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Convergence of utility indifference prices to the superreplication price⋆ Laurence Carassus1 , Mikl´ os R´ asonyi2 1

2

Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Paris 7 Denis Diderot, 16 rue Clisson, 75013 Paris, France (e-mail: [email protected]) Computer and Automation Institute of the Hungarian Academy of Sciences, 1518 Budapest, P. O. Box 63., Hungary (e-mail: [email protected])

Manuscript received: / Final version received:

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 positive axis. Under suitable conditions, we show that the utility indifference prices of a bounded contingent claim converge to its superreplication price when the investors’ absolute risk-aversion tends to infinity. Keywords: Utility indifference price, Superreplication price, Convergence, Utility maximization, Risk aversion.

1 Introduction In the present paper a sequence of investors is considered. Preferences of investor n are expressed via the choice of his or her concave strictly increasing utility function Un . We treat the case dom(Un ) = (0, ∞). The utility indifference price (also called Hodges-Neuberger price or 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 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 a utility free concept. It is the minimal initial wealth needed for hedging the claim without risk. ⋆ The authors thank their laboratories for hosting this research. The visit of L. Carassus was financed by the EU Centre of Excellence programme and the one of M. R´ asonyi by the University of Paris 7, M. R´ asonyi was supported by OTKA grants T 047193 and F 049094.

2

Laurence Carassus, Mikl´ os R´ asonyi

We show that (under appropriate technical conditions) the utility indifference prices of a bounded claim converge to its superreplication price when the absolute risk-aversion −Un′′ /Un′ of the respective agents tends to infinity. Up to now, this result was known essentially for exponential utility functions. See, among others and in various contexts, Rouge and El Karoui (2000), Bouchard (2000), Bouchard et al. (2001), Collin-Dufresne and Hugonnier (2004) and Delbaen et al. (2002). Note that the particular techniques for exponential functions can not be used for general utility functions. Moreover, we do not rely on the duality machinery and treat directly the primal problem. Convergence results can also be found in Jouini and Kallal (2001), for another concept of utility price in a finite probability space and in Carassus and R´ asonyi (2005a) for reservation and Davis prices (in the same framework as the present paper). More precisely, in the latter paper the convergence of those prices was shown when Un tend to some limiting utility function U∞ . This is connected with the main result of this paper noting that the superreplication price can be considered as the utility indifference price for the function U∞ (y) := −∞, 0 < y < x,

U∞ (y) := 0, y ≥ x,

(1)

where x is the agent’s initial capital, see section 3 for details. But we can neither apply directly the results of Carassus and R´ asonyi (2005a) nor the same techniques since they are based on smoothness of U∞ .

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. 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. Trading strategies are given by d-dimensional processes {φt , 1 ≤ t ≤ T }, φit representing the number of asset i held in the portfolio at time t. We suppose that the process φ is predictable (i.e. φt is Ft−1 -measurable), this means that new prices are revealed after the agent’s portfolio readjusment. 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, i.e. changes in the portfolio value are due to price fluctuations only, so the value process of a portfolio φ ∈ Φ is t X z,φ hφj , ∆Sj i, Vt := z + j=1

Convergence of utility indifference prices to the superreplication price

3

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 and asserts that it is not possible to generate a strictly positive wealth out of nothing: (N A) : ∀φ ∈ Φ (VT0,φ ≥ 0 a.s. ⇒ VT0,φ = 0 a.s.). For the purpose of this paper we need to strengthen the N A condition and first to reformulate it (see Proposition 1 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 projection φˆt on Dt without changing the portfolio value since hφt , ∆St i = hφˆt , ∆St i a.s., as easily seen. See also Korn and Sch¨ al (1999), Sch¨ al (2000) and Proposition A.1 of R´ asonyi and Stettner (2005a) for more information about the random set Dt . Define Ξ˜t := {ξ ∈ Ξt : ξ ∈ Dt+1 a.s., |ξ| = 1 on {Dt+1 6= {0}}}. The following Proposition roughly says that (NA) is equivalent to the fact that the convex hull of the conditional distribution of ∆St with respect to Ft−1 contains a ball with a random radius (of the dimension of Dt ) around the origin, see Remark 1 below. Compare to Theorem 3 of Jacod and Shiryaev (1998), consult also Remark 3.5 in Sch¨ al (2000) and Remark 4.2 of Korn and Sch¨ al (1999). Proposition 1 (NA) holds iff there exist Ft -measurable random variables βt > 0, 0 ≤ t ≤ T − 1 such that ess. inf P (hξ, ∆St+1 i < −βt |Ft ) > 0 a.s. on {Dt+1 6= {0}}. ˜t ξ∈Ξ

(2)

Proof The direction (N A) ⇒ (2) is Proposition 3.3 of R´ asonyi and Stettner (2005a). The other direction is clear from the implication (g) ⇒ (a) in Theorem 3 of Jacod and Shiryaev (1998). The following condition is used to derive bounds on trading strategies. It is, in fact, equivalent to the “uniform no-arbitrage” condition introduced in Sch¨ al (2000) (Assumption 4.3) and Korn and Sch¨ al (1999). Without this hypothesis it may happen that the supremum of expected utility is ∞, just like in Example 3.3 of Carassus and R´ asonyi (2005a). Assumption 1 There exists a constant β > 0 such that for 0 ≤ t ≤ T − 1 ess. inf P (hξ, ∆St+1 i < −β|Ft ) > 0 a.s. on {Dt+1 6= {0}}. ˜t ξ∈Ξ

4

Laurence Carassus, Mikl´ os R´ asonyi

Remark 1 Assume Dt = Rd for simplicity and let Ct (ω) denote the closed convex hull of the support of the conditional distribution of ∆St with respect to Ft−1 . The “uniform no arbitrage” condition in Sch¨ al (2000) asserts that there should exist a ball of fixed, deterministic radius around the origin which is contained in Ct a.s. Clearly, this holds iff each halfspace whose bordering hyperplane is closer to the origin than some fixed constant contains some point of Ct (ω), which is Assumption 1. Let G ∈ L∞ + be a random variable which will be interpreted as the payoff of some derivative security at time T . Now the concept of superreplication price is formally introduced as the minimal initial wealth needed for hedging without risk the given contingent claim: π(G) := inf{z ∈ R : VTz,φ ≥ G for some φ ∈ Φ}. We refer to Karatzas and Cvitani´c (1993), El Karoui and Quenez (1995), Kramkov (1996) and F¨ollmer and Kabanov (1998) for more information about this notion. We go on incorporating a sequence of agents in our model. It is standard to model economic agents’ preferences by concave increasing utility functions Un . Concavity of Un is related to risk aversion of agent n. Arrow (1964) and Pratt (1965) introduced a measure of local risk aversion U ′′ (x) with the “absolute risk-aversion” functions rn defined by rn (x) := − Un′ (x) . n Pratt (1965) shows in particular that an investor n has greater absolute risk-aversion than investor m (i.e rn (x) > rm (x) for all x) if and only if investor n is globally more risk averse than m, in the sense that the cash equivalent of every risk (the amount of cash for which he would exchange the risk) is smaller for n than for m. Hence Assumption 2 says that the sequence of agents we consider have asymptotically infinite aversion towards risk. Assumption 2 Suppose that Un : (0, ∞) → R, n ∈ N is a sequence of concave strictly increasing twice continuously differentiable functions such that U ′′ (x) → ∞, n → ∞. ∀x ∈ (0, ∞) rn (x) := − n′ Un (x) We extend each Un to [0, ∞) by continuity (Un (0) may be −∞.) Example 1 Typical examples are the sequences Un (x) = −e−γn x , x > 0 where 0 < γn and γn → ∞. In this case rn (x) = γn constant. Other 2 examples are the utility functions with derivatives Un′ (x) = e−γn x , x > 0 where 0 < γn and γn → ∞. Define for each x > π(G), the set A(G, x) of admissible strategies: A(G, x) := {φ ∈ Φ : VTx,φ ≥ G a.s.}.

Convergence of utility indifference prices to the superreplication price

5

Define the supremum of expected utility at the terminal date when delivering claim G, starting from initial wealth x ∈ (π(G), ∞) : un (G, x) :=

sup φ∈A(G,x)

EUn (VTx,φ − G),

where the expectations exist if S is bounded and Assumption 1 holds, see Lemma 2 below. Remark 2 It would also be possible to extend Un on the negative half-line as −∞. In this case one may work without the admissibility condition on strategies and with arbitrary initial wealth. See also section 4. Definition 1 The utility indifference price pn (G, x) is defined as pn (G, x) = inf{z ∈ R : un (G, x + z) ≥ un (0, x)}. Intuitively, it seems reasonable that if Assumption 2 holds then the utility prices pn (G, x) tend to π(G). We wish to find conditions on S and Un which guarantee that this is indeed so. Our main result is the following Theorem, see also Remark 3 for possible generalizations. Theorem 3 Suppose that x ∈ (0, ∞), S is bounded, Assumptions 1 and 2 hold. Then the utility indifference prices pn (G, x) are well-defined and converge to π(G) as n → ∞. 3 Proof of the main result In Lemmas 1 and 2 we shall show that under our assumptions the utility maximization problem is well-posed and the value processes of admissible portfolios are uniformly bounded. Lemma 4 reduces the problem to the case where Un → U∞ and U∞ is given by (1). Then, using Lemma 3 and the uniform bounds on strategies it is possible to establish that un (G, y) → −∞ for π(G) < y < π(G) + x (see (6) below), and this will imply Theorem 3. Lemma 1 Let x > π(G). Suppose that S is bounded and Assumption 1 holds. Take any strategy φ ∈ A(G, x) satisfying φt ∈ Dt , 1 ≤ t ≤ T . There exist increasing functions Mt (x) ≥ 0 such that Vtx,φ ≤ Mt (x). Proof For t = 0 take M0 (x) := x. Suppose that the statement has been shown up to t − 1. We claim that |φt | ≤

x,φ Vt−1 . β

Indeed, define A :=

(

V x,φ |φt | > t−1 β

)

  φt ∈ Ft−1 , B := h , ∆St i < −β . |φt |

(3)

6

Laurence Carassus, Mikl´ os R´ asonyi

Clearly, {Vtx,φ < 0} ⊃ A ∩ B and P (A ∩ B) = E[E[IA∩B |Ft−1 ]] = E[IA [E(IB |Ft−1 )]]. By Assumption 1, P (B|Ft−1 ) > 0, thus P (A) > 0 would imply that P (Vtx,φ < 0) > 0. But as VTx,φ ≥ G ≥ 0 a.s., the no-arbitrage condition (NA) implies that Vtx,φ ≥ 0 a.s. for all t. This contradiction shows that (3) holds. Thus by the induction hypothesis Vtx,φ ≤ Mt−1 (x) + k∆St k∞ Mt−1 (x)/β =: Mt (x), which defines a suitable Mt (x). Lemma 2 Let x > π(G). If S is bounded and Assumption 1 holds then un (G, x) is well-defined, finite and un (G, x) =

sup φ∈A(G,x), φt ∈Dt

EUn (VTx,φ − G).

˜ Proof Take some strategy φ˜ ∈ A(G, x) such that VTx,φ ≥ G + ε for some ε > 0. Then un (G, x) ≥ Un (ε) > −∞.

Let φ ∈ A(G, x) and φˆt (ω) be the orthogonal projection of φt (ω) on Dt (ω), this is in Ξt−1 by Proposition 4.6 of R´ asonyi and Stettner (2005a). From Lemma 1, ˆ

Un (VTx,φ − G) ≤ Un (MT (x)). By definition of Dt , hφt , ∆St i = hφˆt , ∆St i a.s. and thus un (G, x) =

sup φ∈A(G,x)

EUn (VTx,φ − G) =

sup φ∈A(G,x)

ˆ

EUn (VTx,φ − G) < ∞,

the statements are proved. Denote by L0 the set of all real-valued random variables on (Ω, F, P ) equipped with the topology of convergence in probability. The notation L0+ stands for the set of nonnegative random variables. Define for z ∈ R Kz := {VTz,φ : φ ∈ Φ}. We recall the following fundamental fact, see Kabanov and Stricker (2001) for a proof. Theorem 4 Under (NA), the set Kz − L0+ is closed in probability.

Convergence of utility indifference prices to the superreplication price

7

Lemma 3 Let B ∈ L0 such that B ∈ / Kz − L0+ . Then there exists ε > 0 such that inf P (θ ≤ B − ε) ≥ ε. θ∈Kz

Proof Suppose that the statement is false. Then for all n there is θn ∈ Kz such that P (θn ≤ B − 1/n) ≤ 1/n, hence for κn := [θn − (B − 1/n)]I{θn ≥B−1/n} ∈ L0+ : P (θn − κn = B − 1/n) ≥ 1 − 1/n. This implies θn − κn → B in probability, hence B ∈ Kz − L0+ = Kz − L0+ , a contradiction. Lemma 4 Suppose that Un , n ∈ N satisfy Assumption 2 as well as ∀n ∈ N

Un (x) = 0,

Un′ (x) = 1,

for some fixed x ∈ (0, ∞). Then ∀0 < y < x

Un (y) → −∞, n → ∞,

∀y ≥ x

Un (y) → 0, n → ∞.

Proof First take y < x. As Un′ is decreasing, Un′ (u) ≥ Un′ (x) = 1, for u ≤ x, hence rn (u) ≤ −Un′′ (u). Necessarily Un′ (y)

=

Un′ (x)



Z

x

Un′′ (u)du

y

as n → ∞, by the Fatou-lemma. Also Z Un (y) = Un (x) −

y

x

≥ 1+

Z

x

rn (u)du → ∞, y

Un′ (u)du → −∞,

by the same reasoning, using the previous convergence observation. Now for any y > x we claim that Un′ (y) → 0. If this were not the case, along a subsequence nk , for all k we would have Un′ k (y) ≥ α > 0. By monotonicity Un′ k (u) ≥ α, for all u ≤ y, so rn (u) → ∞ implies that −Un′′k (u) → ∞, k → ∞, u ≤ y. Then necessarily 0 ≤ Un′ k (y) = Un′ k (x) +

Z

y x

Un′′k (u)du = 1 +

Z

y

x

Un′′k (u)du → −∞,

a contradiction proving the second assertion of this Lemma.

8

Laurence Carassus, Mikl´ os R´ asonyi

Proof (of Theorem 3) Fix x > 0. As we have already pointed out in Lemma 2, un (G, x) is well-defined and finite. Notice that Assumption 2 remains true if we replace each Un by αn Un + βn for some αn > 0, βn ∈ R. Also, the utility indifference prices defined by these new functions are the same as the ones defined by the original Un . Hence by choosing αn := 1/Un′ (x) and βn := −Un (x)/Un′ (x), we may and will suppose that for all n ∈ N Un (x) = 0,

Un′ (x) = 1.

(4)

Fix π(G) < y < x + π(G). Then x+G∈ / Ky − L0+ , by the definition of the superreplication price. Take 0 < ε given by Lemma 3 appplied with B := x+G and z = y. Notice that the function MT (·) figuring in Lemma 1 does not depend on the particular choice of the strategy φ and hence can be chosen uniformly for all φ ∈ A(G, y) such that φt ∈ Dt for all t. For such a φ, define the set Aφ := {ω ∈ Ω : VTy,φ (ω) ≤ x + G(ω) − ε}. It follows from Lemma 3 that P (Aφ ) ≥ ε. We get EUn (VTy,φ − G) ≤ EIAφ Un (x − ε) + EIACφ Un (MT (y)) ≤ P (Aφ )Un (x − ε) + Un (MT (y) + x)P (AC φ) ≤ εUn (x − ε) + Un (MT (y) + x).

(5)

Here we used the fact that Un (x − ε) ≤ Un (x) = 0 and that Un (z) ≥ 0 for all z ≥ x. Thus, by Lemma 2, un (G, y) ≤ εUn (x − ε) + Un (MT (y) + x) → −∞,

(6)

by Lemma 4 and (4). We also see from the definition of un (0, x) that lim inf un (0, x) ≥ lim inf Un (x) = 0. n→∞

n→∞

(7)

One may easily check that pn (G, x) ≤ π(G).

(8)

ˆ Indeed, for any δ > 0 we may take a strategy φ(δ) ∈ A(G, π(G) + δ) such that ˆ π(G)+δ,φ(δ) ≥ G. VT Then, as Un is non-decreasing, un (0, x) ≤

ˆ x+π(G)+δ,φ+φ(δ)

sup φ∈A(0,x)



EUn (VT

sup φ∈A(G,x+π(G)+δ)

− G)

x+π(G)+δ,φ

EUn (VT

− G) = un (G, x + π(G) + δ),

Convergence of utility indifference prices to the superreplication price

9

so by the definition of the utility indifference price pn (G, x) ≤ π(G) + δ and (8) follows by letting δ → 0. Now it remains to prove lim inf pn (G, x) ≥ π(G). n→∞

(9)

Suppose that this fails, i.e. for some x > η > 0 and a subsequence nk pnk (G, x) ≤ π(G) − η holds, for all k ∈ N. Again, by Definition 1, unk (G, x + π(G) − η) ≥ unk (0, x), the left-hand side tends to −∞ by (6) applied to y = x + π(G) − η and the liminf of the right-hand side is nonnegative by (7), a contradiction proving (9) and hence the Theorem. Remark 3 It is possible to extend Theorem 3 to certain unbounded price processes and relax Assumption 1, too. Define W as the set of random variables with finite moments of all orders. Suppose Assumption 2, ∆St ∈ W, 1/βt−1 ∈ W, 1 ≤ t ≤ T , where βt satisfies (2). Then pn (G, x) tends to π(G). Lemma 1 can be shown with random variables Mt (x) ∈ W instead of constants. Lemma 2 also follows easily. Then the same arguments work, just like in (5) we get EUn (VTx,φ − G) ≤ εUn (x − ε) + EIACφ Un (MT (y) + x). Here IACφ Un (x + MT (y)) ≤ IACφ [Un (x) + MT (y)Un′ (x)] ≤ MT (y), and this is integrable (in fact, lies in W), hence un (G, y) ≤ εUn (x − ε) + EMT (y), and un (G, y) → −∞ for π(G) < y < π(G) + x. The rest of the proof is identical.

4 Conclusion It is well-known that exponential utility prices converge to superreplication price when agents’ risk aversion tends to infinity. We have generalized this result to concave, strictly increasing, twice differentiable utility functions with domain (0, ∞). What happens when the domain is the whole real axis? The same kind of results can be obtained but this requires different techniques and will be addressed elsewhere, see Carassus and R´ asonyi (2005b).

10

Laurence Carassus, Mikl´ os R´ asonyi

Another natural question is the convergence of the optimal strategies to some superreplication strategy. This is false in general, see the example of Cheridito and Summer (2005). In the setting of the present paper, all considered strategies are superhedging strategies by the definition of A(G, x). In particular, the optimal strategies, which exist under our assumptions by R´ asonyi and Stettner (2005b), are superreplicating ones.

References 1. Arrow K (1965) Essays in the Theory of Risk-Bearing. North-Holland, Amsterdam. 2. Bouchard B (2000) Stochastic control and applications in finance. PhD thesis, Universit´e Paris 9. 3. Bouchard B, Kabanov YuM, Touzi N (2001) Option pricing by large risk aversion utility under transaction costs. Decis. Econ. Finance, 24:127–136. 4. Collin-Dufresne P, Hugonnier J (2004) Pricing and hedging in the presence of extraneous risks. working paper. 5. Cvitani´c J, Karatzas I (1993) Hedging contingent claims with constrained portfolios. Ann. Appl. Probab., 3:652–681. 6. Carassus L, R´ asonyi M (2005a) Optimal strategies and utility-based prices converge when agents’ preferences do. submitted. 7. Carassus L, R´ asonyi M (2005b) Convergence of utility indifference prices to the superreplication price: the whole real line case. preprint. 8. Cheridito P, Summer Ch (2005) Utility-maximizing strategies under incresing risk aversion. Forthcoming in Finance Stoch. 9. Delbaen F, Grandits P, Rheinl¨ ander T, Samperi D, Schweizer M, Stricker Ch (2002) Exponential hedging and entropic penalties. Math. Finance, 12:99–123. 10. Hodges R, Neuberger K (1989) Optimal replication of contingent claims under transaction costs. Rev. Futures Mkts., 8:222-239. 11. El Karoui N, Quenez MC (1995) Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim., 33:29–66. 12. F¨ ollmer H, Kabanov YuM (1998) Optional decomposition and Lagrange multipliers. Finance Stoch., 2:69–81. 13. Jacod J, Shiryaev AN (1998) Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch., 2:259–273. 14. Jouini E, Kallal H (2001) Efficient trading strategies in the presence of market frictions. Review of Financial Studies, 14:343–369. 15. Kabanov YuM, Stricker Ch (2001) A teachers’ note on no-arbitrage criteria. S´eminaire de Probabilit´es, 37:149–152, Springer, Berlin. 16. Korn R, Sch¨ al M (1999) On value preserving and growth optimal portfolios. Math. Methods Oper. Res., 50:189–218. 17. Kramkov DO (1996) Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields, 105:459–479. 18. Pratt J (1964) Risk aversion in the small and in the large. Econometrica, 32:122–136. 19. R´ asonyi M, Stettner L (2005a) On the utility maximization problem in discrete-time financial market models. Ann. Appl. Probab., 15:1367–1395.

Convergence of utility indifference prices to the superreplication price

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20. R´ asonyi M, Stettner L (2005b) On the existence of optimal portfolios for the utility maximization problem in discrete time financial market models. Forthcoming in the Proceedings of the 2nd Bachelier Colloqium. 21. Rouge R, El Karoui N (2000): Pricing via utility maximization and entropy. Math. Finance, 10:259-276. 22. Sch¨ al M (2000) Portfolio optimization and martingale measures. Math. Finance, 10:289–303.

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