Convergence of utility indifference prices to the superreplication price : the whole real line case∗ Laurence Carassus Laboratoire de Probabilit´es et Mod`eles Al´eatoires Universit´e Denis Diderot, Paris 7

Mikl´os R´asonyi Computer and Automation Institute of the Hungarian Academy of Sciences and Vienna University of Technology

October 24, 2006

Abstract A discrete-time financial market model is considered with a sequence of investors whose preferences are described by concave strictly increasing functions defined on the whole real line. Under suitable conditions we prove that, whenever their absolute riskaversion tends to infinity, the respective utility indifference prices of a given bounded contingent claim converge to the superreplication price. We also prove that there exists an accumulation point of the optimal strategies’ sequence which is a superhedging strategy.

Keywords: derivative pricing, utility indifference price, superreplication, utility maximization.

1

Introduction

We consider a sequence of investors indexed by n; preferences of investor n are expressed via the choice of his or her concave strictly increasing utility function Un with dom(Un ) = R. The utility indifference price (also called reservation price) for the seller of a contingent claim has been introduced by Hodges and Neuberger (1989). It is the minimal amount a seller should add to his or her initial wealth so as to reach an optimal expected utility when delivering the claim which is greater than or equal to the one he or she would have obtained trading in the basic assets only. The superreplication price is the minimal initial wealth needed for hedging the claim without risk; this is thus a utility-free pricing concept. ∗

The authors thank their laboratories for hosting this research. The visit of L. Carassus was financed by the EU Centre of Excellence programme and that of M. R´ asonyi by Universit´e Paris 7, M. R´ asonyi was supported by Hungarian National Science Foundation (OTKA) grants T 047193, F 049094; Austria Science Fund (FWF) grant P 15889 and EU Research Training Network HPRN-CT-2002-00281.

1

In the exponential case where Un (x) = −e−αn x for some positive sequence αn , n ∈ N the asymptotic properties of utility indifference prices are well understood. In Becherer (2003) and Stricker (2004) it was shown that if αn → 0 then the reservation prices of a claim tend to its expectation with respect to the so-called minimal entropy martingale measure. If αn converges to ∞ it was found that reservation prices tend to the superreplication price, see e.g. Delbaen et al. (2002). This (economically intuitive) latter result has lacked a generalization for arbitrary utility functions up to now. In such a generalization αn , the risk-aversion parameter of Un ought to be replaced by the absolute risk aversion function rn := −Un00 /Un0 . In Carassus and R´asonyi (2006a) we managed to prove, in appropriate discrete-time market models, that the convergence of utility indifference prices to the superreplication price takes place for bounded contingent claims when rn tends to infinity pointwise. A drawback of that paper was, however, the use of utility functions defined on (0, ∞), hence the theorem did not subsume the well-known exponential case. The aim of the present paper is to prove this convergence result for utility functions defined on the whole real line, in a suitable class of discrete-time market models. We will use ideas of Carassus and R´asonyi (2006a) but, as usual, the corresponding utility maximization problem is more difficult to treat in the whole real line case. Another natural conjecture is: do the optimal strategies converge to some superhedging strategy ? This is false in general, see section 2 of Cheridito and Summer (2006). We will show, however, that some accumulation point of the sequence of optimal strategies is a superhedging strategy. This issue did not arise in the case dom(Un ) = (0, ∞) because the optimal strategies were superreplicating ones by the definition of admissible trading policies. In section 2 we present the model and the main theorems. Section 3 develops a few facts about utility maximization. Section 4 proves the main results. The appendix contains some technical material used in the proofs.

2

Definitions, assumptions and results

Let (Ω, F, (Ft )0≤t≤T , P ) be a discrete-time filtered probability space with time horizon T ∈ N. We assume that F0 coincides with the family of P -zero sets. If Y is a random variable, we denote by supω∈Ω Y its essential supremum (in R ∪ {∞}). Let {St , 0 ≤ t ≤ T } be a d-dimensional adapted process representing the discounted (by some num´eraire) price of d securities in a given economy. The notation ∆St := St −St−1 will often be used. Self-financing trading strategies are given by d-dimensional processes {φt , 1 ≤ t ≤ T } which are supposed to be predictable (i.e. φt is Ft−1 -measurable). The class of all such strategies is denoted by Φ. Denote by L∞ (L∞ + ) the sets of bounded (nonnegative bounded) random variables on (Ω, F, P ), equipped with the supremum norm k · k∞ . Trading is assumed to be self-financing, so the value process of a portfolio φ ∈ Φ is Vtz,φ

:= z +

t X j=1

2

hφj , ∆Sj i,

where z is the initial capital of the agent in consideration and h·, ·i denotes scalar product in Rd . The following absence of arbitrage condition is standard: (N A) : ∀φ ∈ Φ (VT0,φ ≥ 0 a.s. ⇒ VT0,φ = 0 a.s.). However, we need to assume a certain strengthening of the above concept hence an alternative characterization is provided in the Proposition below. Therefore, we introduce the following sets. Let Ξt denote the set of Ft -measurable d-dimensional random variables. Denote by Dt (ω) the smallest affine hyperplane containing the support of the (regular) conditional distribution of ∆St with respect to Ft−1 . If Dt = Rd then, intuitively, there are no redundant assets. Otherwise, one may always replace Φt ∈ Ξt−1 by its orthogonal ˆ t on Dt without changing the portfolio value since a.s. hΦt , ∆St i = hΦ ˆ t , ∆St i. projection Φ See also Korn and Sch¨al (1999), Sch¨al (2000) and Proposition A.1 of R´asonyi and Stettner (2005) for more information about the random set Dt . Define ˜ t := {ξ ∈ Ξt : ξ ∈ Dt+1 a.s., |ξ| = 1 on {Dt+1 6= {0}}}. Ξ The following proposition give a reformulation of the (NA) condition. Compare to Theorem 3 of Jacod and Shiryaev (1998), see also Remark 3.5 in Sch¨al (2000), Remark 4.2 in Korn and Sch¨al (1999) and Remark 1 in Carassus and R´asonyi (2006a). Proposition 2.1 (NA) holds iff there exist Ft -measurable, strictly positive, random variables κt , βt , 0 ≤ t ≤ T − 1 such that ess. inf P (hξ, ∆St+1 i < −βt |Ft ) > κt a.s. on {Dt+1 6= {0}}. ˜t ξ∈Ξ

(1)

Proof. The direction (N A) ⇒ (1) is Proposition 3.3 of R´asonyi and Stettner (2005). The other direction is clear from the implication (g) ⇒ (a) in Theorem 3 of Jacod and Shiryaev (1998). 2 In Carassus and R´asonyi (2006a) a strengthening of (NA) was required to derive the main result: the so-called “uniform no-arbitrage” condition of Korn and Sch¨al (1999) and Sch¨al (2000) which requires βt to be a constant. In order to obtain Theorems 2.4 and 2.6 of the present paper we need an even stronger hypothesis : neither κt is allowed to be an Ft -measurable positive random variable, it also has to be a constant. Assumption 2.2 There exist constants β, κ > 0 such that ess. inf P (hξ, ∆St+1 i < −β|Ft ) > κ a.s. on {Dt+1 6= {0}}. ˜t ξ∈Ξ

Note that if Assumption 2.2 fails, i.e. either βt or κt is not constant, then the utility maximisation problem may be ill posed (see Example 3.3 of Carassus and R´asonyi (2006b)). We go on incorporating a sequence of agents in our model with concave utility functions Un . The functions rn below express the absolute risk-aversion of the respective agents (see Arrow (1965) and Pratt (1964)). Assumption 2.3 Suppose that Un : R → R, n ∈ N is a sequence of concave strictly increasing twice continuously differentiable functions such that ∀x ∈ R

rn (x) := −

Un00 (x) → ∞, n → ∞. Un0 (x) 3

Fix G ∈ L∞ + , a random variable which will be interpreted as the payoff of some derivative security at time T and define un (G, z) :=

sup φ∈Φ(Un ,G,z)

EUn (VTz,φ − G),

(2)

where Φ(Un , G, z) denotes the family of strategies φ ∈ Φ such that EUn (VTz,φ − G) exists. The quantity un (G, z) represents the supremum of expected utility from initial capital z delivering a contingent claim with payoff G at the terminal date. In the following Theorem we show that, due to the specific convergence of Un as n → ∞ (see Lemma 5.4), we can construct optimal strategies ψn∗ (z) for the problem (2), for n large enough, without the usual “asymptotic elasticity” conditions (see e.g. Kramkov and Schachermayer (1999) and Schachermayer (2001)). Theorem 2.4 Suppose that S is bounded, Assumptions 2.2 and 2.3 hold. Then there exists N0 ∈ N such that for all n ≥ N0 and z ∈ R, the utility maximization problems (2) admit optimal solutions ψn∗ (z). We now formally define the utility indifference price which is the minimal amount a seller should add to his or her initial wealth so as to reach an optimal expected utility when delivering the claim which is greater than or equal to the one he or she would have obtained trading in the basic assets only. Definition 2.5 The utility indifference price pn (G, x) is defined as pn (G, x) = inf{z ∈ R : un (G, x + z) ≥ un (0, x)}. The concept of superreplication price is utility free and it is the minimal initial wealth needed for hedging without risk the given contingent claim G: π(G) := inf{z ∈ R : VTz,φ ≥ G a.s. for some φ ∈ Φ}. In fact, under (NA), this infimum is a minimum, see e.g. F¨ollmer and Kabanov (1996, 1998). We wish to find conditions on S and Un which guarantee that pn (G, x) tends to π(G) whenever Assumption 2.3 holds. Theorem 2.6 Suppose that S is bounded, Assumptions 2.2 and 2.3 hold. Then for each x0 ∈ R the utility prices pn (G, x0 ) are well-defined (for n sufficiently large, namely n ≥ N0 ) and converge to π(G) as n → ∞. The last problem we address is the convergence of the optimal strategies of the utility maximization problems (2) from initial capital z ≥ π(G) to some superreplication strategy (i.e some φ ∈ Φ such that VTz,φ ≥ G). In Grandits and Summer (2006) it is shown that if Ω is finite then the optimal strategies indeed converge to a particular superreplication strategy provided that lim inf rn (x) = +∞. n x∈R

4

(3)

This convergence does not hold true on general Ω, see Cheridito and Summer (2006). For the one-step case where T = 1, this problem has been studied by Cheridito and Summer (2005) and Summer (2002). They introduced some special super replication strategies, called balanced strategies, and they showed that the distance between the set of such strategies and the optimal ones goes to zero if (3) holds. Under our weaker Assumption 2.3 we manage to establish a related result for multistep models and general probability space which, as far as we know, is the first of its kind. We assert the existence of an accumulation point of the sequence of optimal strategies which is a superhedging strategy. To this end, we have to clarify what we mean by “an accumulation point of the sequence of optimal strategies” (this is rather obvious in one-step models). Remember that a uniformly bounded sequence of random variables does not necessarily have an a.s. convergent subsequence but it always admits an a.s. convergent random subsequence (i.e. the subsequence should be chosen using the information Ft−1 available at that moment) : see Lemma 2 of Kabanov and Stricker (2001). So the first thing to note is that, at time t, one should take random subsequences, see (i) in the definition below. The second thing is that one should “remember what happened before”, i.e. select the subsequence at time t in such a way that it is nested in the subsequence chosen at the previous time instant, see (ii) below. Definition 2.7 We say that a strategy ζ = {ζt , 1 ≤ t ≤ T } ∈ Φ is an accumulation point of a sequence φn = {φn,t , 1 ≤ t ≤ T } ∈ Φ of strategies if it can be obtained in the following recursive way : for all 1 ≤ t ≤ T there exist N-valued random variables σk,t such that (i) σk,t , k ∈ N are Ft−1 -measurable, σk,t < σk+1,t a.s.; (ii) (σk,t+1 )k≥0 is a subsequence of (σk,t )k≥0 almost surely for all 1 ≤ t ≤ T − 1; (iii) ζt = limk→∞ φσk,t ,t almost surely, for each 1 ≤ t ≤ T . The ensuing result has a clear economic interpretation: if the risk-aversion of Un tends to infinity then one can asymptotically superhedge the given contingent claim using a suitably selected subsequence of the sequence of optimal strategies computed for the Un . Theorem 2.8 Suppose that S is bounded, Assumptions 2.2 and 2.3 hold. Let z ≥ π(G). ∗ (z), 1 ≤ t ≤ T }) The sequence of optimal strategies (ψn∗ (z))n≥N0 = ({ψn,t n≥N0 defined in Theorem 2.4 admits some accumulation point ζ(z) = {ζt (z), 1 ≤ t ≤ T }, which is a superhedging strategy. Remark 2.9 We present a proof of our main results working on the primal problem only. It is possible to prove Theorem 2.6 under very similar assumptions relying on duality techniques. However, the latter method does not show the uniform boundedness of strategies which is crucial in Theorem 2.8. It would also require a analysis of the structure of equivalent martingale measures, which is not straightforward. Hence we opted for the present approach.

3

Utility maximization

In this section we use a dynamic programming procedure to prove the existence of optimal strategies and to derive bounds on them. We first introduce the recursive superreplication 5

price of any contingent claim G ∈ L∞ + : πT (G) := G, πt (G) := ess. inf{X : σ(X) ⊂ Ft , ∃φ ∈ Ξt such that X + hφ, ∆St+1 i ≥ πt+1 (G) a.s.}, for 0 ≤ t ≤ T − 1. Note that π0 (G) can be chosen constant, by the triviality of F0 . It is easy to see that π0 (G) = π(G) (see Proposition 5.1) and 0 ≤ πt (G) ≤ kGk∞ . We will also use the sets VT = {0} and Vt := {

T X

hζj , ∆Sj i,

ζj ∈ Ξj−1 , j = t + 1, . . . , T }, 0 ≤ t ≤ T − 1.

(4)

j=t+1

The following Theorem provides optimal strategies (for n large enough) and derives bounds on them which will be crucial in the sequel, in a spirit similar to that of Theorem 4.1 in Carassus and R´asonyi (2006b). Convergence properties of the value function are also established below (see (12) and (13)). Theorem 3.1 Suppose that S is bounded and Assumptions 2.2, 2.3 and Un (0) = 0, Un0 (0) = 1 hold for all n ∈ N. Then there exist constants Ns , 0 ≤ s ≤ T such that for n ≥ Ns the random functions Un,s below are well defined and finite: Un,T (x) := Un (x − G), Un,s (x) := ess. sup E(Un,s+1 (x + hξ, ∆Ss+1 i)|Fs ),

0 ≤ s ≤ T − 1.

ξ∈Ξs

For all 1 ≤ s ≤ T , i) The functions Un,s−1 have almost surely concave and increasing continuously differentiable versions. ii) For all x ∈ R, y ∈ Rd , −∞ < EUn,s (x + hy, ∆Ss i) < ∞,

(5)

iii) For n ≥ Ns and x ∈ R, there exists ξ˜n,s (x) ∈ Ξs−1 such that ξ˜n,s ∈ Ds a.s. and Un,s−1 (x) = E(Un,s (x + hξ˜n,s (x), ∆Ss i)|Fs−1 ), U0 (x) = E(U 0 (x + hξ˜n,s (x), ∆Ss i)|Fs−1 ). n,s−1

n,s

(6) (7)

iv) For all n ≥ N0 , z ∈ R the utility maximization problems EUn (VTz,ψ − G) → max., ψ ∈ Φ(Un , G, z), admit optimal solutions ψn∗ (z) given by ∗ ψn,1 (z) := ξ˜n,1 (z),

∗ ψn,t+1 (z) := ξ˜n,t+1 (z +

t X

∗ hψn,k (z), ∆Sk i),

k=1

and the value functions of the optimization problems are finite, i.e. un (G, z) = Un,0 (z) < ∞. 6

(8)

v) There exist nondecreasing functions Ms , Hs and Υs : R+ → R+ such that for all x ∈ R, n ≥ Ns : |ξ˜n,s (x)| ≤ Hs (x), Un (x − Ms (|x|)) ≤ Un,s (x) ≤ Un (x + Ms (|x|)),

(9) (10)

and for all n ≥ N0 , ∗ |ψn,t (x)| ≤ Υt (|x|).

(11)

vi) Furthermore, for all 0 ≤ s ≤ T , ε > 0 and x ∈ R sup Un,s (x) → 0 on {x ≥ πs (G)},

(12)

Un,s (πs (G) − ε) → −∞ a.s.

(13)

ω∈Ω

Corollary 3.2 Under the conditions of Theorem 3.1, there exist nondecreasing functions Ft : R+ → R+ , 0 ≤ t ≤ T such that for all n ≥ N0 , ∗ (z) z,ψn

|Vt

| ≤ Ft (|z|) a.s.

for the optimal strategies ψn∗ (z) constructed in the previous Theorem. Proof of Theorem 3.1. Suppose d = 1 for notational simplicity and let R denote a constant bound for the process |∆S|. We will need some standard arguments related to, e.g., the existence of jointly measurable approximations and regular versions of certain functions. These technical tools will be borrowed from R´asonyi and Stettner (2005), namely Propositions 4.4 and 4.6, Lemmas 4.5 and A.5, Propositions 4.9, 4.10 and 6.5. Note that those results do not rely on the “asymptotic elasticity” property which is crucial to prove the main theorem of the cited paper. To achieve the same goal without this hypothesis, we will carry out the estimations in a different way, relying on Assumption 2.3. We shall apply backward induction to prove the statements from (5) to (13). First, for s = T set NT = 0; (5) and (10) are trivial ; (12) and (13) hold by Lemma 5.4. The existence of almost surely concave and increasing continuously differentiable versions for Un,T −1 , (6), (7) and (9) will follow just like in the induction step below. Let us proceed supposing that the induction hypotheses hold for s ≥ t + 1, we want to show them for s = t. We get from (9) for s = t + 1 that x + ξ˜n,t+1 (x)∆St+1 ∈ [x − Ht+1 (|x|)R, x + Ht+1 (|x|)R], and from (10) for s = t + 1 Un,t+1 (x + Ht+1 (|x|)R) ≤ Un (x + Ht+1 (|x|)R + Mt+1 (|x| + Ht+1 (|x|)R)) because Mt+1 and Un are nondecreasing. Also Un,t+1 (x − Ht+1 (|x|)R) ≥ Un (x − Ht+1 (|x|)R − Mt+1 (|x| + Ht+1 (|x|)R)) . 7

Defining Mt (u) := Ht+1 (u)R + Mt+1 (u + Ht+1 (u)R), u ∈ R+ , Mt is nondecreasing as Ht+1 and Mt+1 are. Using (6) for s = t + 1 and the fact that Un,t+1 is nondecreasing, we get that almost surely Un (x − Mt (|x|)) ≤ Un,t (x) ≤ Un (x + Mt (|x|)), showing (10) for s = t. Moreover, as S is bounded, (10) for s = t entails that (5) holds true for s = t. It follows also from (10) for s = t that Un,t−1 is well-defined. It turns out that Un,t (πt (G)) is nonnegative. Indeed, take a strategy φˆ such that πt (G)+ ˆ φ∆St+1 ≥ πt+1 (G) almost surely. Then we have Un,t (πt (G)) = ess. sup E(Un,t+1 (πt (G) + φ∆St+1 )|Ft )

(14)

φ

ˆ t+1 )|Ft ) ≥ E(Un,t+1 (πt+1 (G))|Ft ) ≥ E(Un,t+1 (πt (G) + φ∆S ≥ E(E(Un,t+2 (πt+2 (G))|Ft+1 )|Ft ) ≥ . . . ≥ E(Un,T (G)|Ft ) ≥ Un (0) = 0. Then on {x ≥ πt (G)}, we get from (10) for s = t that 0 ≤ Un,t (x) ≤ Un,t (x + kGk∞ ) ≤ Un (x + kGk∞ + Mt (x + kGk∞ )). Thus supω∈Ω Un,t (πt (G)) → 0 by Lemma 5.4 and (12) holds for s = t. Now we establish the existence of optimal strategies. To this end, we need two auxiliary results. Claim 3.3 For y > kGk∞ ,

0 sup Un,t (y) → 0.

ω∈Ω

Proof Claim 3.3. By the induction hypotheses Un,t is continuously differentiable and we can write that a.s. Z y 0 Un,t (w)dw = Un,t (y). Un,t (kGk∞ ) + kGk∞

We have, by monotonicity of the derivative, Z y 0 0 Un,t (w)dw ≤ |Un,t (y)| + |Un,t (kGk∞ )|, Un,t (y)[y − kGk∞ ] ≤ kGk∞

so the statement follows by (12) for s = t and πt (G) ≤ kGk∞ < y.

2

Claim 3.4 There exist an increasing function Ht (·) : R+ → R+ such that for any ξ ∈ Ξt , ξ ∈ Dt+1 a.s. satisfying |ξ| > Ht (|x|), E(Un,t (x + ξ∆St )|Ft−1 ) < E(Un,t (x)|Ft−1 ), for all x ∈ R and for n ≥ Nt , where Nt is a suitable constant. 8

(15)

Proof Claim 3.4. satisfying

Fix x ∈ R and suppose that n ≥ Nt+1 . Let ξ ∈ Ξt , ξ ∈ Dt+1 a.s |ξ| >

x . β

(16)

Let Vn be the “optimal continuation” from x+ξ∆St for the utility Un,t . To write it formally, we introduce the following φn,t+1 := ξ˜n,t+1 (x + ξ∆St ), φn,j+1

:= ξ˜n,j+1 (x + ξ∆St +

j X

φn,k ∆Sk )

for j = t + 1, . . . , T − 1,

k=t+1

Vn :=

T X

φn,k ∆Sk .

k=t+1

It is clear that Vn ∈ Vt . We note that, by Proposition 4.10 of R´asonyi and Stettner (2005), one can replace x in (6) for s = t + 1, . . . , T by arbitrary Fs−1 -measurable functions. Hence we obtain, applying (6) for s = t + 1, . . . , T that E(Un,t (x + ξ∆St )|Ft−1 ) = E(Un (x + ξ∆St + Vn − G)|Ft−1 ) ≤ E(Un (x − β|ξ|)I{ξ∆St +Vn <−β|ξ|} |Ft−1 ) + E(Un,t (|x| + 2kGk∞ + |ξ|R)I{ξ∆St +Vn ≥−β|ξ|} |Ft−1 ). In fact, we could simply take Un,t (|x| + |ξ|R) in the second term, but the additional 2kGk∞ will be needed for reasons that will become clear later. Note first that Un,t (|x| + 2kGk∞ + |ξ|R) ≥ E(Un,t+1 (|x| + 2kGk∞ + |ξ|R)|Ft ) ≥ E(Un,T (|x| + 2kGk∞ + |ξ|R)|Ft ) ≥ Un (0) = 0, and also Un (x − β|ξ|) ≤ Un (0) = 0, from (16). Thus by Proposition 5.2 we obtain E(Un,t (x + ξ∆St )|Ft−1 ) ≤ κT −t+1 Un (x − β|ξ|) + E(Un,t (|x| + 2kGk∞ + |ξ|R)|Ft−1 ). (17) Continue the estimation of the first term on the right-hand side of inequality (17): κT −t+1 κT −t+1 Un (x − β|ξ|) + Un (x − β|ξ|) 2 2 · ¸ κT −t+1 κT −t+1 β|ξ| β|ξ| 0 β|ξ| ≤ Un ( (x − β|ξ|)) + Un (x − )− U (x − ) , 2 2 2 2 n 2

κT −t+1 Un (x − β|ξ|) =

where we used concavity of Un and Un (0) = 0. Choose ξ such that, in addition to (16), both x−

β|ξ| 2

< −1,

κT −t+1 (x − β|ξ|) < x − Mt (|x|), 2 9

(18) (19)

hold. For the estimation of the second term of the right-hand side of (17), we use concavity of Un,t and (10) for s = t to see that 0 Un,t (|x| + 2kGk∞ + |ξ|R) ≤ Un,t (|x| + 2kGk∞ ) + Un,t (|x| + 2kGk∞ )|ξ|R 0 ≤ Un (|x| + 2kGk∞ + Mt (|x| + 2kGk∞ )) + Un,t (2kGk∞ )|ξ|R 0 ≤ |x| + 2kGk∞ + Mt (|x| + 2kGk∞ ) + Un,t (2kGk∞ )|ξ|R, 0 is monotone decreasing, U (0) = 0 and U 0 (0) = 1. So we get from (18) and (19) as Un,t n n that

E(Un,t (x + ξ∆St )|Ft−1 ) < Un (x − Mt (|x|)) · ¸ βκT −t+1 0 0 +|ξ| Un,t (2kGk∞ )R − Un (−1) 4

+|x| + 2kGk∞ + Mt (|x| + 2kGk∞ ) + (κT −t+1 /2)Un (−1).

Here the first term is ≤ Un,t (x) by (10). Note also that Un (−1) ≤ 0. Now by Claim 3.3 and Lemma 5.4, there exists some Nt ≥ Nt+1 such that for n ≥ Nt , 0 Un,t (2kGk∞ )R −

βκT −t+1 0 Un (−1) ≤ −1. 4

Thus E(Un,t (x + ξ∆St )|Ft−1 ) < E(Un,t (x)|Ft−1 ) − |ξ| + |x| + 2kGk∞ + Mt (|x| + 2kGk∞ ). Choosing Ht so large that if |ξ| > Ht (|x|), then |ξ| > |x| + 2kGk∞ + Mt (|x| + 2kGk∞ ), (16), (18) and (19) are satisfied, we get the Claim. 2 Using Claim 3.4 and a compactness argument, we are now able to prove that a bounded optimal strategy ξ˜n,t (x) exists for x ∈ [0, 1] (repeating the same argument we get the existence on the whole real line). k (x, ω), x ∈ [0, 1], ω ∈ Ω, k ∈ N} attaining the Take a jointly measurable sequence {ξn,t essential supremum in the definition of Un,t−1 (such a sequence is constructed in Lemma 4.5 of R´asonyi and Stettner (2005)). k (x) ∈ D a.s. (see Proposition 4.6 of the same paper). Let We may suppose that ξn,t t k (x)| > H (x)} ∈ F A := {|ξn,t t t−1 . From Claim 3.4, k k E(Un,t (x + ξn,t (x)∆St )|Ft−1 ) ≤ IA E(Un,t (x)|Ft−1 ) + IAc E(Un,t (x + ξn,t (x)∆St )|Ft−1 ) k = E(Un,t (x + ξn,t (x)IAc ∆St )|Ft−1 )

≤ Un (|x| + RHt (|x|) + Mt (|x| + RHt (|x|))), which implies, in particular, that Un,t−1 is not ∞. An estimation like (14) shows Un,t−1 (x) ≥ Un (x − kGk∞ ) hence Un,t−1 is finite and we also get (5) for s = t − 1. Define k k ζn,t (x) := ξn,t (x)IAc , k (x), k ∈ N} by {ζ k (x), k ∈ N} by the above inequalities we may replace the sequence {ξn,t n,t as the latter gives higher (conditional) expected utility.

10

As Ht (.) is increasing,

k |ζn,t (x)| ≤ Ht (1),

Lemma A.5 of R´asonyi and Stettner (compare also to Lemma 2 of Kabanov-Stricker (2001)) σk provides the existence of a random subsequence σk (x, ω) such that almost surely ζn,t (x) converges for all x ∈ [0, 1] to some limit called ξ˜n,t (x). We have ξ˜n,t (x) ∈ Ξt−1 and ∈ Dt a.s., by construction. Now by the Fatou Lemma (which is applicable because of (10) for s = t), σk Un,t−1 (x) = lim sup E(Un,t (x + ζn,t (x)∆St )|Ft−1 ) k→∞

≤ E(Un,t (x + ξ˜n,t (x)∆St )|Ft−1 ), finishing the proof of (6) and (9) for s = t and x ∈ [0, 1]. Repeating the same argument we may conclude for all x ∈ R. Moreover, we get from Propositions 4.4 and 6.5 of R´asonyi and Stettner (2005) that Un,t−1 has almost surely continuously differentiable versions and (7) is satisfied for s = t. To prove (13) for s = t, we first recall that by (6) for s = t, . . . , T , Un,t (πt (G) − ε) = E(Un (πt (G) − ε + Vn − G)|Ft ), for some Vn ∈ Vt (the “optimal continuation” from πt (G) − ε). Now apply Lemma 5.3 and (10) to obtain the estimation Un,t (πt (G) − ε) ≤ E(Un (−ε/2)I{πt (G)−ε/2+Vn
It remains to prove that the strategies defined by (8) are optimal. By standard arguments (just like in Proposition 5.3 of R´asonyi and Stettner (2005)), we obtain that for any trading strategy ψ ∈ Φ(Un , G, z): ∗ (z) z,ψn

E(Un (VTz,ψ )|F0 ) ≤ Un,0 (z) = E(Un (VT

)|F0 ).

As Un,0 (z) is finite and F0 is trivial one gets that un (G, z) = Un,0 (z) < ∞ and for all z,ψ ∗ (z) ψ ∈ Φ(Un , G, z), E(Un (VTz,ψ )) ≤ E(Un (VT n )) = un (G, z) < ∞. Thus ψn∗ (z) is the solution of EUn (VTz,ψ ) → max., ψ ∈ Φ(Un , G, z). and un (G, z) = Un,0 (z) < ∞, i.e. the value functions of the sequence of optimization problems are finite. 2 hP i t Proof of Corollary 3.2. Indeed, define Ft (u) := u + R 2 j=1 Υj (u) . 11

4

Proof of the main results

Proof of Theorems 2.4 and 2.8. By considering Un (x) Un (0) − Un0 (0) Un0 (0) instead of Un (x) one may assume Un (0) = 0, Un0 (0) = 1; this does not affect the validity of Assumption 2.3 and does not change optimal strategies either. From Theorem 3.1, item (8), we get the existence of the optimal strategies ψn∗ (z) for n ≥ N0 . We now turn to the convergence result. Fix z ≥ π(G). We first prove that there exists an increasing sequence (nk )k of natural numbers such that for all k ≥ 0, P (z +

T X

ψn∗ k ,t (z)∆St ≤ G − 1/k) ≤

t=1

1 1 − . k k+1

(20)

As z ≥ π(G), remembering (14) we see that for l such that l ≥ N0 , 0 ≤ Ul,0 (z) = EUl (z +

T X

∗ ψl,t (z)∆St − G).

t=1

Introduce Jl,k := {z +

T X

∗ ψl,t (z)∆St ≤ G − 1/k}.

t=1

Using Corollary 3.2, we get for l ≥ N0 that c Ul (z + 0 ≤ Ul,0 (z) ≤ Ul (−1/k)P (Jl,k ) + E(IJl,k

T X

∗ ψl,t (z)∆St − G))

t=1

≤ Ul (−1/k)P (Jl,k ) + Ul (FT (|z|)).

From Lemma 5.4, we can recursively define n0 := N0 and nk > nk−1 such that Unk (−1/k) ≤ −1 1 1 − . Unk (FT (|z|)) ≤ k k+1 Thus we get that P (Jnk ,k ) ≤

1 1 − , k k+1

so indeed (20) holds. From (11) for t = 1 there exists some subsequence (σk,1 )k≥0 of nk and some constant ζ1 (z), ψσ∗k,1 ,1 (z) → ζ1 (z). Then from (11) for t = 2 and Lemma 2 of Kabanov and Stricker (2001), there exists a random F1 -measurable subsequence (σk,2 )k≥0 of (σk,1 )k≥0 and an F1 -measurable random variable ζ2 (z) such that ψσ∗k,2 ,2 (z) → ζ2 (z) a.s. Then we keep on extracting such random subsequences (σk,t )k satisfying items (i) and (ii) of Definition 2.7 and 12

finding Ft−1 -measurable random variable ζt (z) satisfying Definition 2.7 (iii) : (ζt (z))1≤t≤T ∗ (z)) is an accumulation point of the optimal strategies’ sequence ((ψn,t 1≤t≤T )n≥N0 . Let B be the set of ω ∈ Ω such that (σk,2 (ω))k≥0 is a subsequence of (σk,1 )k≥0 , for all t = 3, . . . , T , (σk,t (ω))k≥0 is a subsequence of (σk,t−1 (ω))k≥0 and ψσ∗k,T (ω),t (z)(ω) → ζt (z)(ω) for all t = 1, . . . , T . By the construction, P (B) = 1. Define A = B ∩ {z +

Ak = B ∩ {z +

T X

t=1 T X

ζt (z)∆St ≤ G} ψσ∗k,T ,t (z)∆St ≤ G − 1/k}.

t=1

Then it is easy to see that A ⊂ lim inf k Ak and by the Fatou lemma, P (A) ≤ P (lim inf Ak ) ≤ lim inf P (Ak ). k

k

By definition, B = ∪∞ j=k {σk,T = σj,1 } and from (20) applied to every j ≥ k P (z +

T X

ψσ∗j,1 ,t (z)∆St ≤ G − 1/k) ≤ P (z +

T X

ψσ∗j,1 ,t (z)∆St ≤ G − 1/j)

t=1

t=1

≤

1 1 − , j j+1

since σk,1 was a subsequence of nk . Thus we arrive at P (Ak ) ≤

∞ X

P ({z +

j=k

T X

ψσ∗j,1 ,t (z)∆St

≤ G − 1/k} ∩ {σk,T = σj,1 }) ≤

t=1

∞ µ X 1 j=k

So we conclude that P (z +

T X

1 − j j+1

¶

=

1 . k

ζt (z)∆St ≥ G) = 1,

t=1

and (ζt (z))1≤t≤T is a superhedging strategy for G.

2

Proof of Theorem 2.6. Fix x0 ∈ R. Suppose that n ≥ N0 , then pn (G, x0 ) is well defined. Un (x0 ) n (x+x0 ) − U . Then Wn (0) = 0, Wn0 (0) = 1 and Consider the sequence Wn (x) := UU 0 0 n (x0 ) n (x0 ) Assumption 2.3 is valid for Wn , n ∈ N. We will prove that the corresponding utility prices pWn (G, 0) tend to π(G). As obviously pUn (G, x0 ) = pWn (G, 0), this will imply the Theorem. From now on we denote pWn (G, 0) by pn (G, 0). One may easily check (see the proof of Theorem 2.6 in Carassus and R´asonyi (2006a)) that pn (G, 0) ≤ π(G). Now it remains to prove lim inf pn (G, 0) ≥ π(G). n→∞

13

(21)

Suppose that this fails, i.e. for some η > 0 and some subsequence nk pnk (G, 0) ≤ π(G) − η holds, for all k ∈ N. By Definition 2.5, unk (G, π(G) − η) ≥ unk (0, 0). The liminf of the right-hand side is nonnegative : lim inf un (0, 0) ≥ lim inf Un (0) = 0. n→∞

n→∞

The left-hand side tends to −∞ by item (13) of Theorem 3.1 for s = 0 and this contradiction proves (21). 2

5

Appendix

We start this section with an alternative characterization of πt (G), recall (4) for the definition of Vt . Proposition 5.1 πt (G) = ess. inf{Y : ∃V ∈ Vt Y + V ≥ G a.s.}.

(22)

In particular, π0 (G) = π(G). Proof. Let us prove it by induction; the case t = T is P trivial. Suppose that the proposition holds for t + 1. Let Y be such that there exists V = Tk=t+1 hφk , ∆Sk i ∈ Vt with Y + V ≥ P G a.s. As Tk=t+2 hφk , ∆Sk i ∈ Vt+1 , Y + hφt+1 , ∆St+1 i ≥ πt+1 (G) a.s, by the induction hypothesis. Now by definition of πt (G) we get that Y ≥ πt (G) hence πt (G) ≤ ess. inf{Y : ∃V ∈ Vt Y + V ≥ G a.s.}. Conversely, fix ε > 0. By definition of πk (G), there exists {φk , t ≤ k ≤ T − 1} such that φk ∈ Ξk−1 and ε + hφk+1 , ∆Sk+1 i ≥ πk+1 (G) a.s. πk (G) + T −t Summing over all k = t, . . . , T − 1, πt (G) + ε +

T X

hφk , ∆Sk i ≥ G a.s.

k=t+1

follows and therefore πt (G) + ε ≥ ess. inf{Y : ∃V ∈ Vt Y + V ≥ G a.s.}, so letting ε → 0 achieves the proof of the Proposition. We now deduce two consequences of the (uniform) absence of arbitrage property. 14

2

˜ t−1 , Proposition 5.2 Under Assumption 2.2, for each 1 ≤ t ≤ T and ξ ∈ Ξ ess. inf P (V ≤ 0|Ft ) ≥ κT −t , V ∈Vt

ess. inf P (hξ, ∆St i + V < −β|Ft−1 ) ≥ κT −t+1 . V ∈Vt

(23) (24)

Proof. We prove inequality (23) by backward P induction. The step t = T is trivial. Assume that (23) holds true for t + 1 and let V = Tj=t+1 hζj , ∆Sj i ∈ Vt . P (V ≤ 0|Ft ) ≥ P (hζt+1 , ∆St+1 i ≤ 0, V − hζt+1 , ∆St+1 i ≤ 0|Ft )

= E[E[I{V −hζt+1 ,∆St+1 i≤0} |Ft+1 ]I{hζt+1 ,∆St+1 i≤0} |Ft ] ≥ κT −t−1 × κ. by Assumption 2.2 and (23) for t + 1. Then (24) is a consequence of (23) : for any V ∈ Vt ˜ t−1 , and ξ ∈ Ξ P (hξ, ∆St i + V < −β|Ft−1 ) ≥ P (hξ, ∆St i < −β, V ≤ 0|Ft−1 ) = E[E[I{V ≤0} |Ft ]I{hξ,∆St i<−β} |Ft−1 ] ≥ κT −t × κ, by (23) and Assumption 2.2.

2

Lemma 5.3 Under (NA), for all ε > 0, we have Zt := ess. inf P (πt (G) − ε/2 + V < G|Ft ) > 0, V ∈Vt

almost surely. Proof. Consider the set L0 of all random variables on (Ω, F, P ) equipped with the topology of convergence in probability, L0+ denotes the set of nonnegative random variables. Suppose that the statement fails and A := {Zt = 0} ∈ Ft has positive probability. Define the set Q := {X ∈ L0 : X = πt (G) − ε/2 + V for some V ∈ Vt } ⊂ L0 . Then there are Vn ∈ Vt such that for Bn := {πt (G) − ε/2 + Vn ≥ G} one has P (Bn |Ft ) → 1 almost surely on A. Consequently, Yn := IA πt (G) − IA ε/2 + IA Vn − IA IBn [πt (G) − ε/2 + Vn − G] tends to GIA in probability: indeed, P (Yn 6= GIA ) ≤ P (BnC ∩ A) = EIA P (BnC |Ft ) → 0, n → ∞. Clearly, Yn ∈ IA (Q − L0+ ). Necessarily, GIA ∈ IA (Q − L0+ ), where closure is taken in the topology of stochastic convergence. But it is well-known (see e.g. arguments of Kabanov 15

and Stricker (2001)) that under (NA) the set IA (Q − L0+ ) is closed in probability. This means that for some V˜ ∈ Vt and l ∈ L0+ we have GIA = IA πt (G) − IA ε/2 + IA V˜ − IA l. Let φˆk , k + t + 1, . . . , T be a super-replication strategy for G starting from πk (G), IA (πt (G) − ε/2) + IAc πt (G) + IA V˜ + IAc

T X

φˆk ∆Sk+1 ≥ G,

k=t+1

which contradicts (22), so the statement is proved. We also recall Lemma 4 of Carassus and R´asonyi (2006a) :

2

Lemma 5.4 Suppose that Un , n ∈ N satisfy Assumption 2.3 as well as Un (0) = 0, Un0 (0) = 1, for all n ∈ N. Then ∀y < 0 Un (y) → −∞, n → ∞, ∀y < 0

Un0 (y)

∀y ≥ 0 Un (y) → 0, n → ∞, ∀y ≥ 0 Un0 (y) → 0, n → ∞.2

→ +∞, n → ∞,

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