Risk-neutral pricing for APT

We consider the problem of super-replication (hedging without risk) for the Arbitrage Pricing Theory. The dual characterization of super-replication cost is provided. It is shown that the reservation prices of investors converge to this cost as their respective risk-aversion tends to infinity.


Introduction
Arbitrage Pricing Theory (APT) was originally introduced by Ross ( [28], [29]) and later extended by [18], [9], [10] and numerous other authors. The APT assumes an approximate factor model and states that the risky asset returns in a "large" financial market are linearly dependent on a finite set of random variables, termed factors, in a way that the residuals are uncorrelated with the factors and with each other.
The APT emphasizes the role of the covariance between asset returns and those exogenous factors, while the Capital Asset Pricing Model (CAPM) of [30,22] is based on the covariance between asset returns and the endogenous market portfolio. One of the desirable aspects of the APT is that it can be empirically tested as argued, for example, in [14]. These remarkable conclusions had a huge bearing on empirical work, see [7], [4]. Papers on the theoretical aspects of APT mainly focused on showing that the model is a good approximation in a sequence of economies when there are "sufficiently many" assets, see for example [9,10,18,19,28,27,1].
Ross derives the APT pricing formula under the assumption of absence of asymptotic arbitrage in the sense that a sequence of asymptotically costless and riskless finite portfolios does not asymptotically yield a positive return. Mathematical finance subsequently took up the idea of a market involving a sequence of markets with an increasing number of assets in the so-called theory of large financial markets (see, among other papers, [20,21]) and mainly study the characterization of a notion of absence of arbitrage, using sequence of portfolio involving finitely many assets where the classical notion of no arbitrage holds true i.e. non-negative portfolios with zero cost should have zero return. For the sake of generality, continuous trading was assumed in the overwhelming majority of related papers. But these generalizations somehow overshadowed the highly original ideas suggested in [28] where a one-step model was considered. They did not answer the following natural questions either: In the APT is there a way to consider strategies involving possibly all the infinitely many assets and to exclude exact arbitrage rather than asymptotic one? A first answer was given in [2] in a measure-theoretical setup. Then [26] proposes a straightforward concept of portfolios using infinitely many assets which we will use in the present paper, see Section 2 below. This notion leads to the existence of equivalent risk-neutral probability measures (also called martingale measures or pricing measures) which are equivalent probability measures under which the asset returns has probability zero.
While questions of arbitrage for APT have been extensively studied, other crucial topics -such as utility maximization or pricing -received little attention though these are important questions in today's markets where there is a vast array of available assets. This is particularly conspicuous in the credit market where bonds of various maturities and issuers indeed constitute an entity that may be best viewed as a large financial market, see [11]. Questions of pricing inevitably arise and current literature on APT does not provide satisfactory answers. A standard problem is calculating the superreplication cost of a claim G. It is the minimal amount needed for an agent selling G in order to superreplicate G by trading in the market. This is the hedging price with no risk and, to the best of our knowledge, it was first introduced in [5] in the context of transaction costs. In complete markets with finitely many assets the superreplication cost is just the cash flows expectation computed under the unique martingale measure. But when such markets are incomplete, there exists a so-called dual representation in terms of supremum of those expectations computed under the different riskneutral probability measures. see e.g. [15]. Our first contribution is such a representation theorem for APT under mild conditions, see Theorem 4.2 below.
We are also able to prove the existence of optimizers for utility functions on the positive real axis, see Theorem 5.2. Such results are standard for finitely many assets, see e.g. [15]. In the context of APT, the case of utility functions defined on the real line (i.e. admitting losses) has been considered in [26]. Finally, we establish that, when risk aversion tends to infinity, the utility indifference (or reservation) prices (see [17]) tend to the superreplication price. This links in a nice way investors' price calculations to the preference-free cost of superhedging, see Theorem 6.2.
The model is presented in Section 2. Concepts of no arbitrage are discussed in Section 3. The dual characterization of superreplication prices in given in Section 4, the utility maximization problem is treated in Section 5. Finally, the asymptotics of reservation prices in the high risk-aversion regime is investigated in Section 6.

The large market model
Let (Ω, F, P ) be a probability space. We consider a one step economy which contains a countable number of tradeable assets. The price of asset i ∈ N is given by (S i t ) {t∈{0,1}} . The returns R i , i ∈ N represent the profit (or loss) created tomorrow from investing one dollar's worth of asset i today, We briefly describe below our version of the Arbitrage Pricing Model, identical to that of [20,25,26], which is a special case of the model presented in [28,18].
We assume that the assets' returns are given by where the ε i are random variables and r, µ i , β j i , β i are constants. Asset 0 represents a riskless investment with a constant rate of return r ∈ R. For simplicity, we set r = 0 i.e. S 0 1 = S 0 0 from now on. The random variables ε i , i = 1, . . . , m serve as factors which influence the return on all the assets i ≥ 1 while ε i , i > m are random sources particular to the individual assets R i , i > m.
The ε i are square-integrable, independent random variables satisfying

Remark 2.2
If the ε = (ε i ) i≥1 fail to be independent then the market will not display good pricing properties (namely, there may not exist a martingale measure having a second moment, see Proposition 4 of [25]).
We further assumeβ i = 0, i ≥ 1 and reparametrize the model by introducing Asset returns take then the following form For some n ∈ N, a portfolio φ in the assets numbered 0, . . . , n is an where x is a given initial wealth. Such a portfolio will have value tomorrow given by using (1), r = 0 and where ψ i = φ i S i 0 is the amount held at time 1 in asset i. Using our parametrization one can easily rewrite that for some point of view, see for example [2]. We require that those strategies belong to the family of square-summable sequences for integrability reasons. Recall that ℓ 2 is a Hilbert space with the norm ||h|| ℓ 2 := ∞ i=1 h 2 i . We denote by Φ the function mapping ℓ 2 to L 2 (Ω, F, P ) := {X : Ω → R, E|X| 2 < ∞} (shortly denoted by L 2 (P ) from now on) -the space of square-integrale random variables, which is again a Hilbert space with the norm ||X|| L 2 := E(|X| 2 ) -and which is defined by First the infinite sum in Φ(h) has to be understood as the limit in L 2 (P ) of ( n i=1 h i ε i ) n≥1 , which are Cauchy sequences. Indeed, under Assumption 2.1, for m > n, which can be arbitrarily small for n large enough, since it is the tail sum of a converging series. Actually, under Assumption 2.1, Φ is even an isometry by the same computation : We would like to give sense (as an L 2 (P ) limit of a sequence of finite sums) to the portfolio value Using again the same kind of computation, for h ∈ ℓ 2 , So without any assumption on b, one can not expect that the finite sums ( n i=1 h i (ε i − b i )) n≥1 converge in L 2 (P ). Hence we stipulate the following as well.
is a Cauchy-sequence in L 2 (P ) and the infinite sum in (3) can be understood as an L 2 (P ) limit of finite sums. Notice furthermore that From now, we will use the notation Under Assumptions 2.1 and 2.3, the value tomorrow that can be attained using infinitely many assets with a strategy in ℓ 2 is thus given by

No arbitrage in large markets
In Arbitrage Pricing Theory, the classical notion of arbitrage is the asymptotic arbitrage in the sense of Ross (1976) and Huberman (1982).
If there exists no such sequence, then we say that there is absence of asymptotic arbitrage (AAA).
We would like to understand the link between AAA and the classical definition of no arbitrage, which says roughly that if the value of a portfolio at time 1 -with value at time zero equal to 0 -is non negative then it should be zero. The no-arbitrage condition on a "small market" with N random sources (called AOA(N )) for some N ≥ 1 holds We prove in Lemma 3.3 that under the following assumption there is absence of arbitrage in any of the markets containing N assets.
Note that (6) implies that Proof. We first prove (7). (7) holds true. Since this implies that the return on every non-zero portfolio is negative with positive probability, AOA(N ) follows for every N ≥ 1. We now prove (6). Introduce the following set for n ≥ 1 (7). Thus n 0 < ∞, we can set α N = 1 n 0 and for every ( ✷ It is well-known that absence of arbitrage in markets with finitely many assets is equivalent to the existence of an equivalent martingale measure, see e.g. [13,15]. In the present setting with infinitely many assets we need to use a concept that is somewhat more technical. We say that EMM2 (equivalent martingale measure with density in L 2 ) holds true if the set of martingale measures having a finite moment of order 2 is not empty i.e.
Using the Cauchy-Schwarz inequality, we get that  [25]). So we also postulate the following.
With this in hand, one can show that AAA implies the classical no arbitrage condition stated with infinitely many assets.
Proof. Under AAA, we have that both Assumption 2.3 and EMM2 holds true (see (8) s. and also P -a.s. since P and Q are equivalent. ✷ Lemma 3.7 Assume that Assumptions 2.1 and 2.3 hold true and that by the definition of uniform integrability. Here the second inequality holds since, for N ≥ 2 on the set {|X n | γ ≥ N − 1} one has 1 ≤ |X n | γ . Thus, under Assumptions 2.1, 2.3 and 3.5, for any c > 0, { V x,h 2 , h ∈ ℓ 2 , h ℓ 2 ≤ c} and also {|V x,h |, h ∈ ℓ 2 , h ℓ 2 ≤ c} are uniformly integrable.
We have concluded that sup n ||h(n)|| ℓ 2 < ∞. Then the Banach-Saks Property (recall that ℓ 2 has the Banach-Saks Property), there exists a subsequence (n k ) k≥1 and some h * ∈ ℓ 2 such that for h(N ) : Hence, using (5), and V z, h(N ) − 1 N N k=1 κ n k converges to B in probability and also a.s. for a subsequence for which we keep the same notation. This implies that 1 N N k=1 κ n k converges a.s. and thus B ∈ C z , a contradiction. ✷ Corollary 3.10 Assume that Assumptions 2.1, 2.3, 3.2 and 3.5 hold true and fix some z ∈ R. Then C z is closed in probability.
Proof. We prove that there is an equivalence between (i) C z is closed in probability and (ii) B ∈ L 2 (P ) \ C z implies that there exists some η > 0 such that Assume that C z is closed in probability and let B ∈ L 2 \ C z . If (12) does not hold true, going through the first paragraph of the proof of Proposition 3.9, one gets that B ∈ C z and the contradiction is immediate. Conversely, assume that C z is not closed in probability. Then one can find some h(n) ∈ ℓ 2 and κ n ∈ L + 2 such that θ n = z + h(n), ε − b − κ n ∈ C z converges in probability to some θ * / ∈ C z . Then which contradicts (12). So (i) and (ii) are equivalent. But Proposition 3.9 shows that (ii) holds true under Assumptions 2.1, 2.3, 3.2 and 3.5. We conclude that (i) holds true: C z is closed in probability. ✷ We now provide a quantitative version of the NA condition (see Assumption 3.2) in the spirit of (6). Proposition 3.11 Assume that Assumptions 2.1, 2.3, 3.2 and 3.5 hold true. Then there exists some α > 0, such that for all h ∈ ℓ 2 satisfying h ℓ 2 = 1 Note that changing h by −h, we find that P ( h, ε > α) > α and thus Proof. We argue by contradiction. Assume that for n ≥ 1, there exists h(n) with h(n) ℓ 2 = 1 and Using Hölder's inequality and the fact that {< h(n), ε >≥ − 1 n } ⊂ {| < h(n), ε > | ≤ 1 n }, we get that using Lemma 3.7. When n → ∞, this contradicts (recall (2)) ✷ The following lemma proves that under the NA condition (see Assumption 3.2) any admissible strategy is bounded.
Proof. Recall α > 0 from Proposition 3.11. As b ∈ ℓ 2 , there exists some ≤ α/2. By the admissibility condition, we get that a.s If all the h i are zero then there is nothing to prove. Assume that h i = 0 for some i ∈ {1, . . . , M α − 1}. Let From the no arbitrage condition in the market with M α − 1 assets (see Assumption 3.2 and (7) in Lemma 3.3) we get that P (A) > 0. Assume that P (B) > 0. As the (ε i ) i≥1 are independent, we get that P (A ∩ B) = P (A)P (B) > 0 which contradicts the admissibility of h (see (13)). Thus a.s.

Then on the set
Thus h ℓ 2 ≤ ȳ α and ✷ Let G ∈ L 0 be a random variable which will be interpreted as the payoff of some derivative security at time T . The superreplication price is the minimal initial wealth needed for hedging G without risk where π(G) = +∞ is the set is empty. We refer to [15] for more information about this preference-free price. Then π(G) > −∞ and there exists h ∈ ℓ 2 such that π(G) + h, ε − b ≥ G a.s.
If π(G) = +∞, the second claim is trivial. So we can assume that, π(G) is finite. Then for all n ≥ 1, there exists h n ∈ ℓ 2 such that π(G) We are now in position to prove our duality result. Proof. Let x be such that there exists h ∈ ℓ 2 such that x + h, ε − b ≥ G a.s. Fix Q ∈ M 2 (which is non-empty by Corollary 1 of [25], see (8)). As G ∈ L 2 (P ), E Q (G) is well-defined by the Cauchy-Schwarz inequality. Using Remark 3.4, E Q (x + h, ε − b ) = x. Thus x ≥ E Q (G) and as this holds true for all Q ∈ M 2 and for all x such that there exists h ∈ ℓ 2 such that For the other inequality, it is enough to prove that G−sup Q∈M 2 E Q (G) ∈ C 0 . Indeed, this will imply that there exists h ∈ ℓ 2 such that sup Assume this is not true. Then {G−sup Q∈M 2 E Q (G)}∩(C 0 ∩L 2 (P )) = ∅. As C 0 is closed in probability (see Corollary 3.10), we can apply the Hahn-Banach theorem and there exists Z ∈ L 2 (P ) \ {0} and θ, γ ∈ R such that for all X ∈ C 0 ∩ L 2 (P ). As C 0 ∩ L 2 (P ) is a cone, one can choose γ = 0. As −1 Z<0 ∈ −L 2 (P ) ⊂ C 0 ∩ L 2 (P ), E(−1 Z<0 Z) ≤ 0 and this implies that P (Z ≥ 0) = 1.
Choose h i to be the infinite sequence which is zero everywhere except in the ith coordinate: Choose someQ ∈ M 2 and let η ∈ (0, 1). Set Q η = (1 − η)Q + ηQ. It is clear that Q η ∈ L 2 and that Q η ∼ P . It is also clear that E Qη (ε i − b i ) = 0 for all i ≥ 1 and Q η ∈ M 2 follows. Moreover, one can always find some η ∈ (0, 1), such that E Qη (G) > sup Q∈M 2 E Q (G). This contradiction completes the proof. ✷

Remark 4.3
One may wonder whether π n (G) the superreplication price of G in the small market with n random sources (ε i ) 1≤i≤n converges to π(G) the superreplication price of G in the large market. The answer is no in general: If G needs to be hedge with all the (ε i ) i≥1 , then π n (G) = +∞. Let us sketch a concrete example: Let ε i , i ∈ N be standard Gaussian, let b i = 0 for all i ∈ N and define G : where the left-hand side is a Gaussian random variable with non-zero variance. It follows that π n (G) = ∞ while π(G) = 0, trivially.

Utility maximisation
The idea of modeling preferences of agents by utility functions goes back to [6]. This approach was revived after the appearance of [23] and led to the axiomatic treatment [31]. We follow this traditional viewpoint and model economic agents' preferences by concave increasing utility functions U , where concavity of U is related to risk aversion. So let us suppose that U : (0, ∞) → R is a concave strictly increasing differentiable function. Note that we extend U to [0, ∞) by (right)-continuity (U (0) may be −∞). We also set U (x) = −∞ for x ∈ (−∞, 0). For a contingent claim G ∈ L 0 and x ∈ R, we define We also introduce the set of strategies which dominate G a.s. starting from a given wealth x ∈ R Note that even for x ≥ π(G), A(U, G, x) might be empty. Indeed, from Lemma 4.1, we know that there exists some h ∈ A(G, x), but h might not belong to Φ(U, G, x). But this holds true under appropriate assumptions, as proved in the lemma below.
Proof. As U is increasing, concave and differentiable with U (x 0 ) = 0 and U ′ (x 0 ) = 1, for all x ∈ R, Then V x,h ≥ G ≥ 0 a.s. and h ∈ A(0, x). Hence, we get that since h ∈ A(0, x). Using (5), we get that Cauchy-Schwarz inequality and Lemma 3.12 imply that ✷ We now define the supremum of the expected utility at the terminal date when delivering claim G, starting from initial wealth x ∈ R : The following result establishes that there exists an optimal investment for the investor we are considering.
Proof. If U is constant there is nothing to prove. Else there exists , we may and will suppose that U (x 0 ) = 0 and U ′ (x 0 ) = 1. Let x ≥ π(G) and let h n ∈ A(G, x) = A(U, G, x) (see Lemmata 4.1 and 5.1) be a sequence such that Hence as ℓ 2 has the Banach-Saks Property, there exists a subsequence (n k ) k≥1 and some h * ∈ ℓ 2 such that forh n := 1 n n k=1 h n k h n − h * ℓ 2 → 0, n → ∞ for some h * ∈ ℓ 2 . Note that sup n∈N h n ℓ 2 ≤ x α < ∞. Using (5), we get that in probability by continuity (right continuity in 0) of U on [0, ∞). We claim that the family U + (V x,hn − G), n ∈ N is uniformly integrable. Indeed using (16), we have that Thus, from Assumption 3.5 and Remark 3.8, we get that {U + (V x,hn − G), h n ∈ ℓ 2 , h n ℓ 2 ≤ x α } is uniformly integrable and thus Fatou's lemma used for −U − implies that By concavity of U , U V x,hn k − G and we get that and the proof is finished. ✷

Convergence of reservation price to the superreplication price
We go on incorporating a sequence of agents in our model. A measure of risk aversion have been introduced by [3] and [24] with the "absolute riskaversion" functions r n defined by r n (x) : Pratt in [24] shows that an investor n has greater absolute risk-aversion than investor m (i.e. r n (x) > r m (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.
Keeping these preliminary considerations in mind, Assumption 6.1 below says that the sequence of agents we consider have asymptotically infinite aversion towards risk. Assumption 6.1 Suppose that U n : (0, ∞) → R, n ∈ N is a sequence of concave strictly increasing twice continuously differentiable functions such that Note that we extend each U n to [0, ∞) by (right)-continuity (U n (0) may be −∞). We also set U n (x) = −∞ for x ∈ (−∞, 0). We introduce the value functions for our sequence of utility functions where u n (G, x) = −∞ if A(U n , G, x) = ∅. The utility indifference (or reservation) price p n (G, x), introduced by [17], is defined as p n (G, x) = inf{z ∈ R : u n (G, x + z) ≥ u n (0, x)}.
Intuitively, it seems reasonable that under Assumption 6.1 the utility prices p n (G, x) tend to π(G). This is proved below under a suitable set of assumptions, relying on the results of Sections 4 and 5.
Theorem 6.2 Assume that Assumptions 2.1, 2.3, 3.2 and 3.5 holds true. Suppose that x > 0 and G ∈ L 0 + . Then the utility indifference prices p n (G, x) are well-defined and converge to π(G) as n → ∞.
Proof. Fix some x > 0. Notice that Assumption 6.1 remains true if we replace each U n by α n U n + β n for some α n > 0, β n ∈ R. Also, the utility indifference prices defined by these new functions are the same as the ones defined by the original U n . Hence by choosing α n := 1/U ′ n (x) and β n := −U n (x)/U ′ n (x), we may and will suppose that for all n ∈ N, U n (x) = 0, and U ′ n (x) = 1.
Indeed, we may take a strategyĥ ∈ A(G, π(G)) (which is non-empty, see Lemma 4.1). Then V π(G),ĥ ≥ G a.s. and as U n is non-decreasing, EU n (V x+π(G),h − G) = u n (G, x + π(G)), where the second inequality follows for the fact that if h ∈ A(U n , 0, x) ⊂ A(0, x) then h +ĥ ∈ A(G, x + π(G)) = A(U n , G, x + π(G)) (see Lemma 5.1). So by definition of the utility indifference price (18) follows and we have that p n (G, x) ≤ π(G) < ∞ for all n ≥ 1. Thus, to prove that lim n→∞ p n (G, x) = π(G) it is enough to show that lim inf n p n (G, x) ≥ π(G). Assume that this is not the case. Hence we can find a subsequence (n k ) k≥1 and some η > 0 such that p n k (G, x) ≤ π(G) − η for all k ≥ 1. We may and will assume that x ≥ η. By definition of p n k (G, x) we have that u n k (G, x + π(G) − η) ≥ u n k (0, x).
But lim inf k→+∞ u n k (0, x) ≥ lim inf k→+∞ EU n k (x) = 0, a contradiction. It remains to prove that lim k→+∞ u n k (G, y) = −∞ with y = x + π(G) − η < x + π(G). For ease of notation, we will prove that lim n→+∞ u n (G, y) = −∞. First we show that x + G / ∈ C y . Indeed if this is not the case, there exists some X ∈ L 0 + and h ∈ ℓ 2 such that x+ G = V y,h − X a.s., hence G ≤ V y−x,h T a.s. Therefore we must have y − x ≥ π(G): A contradiction. Applying Proposition 3.9, we get some γ > 0 such that inf h∈ℓ 2 P (A h ) > γ, where Note that we can always assume that x ≥ γ. As y ≥ π(G), Lemma 4.1 implies that A(U n , G, y) = ∅. Hence for all h ∈ A(U n , G, y), we get that Here we used the fact that U n (x − γ) ≤ U n (x) = 0. Using (17), we get that Thus, u n (G, y) ≤ γU n (x − γ) + y + ŷ α when n goes to infinity, by Lemma 4 of [8]. ✷