQ[B−1

T H].

Proof. First, we prove that Y exists. Clearly, U ′ is nonnegative, strictly decreasing,and continuous on (0, H), while U ′(z) = 0 for z > H. Therefore, I is continuous andstrictly decreasing on (U ′(H−),∞) with I(+∞) = 0. If U ′(H−) > 0, then I(y) = ∞for y < U ′(H−), while it will be H otherwise. Then, it is clear that the function:

T : y → I(yξT B−1

T

) ∧H

is almost surely nonincreasing, continuous such that T (0+) = H(ω), and T (+∞) = 0.By the monotone convergence theorem, the mapping

T : y → E[ξT B−1

T

(I

(yξT B−1

T

) ∧H)]

inherits monotonicity and continuity. Moreover,

T (0+) = E[ξT B−1

T H]

= EQ[B−1

T H]

and T (+∞) = 0.

Then, there exists Y(z) so that (17) holds. Plugging y = Y(c) in (13), it follows

uQ(c) ≤ E [I

(Y(c)ξT B−1T , ω

) ∧H].

Because EQ [H2] < ∞, Z = I(Y(c)ξT B−1

T , ω)∧H is Q−square-integrable, and hence,

(11) is attainable at this Z.

9

6 About the structure preserving equivalent mar-

tingale measures

In this part, we specify the class of equivalent martingale measures that preserve thestructure of the model, in the sense that the process Z remains a Levy process. Someof the subsequent observations were pointed out in [2]. Let us first recall the followingwell-known result (see e.g. Section 33 in [17] or [16]).

Theorem 3. Let Z = Zt, 0 ≤ t ≤ T be a Levy process with Levy triple (σ2, ν(dx), α)under some probability measure P.

1. The following two conditions are equivalent:

(a) There is a probability measure Q equivalent to P such that Z is a Levyprocess with triplet (σ2, ν, α)

(b) The triplet (σ2, ν, α) satisfy

i. ν(dx) = H(x)ν(dx) for some Borel function H : R→ R+ with

∫ ∞

−∞

(1−

√H(x)

)2

ν(dx) < ∞. (18)

ii. α = α +∫|x|<1

x(H(x)− 1)ν(dx) + Gσ, for some G ∈ R.

iii. σ = σ.

2. Suppose that the equivalent conditions above are satisfied. Then, the densityprocess is given by

ξ(t) :=dQt

dPt

= exp

GWt − 1

2G2t

+ limε→0

(∫ t

0

∫

|x|>ε

log H(x)N(ds, dx)− t

∫

|x|>ε

(H(x)− 1) ν(dx)

),

where N is the Poisson random measure on R+ × R associated with the jumpsof Z. Moreover, the convergence in the second term above is uniform in t onany bounded interval.

Remark 6. The Levy-Ito decomposition asserts that Z = Zt, 0 ≤ t ≤ T has Levytriple (σ2, ν(dx), α) if and only if, almost surely, Z has the decomposition

Zt = σWt + Xt, ∀t ≥ 0,

where W is a standard Brownian motion and

Xt :=

∫ t

0

∫

|x|<1

x (N(ds, dx)− dsν(dx)) +

∫ t

0

∫

|x|≥1

xN(ds, dx) + αt, (19)

10

for a Poisson random measure N on R+ ×R with intensity ν(dx)× dt (see e.g. [6]).Furthermore, if

∫|x|≥1

|x|ν(dx) < ∞, then

Lt := Xt −(

α +

∫

|x|≥1

xν(dx)

)t =

∫ t

0

∫

R0

x (N(ds, dx)− dsν(dx)) ,

is a martingale. Similarly, if Z has triplet (σ2, ν, α) under Q, then we have:

1. The random measure N associated with the jumps of Z is a Poisson randommeasure with intensity H(x)ν(dx)× dt under Q.

2. The process Zt−Xt, is a Brownian motion with variance σ2, where X is definedas X in (19) with ν and α instead of ν and α. It is easy to check that Z − Xsimplifies to σ (Wt −Gt), and hence,

Wt := Wt −Gt,

is a standard Brownian motion under Q.

3. The process

Lt := Xt −(

α +

∫

|x|≥1

x ν(dx)

)t,

= Xt −(

α +

∫

|x|<1

x(H(x)− 1)ν(dx) +

∫

|x|≥1

xH(x)ν(dx)

)t

is a martingale provided that∫|x|≥1

|x|ν(dx) < ∞.

We are now in position to introduce a family of equivalent martingale measures thatpreserves the structure of the model.

Proposition 4. Let H : R→ R+ be a Borel function satisfying (18) and

∫

|x|≥1

|x|H(x)ν(dx) < ∞. (20)

Define G such that

b− r + σG + α +

∫

|x|<1

x(H(x)− 1)ν(dx) +

∫

|x|≥1

xH(x)ν(dx) = 0. (21)

Then, under the probability measure Q := QH

defined by H and G via Theorem 3,the following is true:

1. Zt := Zt − (r − b)t is a Levy process and a martingale.

11

2. The discounted stock price

St := B−1t St

0≤t≤T

satisfies the equation

dSt = St−dZt,

and hence, it is a local martingale. Moreover, it is a square-integrable martingaleif ∫

|x|≥1

x2H(x)ν(dx) < ∞.

Proof. Let W and L be the processes defined in the Remark 6. Using (21), it can beseen that

σWt + Lt = Zt − (r − b)t = Zt,

and thus, Z is a martingale under Q because of points 2 and 3 in Remark 6. DenoteK the left hand side of equality (21). We can write

St = S0 exp

−rt + σWt + Xt + bt− σ2

2t +

∑0<s≤t

[log (1 + ∆Zs)−∆ Zs]

= S0 exp

Kt + σWt + Lt − σ2

2t +

∑0<s≤t

[log

(1 + ∆Ls

)−∆ Ls

].

Therefore, if K = 0, then S = S0E(σW + L

), the stochastic exponential of the

normal martingale σWt + Lt (see e.g. [15]). It follows that S is a local martingalebecause it satisfies the SDE:

St = S0 +

∫ t

0

Su−dZu. (22)

Moreover, it can be seen that

S2t = S2

0 E(

2σWt +

∫ t

0

∫ ∞

−∞

((1 + x)2 − 1

)NQ(ds, dx)

)e(σ2+

Rx2ν(dx))t,

where NQ is the compensated Poisson process associated with the jumps of Z underthe probability measure Q). Since the stochastic exponential in the previous equationis a local martingale and a supermartingale, we have

sup0≤t≤T

EQ[S2

t

]≤ sup

0≤t≤TS2

0e(σ2+

Rx2ν(dx))t < ∞,

and hence, Stt≤T is a square-integrable martingale.

Remark 7. In fact, we can see that (21) is also a necessary condition for both con-dition 1 and 2 in the previous theorem to hold true. A much more larger class ofmartingale equivalent measures can be devised, but Z will not longer be a Levy process.

12

7 Determining the minimum endowment needed

to replicate a contingent claims

Throughout this part, Q will stand for the family of equivalent probability measuresQ defined in Theorem 3 with H : R→ R+ and G satisfying the conditions below.

Condition set 3.

(i)∫R H(x)− 1− log H(x) ν(dx) < ∞,

(ii)∫|x|≥1

|x|H(x)ν(dx) < ∞,

(iii) b− r + σG + α +∫|x|<1

x(H(x)− 1)ν(dx) +∫|x|≥1

xH(x)ν(dx) = 0.

Notice that if Q ∈ Q then its underlying density process ξ(t) can be written in theform

ξ(t) :=dQt

dPt

= exp

GWt − 1

2G2t +

∫ t

0

∫

R0

log H(x)N(ds, dx) (23)

−t

∫

R0

(H(x)− 1− log H(x)) ν(dx)

,

where N(dt, dx) := N(dt, dx)−ν(dx)dt is the compensated Poisson process associatedwith the jumps of Z (see Theorem 3).Let Vtt≤T be the value process of a portfolio with initial endowment w and tradingstrategy for the stock given by a predictable and locally bounded process β = βtt≤T :

B−1t Vt := w +

∫ t

0

βudSu. (24)

By Proposition 4, the processVt := B−1

t Vt,

is a local martingale under each probability measure Q ∈ Q. In particular, if theportfolio is admissible, then V is a supermartingale and the “budget constraint”

EQ[B−1

T VT

] ≤ w.

is satisfied. Let us give a different interpretation to the above constraint. Suppose wewish to hedge the risk inherent in a contingent claim X. In other words, we wish toconstruct an admissible portfolio with value process Vt : t ≤ T such that VT ≥ X.What is the minimum endowment needed to hedge against X? The budget constraintabove implies that we need at least

w∗ := supQ∈Q

EQ[B−1

T X]

(25)

to hedge against X. That is to say, if we were able to construct an admissible portfoliowith value process V such that VT ≥ X, then V0 ≥ w∗. It is natural to wonder if

13

w ≥ w∗ is a sufficient condition to construct a portfolio with initial endowment w tohedge against X. We will see that under some assumption on X, this is true if themaximum (25) is attainable.Variations of the problem just described have been considered in the literature underdifferent assumptions. For instance, in the general model where the asset price processis a semimartingales and for an arbitrary contingent claim X, if

EQ[B−1

T X] ≤ w,

for all equivalent martingale measures Q, then X is “attainable” with an initial en-dowment w, in the sense that there exists an admissible strategy β for w such that

B−1T X ≤ w +

∫ T

0

βudSu,

(see for instance [9] and [10]). In the context of non-Markovian models driven by Itoprocesses, [8] constructs a family Ξ of positive supermartingales, each associated withcertain fictitious completion of the market, such that if there exists a ξ∗ satisfying

supξ∈ΞE

[B−1

T XξT

]= w∗ = E

[B−1

T Xξ∗T

],

then there exits an admissible portfolio β for w such that

B−1T X = w +

∫ T

0

βudSu

(see their Theorem 8.5 for details). The strategy to get this result is a formula forthe “directional derivative” of the “value of the claim” E

[B−1

T XξT

]. We analyze the

analog of such a formula in our context.

Proposition 5. Let H : R→ R+ be a function satisfying the Condition Set 3 and letξ and Q be its resulting density process and probability measure, respectively, definedby (23). Let D : R→ R be such that

supx∈R

|D(x)| < 1, and lim supx→0

|D(x)||x| < ∞. (26)

Then, the following is true:

1. The functionHε(x) := H(x) (1 + εD(x)) ,

satisfies (i)-(ii) in the Condition Set 3 for each ε ∈ (−1, 1).

2. Let Gε be such that (iii) in the Condition Set 3 is met when replacing G andH by Gε and Hε. Define the density process ξε and probability measure Qε

by (23) with G and H replaced by Gε and Hε. Then, for any FT -measurable,

Q−square-integrable X,

limε↓0

1

ε

(EQε

[X

]− EQ

[X

])= EQ

[X

σ

DW

QT +

∫ T

0

∫

RD(x)N

Q(ds, dx)

],

14

where

σD

:=−1

σ

∫

RxD(x)H(x)ν(dx), (27)

WQt := Wt − tG, (28)

NQ(ds, dx) := N(ds, dx)−H(x)ν(dx)ds. (29)

Proof. Notice that the integrals∫∞−∞ |xD(x)|H(x)ν(dx),

∫∞−∞ D2(x)H(x)ν(dx), and∫∞

−∞ ±D(x)− log (1±D(x)) ν(dx) are finite. Indeed, the integrals are clearly finite

outside any neighborhood of the origin because∫|x|≥1

|x|H(x)ν(dx) < ∞, and the

integrands are bounded in the last two cases. There is also a neighborhood O ofthe origin, such that |D(x)| ≤ K|x|, for x ∈ O and some constant K, implying that∫

O|xD(x)|H(x)ν(dx) < ∞ (same for the other two integrals). Let us verify that Hε

satisfies (i) in the Condition Set 3 (condition (ii) is straightforward). We have that∫ ∞

−∞Hε(x)− 1− log Hε(x) ν(dx) ≤

∫ ∞

−∞H(x)− 1− log H(x) ν(dx)

+ ε

∫ ∞

−∞|D(x)| |H(x)− 1| ν(dx)

+

∫ ∞

−∞εD(x)− log (1 + εD(x)) ν(dx),

and the three integrals above are finite because ε ∈ (−1, 1) and∫

|x|<1

|x| |H(x)− 1| ν(dx) < ∞.

Before proving the second part, let us recall that

WQt

t≤T

is a standard Brownian

motion and NQ

is a compensated Poisson process under the probability measure Q(see Remark 6). In particular, under Q, the process

Yt := σDW

Qt +

∫ t

0

∫

RD(x)N

Q(ds, dx) , 0 ≤ t ≤ T,

is a square-integrable (normal) martingale with quadratic variation

〈Y 〉t = t

(σ2

D+

∫ ∞

−∞D2(x)H(x)ν(dx)

),

(see e.g. Theorem II.1.33 in [5]). Let us denote γ := σ2D

+∫∞−∞ D2(x)H(x)ν(dx). We

can write

ζε(t) :=ξε (t)

ξ (t)= exp

εσ

DW

Qt −

1

2(εσ

D)2 t

+

∫ t

0

∫

Rlog (1 + εD(x)) N

Q(ds, dx)

−t

∫

RεD(x)− log (1 + εD(x))H(x)ν(dx)

.

15

Then, ζε = E (εY ), the stochastic exponential of the normal martingale εY . So,

ζε(t) = 1 + ε

∫ t

0

ζε(s−)dYs, (30)

and

1

ε

(EQε

[X

]− EQ

[X

])= EQ

[X

1

ε(ζε(T )− 1)

]

= EQ

[X

∫ T

0

ζε(t−)dYt

].

Then, we only need to prove that limε→0 EQ[X

∫ T

0(ζε(t

−)− 1) dYt

]= 0. By Cauchy’s

inequality,

E2Q

[X

∫ T

0

(ζε(t

−)− 1)dYt

]≤ γEQ

[X2

]EQ

[∫ T

0

(ζε(t)− 1)2 dt

],

where we used thatEQ

[ζ2ε (t)

]= etε2γ, (31)

as we will see in a moment, and thus,∫ t

0(ζε(s

−)− 1) dYs is a Q−square-integrable

martingale with quadratic variation γ∫ t

0(ζε(s)− 1)2 ds. This is enough because

∫ T

0

EQ[(ζε(t)− 1)2] dt =

∫ T

0

(etε2γ − 1

)dt → 0,

as ε → 0 (recall that Eζε(t) = 1). Equation (31) can be deduced as follows. It is nothard to see that

ζ2ε (t) = Y ′

ε (t)etε2γ,

where

Y ′ε := E

(2εσ

DW

Q· +

∫ ·

0

∫

R

(1 + εD(x))2 − 1

NQ(ds, dx)

).

The driven process of the above stochastic exponential is a well-defined local mar-tingale (see Theorem II.1.33 (d) in [5]), and thus, Y is a local martingale and asupermartingale. Then,

EQ[ζ2ε (t)

] ≤ etε2γ.

In view of (30), the process ζε(t)t≤T is a square-integrable martingale with quadratic

variation γε2∫ t

0ζ2ε (s)ds. In particular,

EQ[ζ2ε (t)

]= 1 + ε2γ

∫ t

0

EQ[ζ2ε (s)

]ds,

implying (31).

The result above has interesting implications. The following is a direct corollary.

16

Corollary 6. Let Q∗ be an equivalent martingale measure in Q with an associatedfunction H : R→ R+ satisfying the Condition Set 3, and let X be an FT -measurable,Q∗−square-integrable contingent claim such that

EQ[X

]≤ EQ∗

[X

],

for all Q ∈ Q. Then, for any function D : R→ R satisfying (26),

EQ∗[X

σ

DW

Q∗

T +

∫ T

0

∫

RD(x)N

Q∗(ds, dx)

]= 0, (32)

where σD, W

Q∗, and N

Q∗are defined by (27)-(29).

Remark 8. We can extend the previous result to a much larger class of functions D.For instance, if D0 is the class of functions D : R→ R+ that satisfy (26), then (32)is true for any D ∈ span (D0). Define the vector space D of functions D : R → Rsuch that

limn→0

∫

R(D(x)−Dn(x))2 H(x)ν(dx) = 0,

for a sequence Dnn≥1 ⊂ span (D0). Then, it is not hard to see that equation (32)holds for every D ∈ D, under the additional condition that

∫R x2H(x)ν(dx) < ∞.

Also, the polynomials x, x2, . . . belong to D if∫|x|>1

|x|iH(x)ν(dx) < ∞.

Suppose that Q∗ and X satisfy the condition in the previous Corollary. It is naturalto ask what implications have that (32) holds for such a wide class of functions

D : R → R. We proceed to characterize X using its Predictable Representation interms of the orthonormalized Teugel’s martingales T (i) : i ≥ 1 constructed fromthe Levy process Z with respect to probability measure Q∗ (see Section 2 for thenotation and [13] for details on the Predictable Representation). In order for suchconstruction to be possible, we need that

∫

(−ε,ε)c

eλ|x|H(x)ν(dx) < ∞, (33)

for some λ, ε > 0. In particular,∫R x2H(x)ν(dx) < ∞. We assume this from now on.

Below, we use the notation introduced in the point 4 of Section 2.

Proposition 7. Let Q∗ be an equivalent martingale measure in Q and let X be anFT -measurable, Q∗−square-integrable contingent claim such that

EQ[X

]≤ EQ∗

[X

],

for all Q ∈ Q. Define WQ∗

and NQ∗

as in (28)-(29). The following two statementsare true:

17

1. If D : R→ R satisfies ∫

RD2(x)H(x)ν(dx) < ∞

and σD

is as in (27), then

σD

WQ∗

t +

∫ t

0

∫

RD(x)N

Q∗(ds, dx) =

∞∑i=2

di (D) T(i)t , (34)

where

di(D) :=

∫

RD(x) (qi−1(x)− ci,1) xH(x)ν(dx).

2. If T (i) : i ≥ 1 are the orthonormalized Teugel’s martingales constructed fromthe Levy process Z with respect to probability measure Q∗, then

EQ∗[X T (i)

T

]= 0,

for all i ≥ 2.

Proof. Define

Mt := σD

WQ∗

t +

∫ t

0

∫

RD(x)N

Q∗(ds, dx).

By Remark 6 and the conditions on D, the process Mt : 0 ≤ t ≤ T is a square-integrable martingale with respect to Q∗ and thus, it enjoys a predictable martingalerepresentation of the form

Mt =∞∑i=1

∫ t

0

β(i)s d T (i)

s ,

for predictable processes β(i) : i ≥ 1 (see [13]). Because of the strong pair-wiseorthogonality of the Teugels martingales and the fact that

⟨T (i), T (i)

⟩t= t,

⟨M, T (i)

⟩t=

∫ t

0

β(i)s ds.

On the other hand, using (4), Remark 6 and the definition of T (i) given in Section 2,

[M, T (i)

]t=

⟨M c,

(T (i)

)c⟩

t+

∑s≤t

∆Ms∆T (i)s

= ci,1σσDt +

∑s≤t

D (∆Zs) pi (∆Zi) .

Then, being the predictable compensator of[M,T (i)

],

⟨M,T (i)

⟩t= ci,1σσ

Dt +

∫ t

0

∫

RD(x)pi(x)H(x)ν(dx)ds

= t

∫

RD(x) (qi−1(x)− ci,1) xH(x)ν(dx)

18

because pi(x) = xqi−1(x). Clearly,⟨M,T (1)

⟩t

= 0. This concludes the proof of thefirst part. For the second part, we use equations (32), (34), Remark 8, and the fact

that the T (i) : i ≥ 1 are pair-wise strongly orthogonal. Indeed, if X has predictablerepresentation

X = EQ∗[X

]+

∞∑i=1

∫ T

0

β(i)s d T (i)

s ,

then

EQ∗[XMT

]=

∞∑i=2

di (D)EQ∗[∫ T

0

β(i)s ds

]= 0,

for all D satisfying the conditions in Remark 8. In particular, if D(j)(x) := xqj−1(x)with j ≥ 2, then for 2 ≤ i 6= j

di

(D(j)

)=

∫

Rqj−1(x) (qi−1(x)− ci,1) x2H(x)ν(dx)

= (qi−1, qj−1)eν − ci,1 (qj−1, 1)eν = 0,

because the polynomial qi−1 : i ≥ 1 form an orthonormal system with respect to

the inner product (·, ·)eν defined in (3) and q0(x) = c1,1. Then, EQ∗[∫ T

0β

(i)s ds

]= 0,

for i ≥ 2.

Corollary 8. Under the assumptions of Proposition 7, if X enjoys a predictablerepresentation of the form

X = EQ∗[X

]+

∫ T

0

β(1)s d T (1)

s +∞∑i=2

γi

∫ T

0

φ(i)s d T (i)

s , (35)

for some reals γii≥2 and non-negative processes φ(i) : i ≥ 2, then X can be

replicated in the real market with an initial endowment EQ∗[X

]. In other words,

γi = 0 for all i ≥ 2, and the fictitious stocks are superfluous to replicate the contingentclaim.

8 Optimal investment in the real market

Let us consider the problem of optimally investing in the real market consisting of thebond and stock, subject to an initial endowment c. Denote u(c) the optimal expectedutility:

u(c) = sup E [U (V πT (ω), ω)] : V0 ≤ c , (36)

where the maximum is over all admissible trading strategies π = (αt, βt) for the bondand stock, and

V πt = V0 +

∫ t

0

αudBu +

∫ t

0

βudSu.

19

Notice that due to the self-financing condition, the only “controls” are in fact β andV0. Since the stock and the bond are embedded in any of the augmented market MQ,we have

u(c) ≤ uQ(c),

for any structure preserving equivalent martingale measure Q. Therefore,

u(c) ≤ infQ∈Q

E[U

(I

(Y(c)

dQT

dPT

B−1T

)∧H, ω

)]. (37)

It is natural to ask whether or not there is a gap in the inequality above. The follow-ing trivial lemma is a direct consequence of the uniqueness of replicating portfolios.Essentially, it says that if the optimal expected final wealth in a completion MQ isthe same as that of a portfolio consisting only of stocks and bonds, then the gap is 0and the fictitious stocks are superfluous.

Lemma 9. If there exist a portfolio π = (α, β) and a structure preserving measureQ ∈ Q such that

E [U (V πT (ω), ω)] = E

[U

(I

(Y(c)

dQT

dPT

B−1T

)∧H, ω

)],

then equality holds in (37),

V πT (ω) = I

(Y(c)

dQT

dPT

B−1T

)∧H, a.s.,

and the optimal portfolio in MQ does not invest in the fictitious stocks.

Much more interesting is the reciprocal problem: What are necessary conditions forthe gap in (37) to be 0?

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