Asymptotic pricing in large financial markets
aa r X i v : . [ q -f i n . M F ] D ec Asymptotic pricing in large financial markets
Micha l Barski
Faculty of Mathematics and Computer Science, University of Leipzig, GermanyFaculty of Mathematics, Cardinal Stefan Wyszy´nski University in Warsaw, Poland
September 6, 2018
Abstract
The problem of hedging and pricing sequences of contingent claims in large financialmarkets is studied. Connection between asymptotic arbitrage and behavior of the α - quantileprice is shown. The large Black-Scholes model is carefully examined. Key words : large financial market, pricing, quantile hedging, risk measures.
AMS Subject Classification : 60G42, 91B28, 91B24, 91B30.
GEL Classification Numbers : G11, G13
A large financial market is a sequence of small arbitrage-free markets. Absence of arbitrageopportunity on each element of the sequence does not guarantee that there is no arbitrage ”inthe limit”. Different concepts of asymptotic arbitrage were introduced in [7] and [8] and theirconnections with some properties of measure families: contiguity and asymptotic separation wereshown. For other similar results in this field see also [9], [10]. For other notions as asymptoticfree lunch and its relation with existence of a martingale measure for the whole market see [9],[12].Another problem arises as a natural consequence of asymptotic arbitrage theory: how one cancalculate the price of a contingent claim and what is the connection between the price andthe no-arbitrage property of the market. We formulate the problem of pricing not for a singlerandom variable but for the sequence of random variables instead. Motivation for such problemstating is presented in section 3. For such a sequence we define different types of sequences ofhedging strategies. The first of them hedges each element of the sequence, thus it carries norisk at all. Basing on that property we define a strong price, which is strictly related to theprice known from the classical theory of financial markets. The other type hedges the sequencewith some risk which does not exceed a fixed level in infinity. For this case we introduce the α -quantile price. In particular, the risk can vanish in infinity indicating the 1-quantile pricewhich is called a weak price. These definitions are presented in section 3.In section 4 we provide characterization theorems for the prices mentioned above for generallarge financial markets. This general description uses the no-arbitrage property of each smallmarket only. The question arises how the prices are related to each other, in particular thestrong and the weak one, under different types of asymptotic arbitrage. Example 4.6 showsthat asymptotic arbitrage actually does affect this relation. We study this problem and showa relevant theorem for the sequence of complete markets. Analogous theorem for incomplete1arkets remains an open problem.A significant part of the paper is section 5 devoted to the large Black-Scholes market withconstant coefficients. In these particular settings we improved previous results and establishedmore precise characterization theorems which includes widely used derivatives such as call andput options. In this section we also provide an alternative proof of the theorem describingdifferent kinds of asymptotic arbitrage which comes from [8]. The method of proving is lessgeneral then in [8], but using Neyman-Pearson lemma provides more indirect insight into theconstruction of relevant sets. Moreover, similar methods based on non-randomized tests aresuccessfully used in other proofs in this section.The paper is organized as follows. In section 2 we present definitions of some propertiesof measure families and known facts about asymptotic arbitrage. For a more comprehensiveexposition see [5] for the statistical part and [7], [8], [9], [10] for the financial part. In section3 we formulate precisely the problem of pricing. Section 4 provides characterization theoremswhich are used and generalized in section 5 describing the large Black-Scholes model.In general, the main idea in the α -quantile price characterization theorem has its originin the paper on quantile hedging [4] . Thus the results presented here can be treated as anextension or further development in this field. By a large financial market we mean a sequence of small markets. Let (Ω n , F n , ( F nt ) , P n ),where t ∈ [0 , T n ] or t ∈ { , , ..., T n } be a sequence of filtered probability spaces and ( S in ( t )) , i =1 , , ..., d n a sequence of semimartingales describing evolution of d n stock prices. A large financialmarket will be called stationary if S in +1 ( t ) = S in for i = 1 , , ..., d n . This means that eachsubsequent small market contains the previous one. To shorten notation assume that all themarkets have the same time horizon, i.e. T n = T for n = 1 , , ... .As a trading strategy on the n -th small market we admit a pair ( x n , ϕ n ), where x n ≥ ϕ n is an R d n valued predictable process integrable with respect to ( S n ( t )). The value of x n is aninitial endowment and ϕ in ( t ) is a number of units of the i -th stock held in the portfolio at time t .The wealth process corresponding to the strategy ( x n , ϕ n ) defined as V x n ,ϕ n t = P d n i =1 ϕ in ( t ) S in ( t )is assumed to satisfy a self-financing condition, that is: V x n ,ϕ n t = x n + Z t ϕ n ( t ) dS n ( t ) for continuous time models V x n ,ϕ n t = x n + t X s =1 d n X i =1 ϕ in ( s )( S in ( s ) − S in ( s − . Definition 2.1
A pair (0 , ϕ n ) is an arbitrage strategy on the n -th small market if V ,ϕ n t ≥ a.s. for each t and P n ( V ,ϕ n T > > . For the n -th small market we recall the definition of the set Q n of all martingale measures. Definition 2.2 Q ∈ Q n ⇐⇒ ( S in ( t )) is a local martingale on [0 , T ] with respect to Q for i =1 , , ..., d n Theorem 2.3 If Q n = ∅ then there is no arbitrage strategy on the n -th small market. Q n = ∅ for all n = 1 , , ... . The fact that there is no arbitrage on each small market does not guarantee that there is noasymptotic arbitrage opportunity. For the large financial markets we have the following conceptsof asymptotic arbitrage which comes from [8].
Definition 2.4
A sequence of strategies ( x n , ϕ n ) realizes the asymptotic arbitrage of the firstkind (AA1) if: V x n ,ϕ n t ≥ for all t ∈ [0 , T ]lim n x n = 0 , lim n P n ( V x n ,ϕ n T ≥ > . Definition 2.5
A sequence of strategies ( x n , ϕ n ) realizes the asymptotic arbitrage of the secondkind (AA2) if: V x n ,ϕ n t ≤ for all t ∈ [0 , T ]lim n x n > , lim n P n ( V x n ,ϕ n T ≥ ε ) = 0 for any ε > . Definition 2.6
A sequence of strategies ( x n , ϕ n ) realizes the strong asymptotic arbitrage of thefirst kind (SAA1) if: V x n ,ϕ n t ≥ for all t ∈ [0 , T ]lim n x n = 0 , lim n P n ( V x n ,ϕ n T ≥
1) = 1 . Definition 2.7
A sequence of strategies ( x n , ϕ n ) realizes the strong asymptotic arbitrage of thesecond kind (SAA2) if: V x n ,ϕ n t ≤ for all t ∈ [0 , T ]lim n x n = 1 ,P n ( V x n ,ϕ n T ≥ ε ) = 0 for any ε > . We say that the large financial market does not admit the asymptotic arbitrage of the first kind(second kind, strong asymptotic arbitrage of the first kind, strong asymptotic arbitrage of thesecond kind ) and denote this property by
N AA
1, (
N AA
N SAA N SAA
2) if for any sequence( n k ) there are no trading strategies ( x n k , ϕ n k ) realizing the corresponding kind of asymptoticarbitrage.For characterization of the asymptotic arbitrage and for later purposes we introduce some defi-nitions from mathematical statistics. 3 efinition 2.8 Let (Ω n , F n ) , n = 1 , , ... be a sequence of measurable spaces and G n , H n : F n −→ R + a sequence of set functions.
1) ( G n ) is contiguous with respect to ( H n ) (notation: ( G n ) ⊳ ( H n ) ) if for every sequence A n ∈ F n we have H n ( A n ) −→ ⇒ G n ( A n ) −→
02) ( G n ) is asymptotically separable from ( H n ) (notation: ( G n ) △ ( H n ) ) if there exists asequence A n ∈ F n such that H n ( A n ) −→ and G n ( A n ) −→ Q n we consider the following set functions:¯ Q n ( A ) = sup Q ∈Q n Q ( A ) , A ∈ F n - the upper envelope of Q n Q n ( A ) = inf Q ∈Q n Q ( A ) , A ∈ F n - the lower envelope of Q n . The following result provides characterization of asymptotic arbitrage in terms of sequences ofsets. For the proofs see [7], [8], [10].
Theorem 2.9
The following conditions hold1. (NAA1) iff ( P n ) ⊳ ( ¯ Q n )
2. (NAA2) iff ( Q n ) ⊳ ( P n )
3. (SAA1) iff (SAA2) iff ( P n ) △ ( ¯ Q n ) iff ( Q n ) △ ( P n ) . Below we present a standard tool from mathematical statistics for searching optimal tests.It is useful to solve the following problem. Let Q and Q be two probability measures withdensity dQ dQ on a measurable space (Ω , F ). We are interested in finding set ˜ A , which is a solutionof the problem A ∈ F : ( Q ( A ) −→ max Q ( A ) ≤ γ with γ ∈ [0 , Lemma 2.10 ( Neyman-Pearson )If there exists constant β such that Q { dQ dQ ≥ β } = γ then Q { dQ dQ ≥ β } ≥ Q ( B ) for any set B satisfying Q ( B ) ≤ γ . We recall also the pricing theorem, which has its origin in the theorem on optional decom-position of the supermartingales. For more details see [11] and for later extensions [2], [3].
Theorem 2.11 (Price characterization)
Let Q be a set of martingale measures for the semi-martingale ( S t ) describing evolution of the stock prices. Let H be a non negative contingentclaim. Then there exists a trading strategy (˜ x, ˜ ϕ ) , where ˜ x = sup Q ∈Q E Q [ H ] s.t. ˜ x + Z t ˜ ϕ ( s ) dS ( s ) ≥ ess sup Q ∈Q E Q [ H | F t ] . The pair (˜ x, ˜ ϕ ) is thus a hedging strategy and ˜ x is the price of H . Problem formulation
Definition 3.1
A contingent claim H on a large financial market is a sequence of randomvariables H , H , ... satisfying the following conditions For each n = 1 , ... H n : Ω n −→ R + is an F n measurable, non negative random variable. For each n = 1 , , ... sup Q ∈Q n E Q [ H n ] < ∞ holds. In classical market models we have always one random variable which we want to price andhedge. The question arises for justification of considering a sequence of random variables. Wepresent two motivations for this fact.1) Assume that we have one random variable G which is measurable with respect to the σ -field σ ( F , F , ... ). Then H n can be defined as projections of G on the spaces (Ω n , F n , P n ), i.e. H n = E P n [ G | F n ]. Thus, we want to price a derivative which depends on infinitely manyassets but taking into account information which is provided by the few coming first.2) Let G be a random variable which depends on the price of the first asset (or some first assetsas well) only. Then we can define H n = G for each n and consider opportunity arising fromthe fact that the number of assets which can be traded is increasing. We examine how theincreasing number of investments possibilities affects the price of G .Below we present two concepts of asymptotic hedging and prices definitions of H . Definition 3.2
A sequence ( x n , ϕ n ) n is a sequence of hedging strategies if V x n ,ϕ n T ≥ H n ∀ n = 1 , , .... Such class of sequences we denote by H . A strong price of H is defined as v ( H ) = inf ( x n ,ϕ n ) ∈H lim n →∞ x n . Throughout the whole paper we assume that α is any number from the interval [0 , Definition 3.3
A sequence ( x n , ϕ n ) n is a sequence of α -hedging strategies if lim n →∞ P n ( V x n ,ϕ n T ≥ H n ) ≥ α. Such class of sequences we denote by H α . An α -quantile price of H is defined as v α ( H ) = inf ( x n ,ϕ n ) ∈H α lim n →∞ x n . A weak price of H is the -quantile price, i.e. ˜ v ( H ) := v ( H ) . As follows from the definition above, we consider sequences of strategies which do not allow toexceed a fixed level of risk when n tends to infinity. If α = 1, then the risk vanishes in infinity.This particular case is distinguished to compare with classical concept of pricing suggested byDefinition 3.2, where there is no risk for any n = 1 , , ... .At this stage it is clear that v α ( H ) ≤ v β ( H ) ≤ ˜ v ( H ) ≤ v ( H ) for α < β since the following inclu-sions hold : H α ⊇ H β ⊇ H ⊇ H . The main goal of the paper is to provide the characterizationof the prices and solve the problem of equality between the strong and the weak price.5 Prices characterization
Using the price characterization Theorem 2.11 on a classical market it is simple to show thefollowing.
Proposition 4.1
The strong price is given by v ( H ) = lim n sup Q ∈Q n E Q [ H n ] . Proof :
Let g := lim n sup Q ∈Q n E Q [ H n ]. By Theorem 2.11, for any ( x n , ϕ n ) ∈ H we get x n ≥ sup Q ∈Q n E Q [ H n ] and thus v ( H ) ≥ g .Taking ˜ x n := sup Q ∈Q n E Q [ H n ], from Theorem 2.11 we know that there exists a sequence ofstrategies ( ˜ ϕ n ) s.t. (˜ x n , ˜ ϕ n ) ∈ H and thus v ( H ) ≤ g . (cid:3) To characterize the weak price we introduce first some definitions.
Definition 4.2 (The class A α )A sequence of sets ( A n ) belongs to the class A α if A n ∈ F n for n = 1 , , ... and lim n →∞ P n ( A n ) ≥ α .In particular ( A n ) belongs to the class A if P n ( A n ) −→ n . The following remarks state the correspondence between the class of α -hedging sequences H α and the class of A α sets. Remark 4.3 ( A H α ⊆ A α )Each element in H α indicates an element in A α . Indeed, for ( x n , ϕ n ) ∈ H α let us define A x n ,ϕ n n := { V x n ,ϕ n T ≥ H n } . By definition of H α we obtain that ( A x n ,ϕ n n ) ∈ A α . Thus, if wedenote the sequences of sets above by A H α the following inclusion holds : A H α ⊆ A . Remark 4.4 ( H A α ⊆ H α )For the sequence ( A n ) ∈ A α let us consider a sequence of strategies s.t. for a fixed number n strategy ( x An , ϕ An ) satisfies : x An = sup Q ∈Q n E Q [ H n A n ] and ϕ An hedges the contingent claim H n A n (on a small market with index n ). It follows that ( x An , ϕ An ) ∈ H α since ( A n ) ∈ A α . Ifwe denote the sequences of strategies of the form above by H A α the following inclusion holds: H A α ⊆ H α . Theorem 4.5
The α -quantile price is given by v α ( H ) = inf ( A n ) ∈A α lim n sup Q ∈Q n E Q [ H n A n ] . Proof :
We show successively two inequalities: ( ≥ ) and ( ≤ ).( ≥ ) Let us consider ( x n , ϕ n ) ∈ H α . Then using the notation of Remark 4.3 we have: x n ≥ sup Q ∈Q n E Q [ H n A xn,ϕnn ]and therefore lim n x n ≥ lim n sup Q ∈Q n E Q [ H n A xn,ϕnn ] .
6y the definition of the α -quantile price and by Remark 4.3 we obtain v α ( H ) = inf ( x n ,ϕ n ) ∈H α lim n x n ≥ inf ( x n ,ϕ n ) ∈H α lim n sup Q ∈Q n E Q [ H n A xn,ϕnn ]= inf ( A n ) ∈A H α lim n sup Q ∈Q n E Q [ H n A n ] ≥ inf ( A n ) ∈A α lim n sup Q ∈Q n E Q [ H n A n ]( ≤ ) Consider an arbitrary element ( A n ) ∈ A α and a corresponding strategy described in Remark4.4. Following the notation of Remark 4.4 we havesup Q ∈Q n E Q [ H n A n ] = x An and therefore lim n sup Q ∈Q n E Q [ H n A n ] = lim n x An By Remark 4.4 we obtaininf ( A n ) ∈A α lim n sup Q ∈Q n E Q [ H n A n ] = inf ( A n ) ∈A α lim n x An = inf ( x n ,ϕ n ) ∈H A α lim n x n ≥ inf ( x n ,ϕ n ) ∈H α lim n x n = v α ( H ) (cid:3) We examine the problem of asymptotic pricing studying the following example.
Example 4.6
Let us consider the stationary large financial market with the following settings:
Ω = [0 , , S in (1) = S in (0)(1 + ξ i ) , i = 1 , , ..., n, n = 1 , , ... where ( ξ i ) is a sequence of random variables given by ξ i = ( − on [0 , − i ] := E iδ (2 i − i − δ (2 i − on (1 − i ,
1] := F i , δ ∈ (0 , . Sigma fields are assumed to be generated by the sequence ( ξ i ) , i.e. F n = σ ( ξ , ξ , ..., ξ n ) , and the n -th objective probability measure P n is a restriction of the Lebesgue’s measure P on [0 , to thesigma-field F n , i.e. P n = P |F n . Each martingale measure Q n on the n -th market is describedby the property : E Q n [ ξ ] = 0 , E Q n [ ξ ] = 0 , ..., E Q n [ ξ n ] = 0 . Thus Q n is indicated by its valueson the intervals E , E , ..., E n and one can check that Q n ( E ) = δ (cid:18) − (cid:19) Q n ( E ) = δ (cid:18) − (cid:19) ...Q n ( E n ) = δ (cid:18) − n (cid:19) . t follows from the above that we have constructed a sequence of complete markets.We shall find an α -quantile price of a trivial contingent claim H ≡ . Proposition 4.7
In the model specified above we have: v α (1) = δα. Proof :
We shall construct explicitly a sequence of sets ( ˜ A n ) ∈ A α satisfying: lim Q n ( ˜ A n ) = inf ( A n ) ∈A α lim Q n ( A n ) . Let: G := E , G n := E n \ E n − for n=2,3,....Then P n ( G n ) = P n ( F n ) = 12 n and δ n = Q n ( G n ) < Q n ( F n ) = 1 − δ (cid:16) − n (cid:17) Consider a series expansion of α : α = ∞ X i =1 γ i i , where γ i ∈ { , } . Define ˜ A n as follows ˜ A n := n [ i =1 { γ i =1 } G i and notice, that P n ( ˜ A n ) = P ni =1 γ i P n ( G i ) = P ni =1 γ i i and therefore lim n →∞ P n ( ˜ A n ) = P ∞ i =1 γ i i = α , so ( ˜ A n ) ∈ A α . For any ( A n ) ∈ A α we have lim P n ( ˜ A n ) ≤ lim P n ( A n ) and lim Q n ( ˜ A n ) ≤ lim Q n ( A n ) .Thus v α (1) = inf ( A n ) ∈A α lim n E Q n [ A n ] = lim n Q n [ ˜ A n ] = lim n n X i =1 { γ i =1 } Q n ( G i )= ∞ X i =1 γ i δ i = δ ∞ X i =1 γ i i = δα. (cid:3) Notice that δ = ˜ v (1) < v (1) = lim n E Q n [1] = 1 , so this example shows that strict inequalitybetween the strong and the weak price is possible.Notice also that this model admits AA2 and does not satisfy AA1. Indeed, taking the sequence ( F n ) , we get: P n ( F n ) = n −→ and Q n ( F n ) = 1 − δ (cid:0) − n (cid:1) −→ − δ > and thus there isAA2. Let ( A n ) be a sequence s.t. Q n ( A n ) −→ . This means that for any l > , Q n ( A n ) < δ l holds for all large n and one can check, that this implies that A n ⊆ (1 − l , for all large n .As a consequence we obtain lim n P n ( A n ) < l and letting l to ∞ we get lim n P n ( A n ) = 0 . Thismeans that NAA1 and also NSAA1, NSAA2 hold.This example shows that NAA1, NSAA1, NSAA2 is insufficient for the equality of the strongand the weak price. Remark 4.8
If we require that ˜ v ( H ) = v ( H ) even for H of simple structure then the marketmust satisfy N AA . Indeed, suppose that AA holds. It implies that for any ( A n ) ∈ A , ¯ Q n ( A n ) holds. Taking H ≡ we obtain ˜ v (1) = inf ( A n ) ∈A lim n ¯ Q n ( A n ) < v (1) . Remark 4.9
If there is
SAA or equivalently SAA , then for any H bounded, i.e. H n ≤ K for some constant K > , we have v α ( H ) = 0 for any α ∈ [0 , . Indeed, by Theorem 2.9 thereexists a sequence ( ˜ A n ) s.t. P n ( ˜ A n ) −→ and ¯ Q n ( ˜ A n ) −→ . Then we have v α ( H ) ≤ ˜ v ( H ) ≤ lim n sup Q ∈Q n E Q n [ H n ˜ A n ] ≤ lim n K ¯ Q n ( ˜ A n ) = 0 . The next theorem provides some insight into the problem of asymptotic pricing for completemodels.
Theorem 4.10
Under the following assumptions:a) (NAA2) ,b) the large market is complete, i.e. Q n = { Q n } is a singleton for each n ,c) H is bounded, i.e. H n ≤ K , for all n , where K is a positive constant,we have v ( H ) = ˜ v ( H ) . Proof :
First notice, that for any fixed ( A n ) ∈ A by N AA P n ( A n ) −→ ⇐⇒ P n ( A cn ) −→ ⇒ Q n ( A cn ) −→ . Now consider two sequences: x n := E Q n [ H n ] y n := E Q n [ H n A n ] . The following holds: x n − y n = E Q n [ H n − H n A n ] = E Q n [ H n A cn ] ≤ K · Q n ( A cn ) −→ n x n = lim n y n . Taking infimum over all ( A n ) ∈ A we obtain the required result. v ( H ) = lim n x n = inf ( A n ) ∈A lim n y n = ˜ v ( H ) (cid:3) emark 4.11 Assume that
N AA holds. For incomplete market we can define the analogoussequences as in Theorem 4.10: x n := sup Q ∈Q n E Q [ H n ] y n := sup Q ∈Q n E Q [ H n A n ] and for these sequences we obtain analogous inequality x n − y n ≤ sup Q ∈Q n E Q [ H n − H n A n ] = sup Q ∈Q n E Q [ H n A cn ] ≤ K · ¯ Q n ( A cn ) . However, we do not know if the last term goes to as n −→ ∞ . We know that Q n ( A cn ) −→ only and this is insufficient to perform the above proof for incomplete markets. Let W t , W t , ... be a sequence of independent standard Brownian motions on a filtered probabilityspace (Ω , F t , F , P ) , t ∈ [0 , T ]. We will consider a stationary market, where the n -th small markethas its natural filtration i.e. F nt = σ (( W s , ..., W ns ) s ∈ [0 ,t ] ) and F n = F nT . The n -th objectivemeasure is an adequate restriction of P i.e. P n = P | F n and the discounted price processes aregiven by dS it = S it ( b i dt + σ i dW it ) i = 1 , , ..., n, t ∈ [0 , T ]where b i ∈ R , σ i > i = 1 , , ..., n . Such sequence forms a complete large market withmartingale measures given by densities dQ n dP n = Z n = e − ( θ n , W nT ) − k θ n k T where θ n = ( b σ , ..., b n σ n ) and W nt = ( W t , ..., W nt ). Recall, that W ∗ nt = W nt + θ n t is a Brownianmotion under Q n . In this setting we show more indirect proofs for the absence of asymptoticarbitrage using methods of mathematical statistics for searching optimal non-randomized tests(see Lemma 2.10). The shortcoming of this approach is that it works for deterministic coeffi-cients only. In this section we show also, that Theorem 4.10 and Remark 4.9 remain true forrandom variables satisfying some integrability conditions, which are satisfied for widely usedderivatives.For this section use let us introduce a class of sequences ( ε n ) which take values in the interval[0,1] and converging to 0. Such class will be denoted by E . Theorem 5.1
For ε > let A nε denote a solution of the problem A ∈ F n : ( P n ( A ) −→ max Q n ( A ) ≤ ε. Then the following conditions are equivalent1)
N AA ( P n ) ⊳ ( Q n ) 10 ) For any sequence ( ε n ) ∈ E , P n ( A nε n ) −→ holds.4) P ∞ i =1 ( b i σ i ) < ∞ Proof :
Equivalence of (1) and (2) is proved in [8].(2) = ⇒ (3) Let ( ε n ) be any element of E . Then Q n ( A nε n ) ≤ ε n −→ P n ( A nε n ) −→ ⇒ (2) Let A n ∈ F n be s.t. Q n ( A n ) −→
0. Then ε n := Q n ( A n ) belongs to class E and by(3), P n ( A n ) ≤ P n ( A nε n ) −→ ⇐⇒ (4) Statistical methods provide an explicit form of the set A nε . According to the Neyman-Pearson Lemma 2.10 it is of the form A nε = { dP n dQ n ≥ γ } , where γ is a constant s.t. Q n ( A nε ) = ε .This construction provides A nε = n e ( θ n , W nT )+ k θ n k T ≥ γ o = (cid:26) ( θ n , W nT ) ≥ ln γ − k θ n k T (cid:27) = (cid:26) ( θ n , ( W ∗ nT − θT )) ≥ ln γ − k θ n k T (cid:27) = (cid:26) ( θ n , W ∗ nT ) ≥ ln γ + 12 k θ n k T (cid:27) . Solving the following equation: Q n ( A nε ) = Q n (cid:26) ( θ n , W ∗ nT ) ≥ ln γ + 12 k θ n k T (cid:27) = 1 − Φ lnγ + k θ n k T k θ n k √ T ! = ε we obtain γ = e k θ n k√ T Φ − (1 − ε ) − k θ n k T . We calculate the value P n ( A nε ). P n ( A nε ) = P n (cid:18) ( θ n , W nT ) ≥ ln γ − k θ n k T (cid:19) = 1 − Φ lnγ − k θ n k T k θ n k √ T ! =1 − Φ k θ n k √ T Φ − (1 − ε ) − k θ n k T k θ n k √ T ! = 1 − Φ (cid:16) Φ − (1 − ε ) − k θ n k √ T (cid:17) Now observe that if P ∞ i =1 ( b i σ i ) < ∞ then for any ( ε n ) ∈ E − Φ (cid:16) Φ − (1 − ε n ) − k θ n k √ T (cid:17) −→ . If P ∞ i =1 ( b i σ i ) = ∞ then ε n := 1 − Φ(1+ k θ n k √ T ) −→ − Φ (cid:16) Φ − (1 − ε n ) − k θ n k √ T (cid:17) = 1 − Φ(1) . (cid:3) The next two theorems provide characterization of
N AA SAA
SAA
2. The proofs aresimilar and therefore we sketch some parts of them only.
Theorem 5.2
For ε > let A nε denote a solution of the problem A ∈ F n : ( Q n ( A ) −→ max P n ( A ) ≤ ε. Then the following conditions are equivalent ) N AA ( Q n ) ⊳ ( P n )
3) For any sequence ( ε n ) ∈ E , Q n ( A nε n ) −→ holds.4) P ∞ i =1 ( b i σ i ) < ∞ Proof : (3) ⇐⇒ (4) The set A nε is of the form A nε = (cid:26) dQ n dP n ≥ γ (cid:27) where γ is s.t. P n ( A nε ) = ε . This procedure yields A nε = ( ( θ n , W nT ) ≤ Φ ln γ − k θ n k T k θ n k √ T !) γ = e − [ Φ − ( ε ) k θ n k√ T + k θ n k T ]and Q n ( A nε ) = Φ (cid:16) Φ − ( ε )+ k θ n k √ T (cid:17) .If P ∞ i =1 ( b i σ i ) < ∞ then for any ( ε n ) ∈ E Φ (cid:16) Φ − ( ε n )+ k θ n k √ T (cid:17) −→ . If P ∞ i =1 ( b i σ i ) = ∞ then taking ε n := Φ(1 − k θ n k √ T ) −→ (cid:16) Φ − ( ε n )+ k θ n k √ T (cid:17) = Φ(1) . (cid:3) Theorem 5.3
For ε > let A nε denote a solution of the problem A ∈ F n : ( P n ( A ) −→ max Q n ( A ) ≤ ε. Then the following conditions are equivalent1)
SAA SAA P n △ Q n Q n △ P n
5) There exists a sequence ( ε n ) ∈ E s.t. P n ( A nε n ) −→ P ∞ i =1 ( b i σ i ) = ∞ . A nε are based on property P n △ Q n . One can base theproof on the property Q n △ P n . This requires replacing measures P n and Q n in the conditionsfor A nε . The first four conditions are proved in [8] and are included in the formulation abovefor the clarity of exposition only. Equivalence of (3) and (5) are easy to prove. Proof : (5) ⇐⇒ (6) We use the construction of A nε found in the proof of Th. 5.1 A nε = (cid:26) ( θ n , W ∗ nT ) ≥ ln γ + 12 k θ n k T (cid:27) γ = e k θ n k√ T Φ − (1 − ε ) − k θ n k T P n ( A nε ) = 1 − Φ (cid:16) Φ − (1 − ε ) − k θ n k √ T (cid:17) If P ∞ i =1 ( b i σ i ) < ∞ then for any ( ε n ) ∈ E , 1 − Φ (cid:16) Φ − (1 − ε n ) − k θ n k √ T (cid:17) −→ P ∞ i =1 ( b i σ i ) = ∞ then ε n := 1 − Φ( k θ n k √ T ) −→ − Φ (cid:16) Φ − (1 − ε n ) − k θ n k √ T (cid:17) −→ (cid:3) In the sequel we will characterize the weak price of H satisfying some integrability conditions.If H = H , where H is one fixed random variable measurable with respect to F , then it is clearthat E Q n [ H ] does not depend on n and thus indicates the strong price. This means that theinvestor doesn’t have any profits from the fact that the market is getting large and that hecan use greater and grater number of strategies. It turns out that he can not make any profitsunless he uses 1-quantile hedging strategies. In this case, but if P ∞ i =1 ( b i σ i ) = ∞ , the initialendowment can be reduced to 0, i.e. the weak price is equal to 0. The condition P ∞ i =1 ( b i σ i ) < ∞ guaranties that the investor is not able to make any profits at all, no matter what strategies heuses, because then v ( H ) = ˜ v ( H ). Theorem 5.4
Let H be a contingent claim on a large Black-Scholes market with constant coef-ficients. Then1) if P ∞ i =1 ( b i σ i ) < ∞ and lim n E [ H δn ] < ∞ for some δ > then ˜ v ( H ) = v ( H ) .2) if P ∞ i =1 ( b i σ i ) = ∞ and lim n E [ H δn ] < ∞ for some δ > then ˜ v ( H ) = 0 . Proof : (1) For any sequence ( A n ) ∈ A define x n := E Q n [ H n ], y n := E Q n [ H n A n ]. Let p, q, p ′ , q ′ > p + q = 1, p ′ + q ′ = 1. Using H¨older inequality twice tothe difference x n − y n we obtain: x n − y n = E Q n [ H n A cn ] = E [ Z n H n A cn ] ≤ (cid:16) E ( Z n H n ) p (cid:17) p (cid:16) P ( A cn ) (cid:17) q ≤ (cid:18)(cid:16) E ( Z pp ′ n ) (cid:17) p ′ (cid:16) E ( H pq ′ n ) (cid:17) q ′ (cid:19) p (cid:16) P ( A cn ) (cid:17) q = (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ (cid:16) E ( H pq ′ n ) (cid:17) pq ′ (cid:16) P ( A cn ) (cid:17) q . Straightforward calculations yields (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ = e k θ n k T ( pp ′ − (5.4.1)13nd thus N AA n →∞ (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ < ∞ . Taking p, p ′ s.t. pq ′ = 1 + δ and using fact that lim n →∞ (cid:16) P ( A cn ) (cid:17) q = 0 we conclude that lim n →∞ ( x n − y n ) = 0. Thuslim x n = lim y n and taking infimum over all ( A n ) ∈ A we get ˜ v ( H ) = v ( H ).(2) For any ( A n ) ∈ A , p, p ′ > q, q ′ s.t. p + q = 1, p ′ + q ′ = 1 using H¨older inequalitieswe obtain: E Q n [ H n A n ] ≤ (cid:16) E Q n ( H pn ) (cid:17) p (cid:16) Q n ( A n ) (cid:17) q = (cid:16) E ( Z n H pn ) (cid:17) p (cid:16) Q n ( A n ) (cid:17) q ≤ (cid:18)(cid:16) E Z p ′ n (cid:17) p ′ (cid:16) E H pq ′ n (cid:17) q ′ (cid:19) p (cid:16) Q n ( A n ) (cid:17) q = (cid:16) E Z p ′ n (cid:17) pp ′ (cid:16) E H pq ′ n (cid:17) pq ′ (cid:16) Q n ( A n ) (cid:17) q Now, similarly to the previously used methods let us solve an auxiliary problem of finding set A nε s.t. A ∈ F n : ( Q n ( A ) −→ min P n ( A ) ≥ − ε. Analogous calculations provide: A nε = (cid:26) dQ n dP n ≤ γ (cid:27) = (cid:26) ( θ n , W n ) ≥ − ln γ k θ n k T (cid:27) γ = e − [Φ − ( ε ) k θ n k√ T + k θ n k T ] Q n ( A nε ) = Φ (cid:16) − Φ − ( ε ) − k θ n k √ T (cid:17) . Taking p = 2 + δ, p ′ = 2 , ε n = Φ (cid:16) − ln( k θ n k √ T ) (cid:17) (AA2 guaranties that ε n →
0) we getlim E [ H pq ′ n ] = lim E [ H δn ] < ∞ and (cid:18)(cid:16) E Z p ′ n (cid:17) pp ′ (cid:16) Q n ( A nε n ) (cid:17) q (cid:19) q = e p ′− p − k θ n k T Φ (cid:16) − Φ − ( ε n ) − k θ n k √ T (cid:17) = e δ k θ n k T Φ (cid:16) ln( k θ n k √ T ) − k θ n k √ T (cid:17) (5.4.2)Replacing k θ n k √ T by x for the sake of convenience, we calculate the following limit usingd’Hospital formula.lim x →∞ e δ x Φ(ln x − x ) = lim x →∞ √ π e − (ln x − x ) ( x − e − δ x ( − δ )2 x = lim x →∞ − δ √ π " e x ( δ − ) − ln x + x ln x x − e x ( δ − ) − ln x + x ln x x = 014he limit is equal to 0 since: lim x ( δ − ) − ln x + x ln x = −∞ .Summarizing, we have shown that lim n →∞ (cid:16) E Z p ′ n (cid:17) pp ′ (cid:16) E H pq ′ (cid:17) pq ′ (cid:16) Q n ( A nε n ) (cid:17) q = 0 for the ad-equate parameters and thus ˜ v ( H ) = 0. (cid:3) Remark 5.5
The integrability conditions imposed on H in the second item of Theorem 5.4 canbe a little bit weakened. It follows from 5.4.2 that we have to find parameters p, p ′ > s.t. p ′ − p − = δ . We can impose additional requirement: pq ′ → min . Then it can be checked, thatthe solution is: p = 1 + q δ , p ′ = q δ + δ δ and pq ′ = √ + (4 + δ ) + √ √ δ . If δ → then pq ′ is arbitrarily close to √ + 2 + 1 < . Thus, we can assume that lim n E [ H √ + (4+ δ )+ √ √ δn ] < ∞ for some δ > . The next theorem provides a more precise characterization of the α -quantile price. But first letus impose a regularity assumption on the random variables H n Z n . Assumption 5.6
The random variable H n Z n has a continuous distribution function with re-spect to the measure P n . By q n ( α ) we denote the α -quantile of H n Z n , i.e. q n ( α ) = { inf x : P n ( H n Z z ≤ x ) ≥ α } .Denote by B α a set of sequences satisfyinglim n −→∞ β n ≥ α. Theorem 5.7
Let H be a contingent claim on a large Black-Scholes model with constant coef-ficients.1) Under assumption 5.6 the α -quantile price is given by the formula v α ( H ) = inf ( β n ) ∈B α lim n →∞ E h H n Z n { H n Z n ≤ q n ( β n ) } i .
2) Let assumption 5.6 be satisfied. If lim n E [ H δn ] < ∞ for some δ > and P ∞ i =1 ( b i σ i ) < ∞ then v α ( H ) = lim n →∞ E h H n Z n { H n Z n ≤ q n ( α ) } i . Moreover, v α ( H ) is a Lipschitz, increasing function of α taking values in the interval [0 , v ( H )] .3) If lim n E [ H δn ] < ∞ for some δ > and P ∞ i =1 ( b i σ i ) = ∞ then v α ( H ) = 0 for each α ∈ [0 , . Proof: (1) By Theorem 4.5 the α -quantile price is given by the formula: v α ( H ) = inf ( A n ) ∈A α lim n sup Q ∈Q n E Q [ H n A n ] . A n ) ∈ A α and define β n := P n ( A n ). Denote by ˜ A n a solution of thefollowing problem: ˜ A n : ( E Q n [ H n A n ] −→ min P n ( A n ) ≥ β n . If we introduce measure ˜ Q n by the density d ˜ Q n dQ n := H n E Qn [ H n ] , then the above problem can bewritten in the equivalent form: ˜ A n : ( ˜ Q n ( A n ) −→ min P n ( A n ) ≥ β n . Therefore by Lemma 2.10 we conclude that ˜ A n is of the form: { H n Z n ≤ γ } , where γ is aconstant s.t. P n ( H n Z n ≤ γ ) = β n . By Assumption 5.6 we know that there exists such γ and itis equal to q n ( β n ). Thus ˜ A n = { H n Z n ≤ q n ( β n ) } and E Q n [ H n A n ] ≥ E Q n [ H n { H n Z n ≤ q n ( β n ) } ] . Letting n → ∞ and taking infimum over all ( A n ) ∈ A α we obtain: v α ( H ) ≥ inf ( β n ) ∈B α lim n →∞ E h H n Z n { H n Z n ≤ q n ( β n ) } i . (5.7.3)However, P n ( H n Z n ≤ q n ( β n )) = β n , so { H n Z n ≤ q n ( β n ) } ∈ A α and this implies equality in5.7.3.(2) Let α, β ∈ [0 ,
1] be two real numbers s.t. β < α . For p, q, p ′ , q ′ > p + q = 1 , p ′ + q ′ = 1we have the following inequality: E [ H n Z n { H n Z n ≤ q n ( α ) } ] − E [ H n Z n { H n Z n ≤ q n ( β ) } ] = E [ H n Z n { q n ( β ) ≤ H n Z n ≤ q n ( α ) } ] ≤ (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ (cid:16) E ( H pq ′ n ) (cid:17) pq ′ (cid:16) P ( q n ( β ) ≤ H n Z n ≤ q n ( α )) (cid:17) q = (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ (cid:16) E ( H pq ′ n ) (cid:17) pq ′ ( α − β )However, by 5.4.1 we have (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ ≤ lim n →∞ (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ < ∞ . Taking p, q ′ s.t. pq ′ =1 + δ and denoting K := lim n →∞ (cid:16) E ( Z pp ′ n ) (cid:17) pp ′ and K := (cid:16) E ( H pq ′ n ) (cid:17) pq ′ , we obtain E [ H n Z n { H n Z n ≤ q n ( α ) } ] − E [ H n Z n { H n Z n ≤ q n ( β ) } ] ≤ K K ( α − β ) . and interchanging the role of α and β we obtain | E [ H n Z n { H n Z n ≤ q n ( α ) } ] − E [ H n Z n { H n Z n ≤ q n ( β ) } ] |≤ K K | α − β | . (5.7.4)Now consider ( β n ) ∈ B α . If lim n −→∞ β n > α then E [ H n Z n { H n Z n ≤ q n ( β n ) } ] > E [ H n Z n { H n Z n ≤ q n ( α ) } ]and thus lim E [ H n Z n { H n Z n ≤ q n ( β n ) } ] ≥ lim E [ H n Z n { H n Z n ≤ q n ( α ) } ]. If lim n −→∞ β n = α then by 5.7.4we have | E [ H n Z n { H n Z n ≤ q n ( α ) } ] − E [ H n Z n { H n Z n ≤ q n ( β n ) } ] |≤ K K | α − β n | and letting n → ∞ we obtain lim E [ H n Z n { H n Z n ≤ q n ( β n ) } ] = lim E [ H n Z n { H n Z n ≤ q n ( α ) } ]. The conclusion from thesetwo cases is that v α ( H ) ≥ lim E [ H n Z n { H n Z n ≤ q n ( α ) } ]. However, { H n Z n ≤ q n ( α ) } ∈ A α andtherefore v α ( H ) = lim n →∞ E h H n Z n { H n Z n ≤ q n ( α ) } i . (5.7.5)16etting n → ∞ in 5.7.4 and using 5.7.5 we obtain: | v α ( H ) − v β ( H ) |≤ K K | α − β | , which proves that v α ( H ) is Lipschitz. It is clear by 5.7.5 that v α ( H ) is increasing and that v ( H ) = 0. By Theorem 5.4 v ( H ) = v ( H ).(3) It is an immediate consequence of Theorem 5.4 (2), since v α ( H ) ≤ ˜ v ( H ). (cid:3) Remark 5.8
Consider the prices of a call option, i.e. H ≡ ( S T − K ) + . The distribution of ( S T − K ) + Z n is discontinuous in . Let α := P n (( S T − K ) + = 0) . It is clear, that for α ≤ α , v α ( H ) = 0 holds. On the interval (0 , ∞ ) the distribution function is continuous, thus for α > α Theorem 5.7 can be applied.
Conclusion
In this paper we have introduced and characterized two types of asymptotic prices. They arebased on different treating of hedging risk which disappears in infinity. Relations between themstrictly depend on the asymptotic arbitrage on the market. In case of the large Black-Scholesmodel with constant coefficients it was possible to find more indirect formula for the α -quantileprice and state some properties of it. On this market there are two situations possible:1) there is no asymptotic arbitrage of any kind - then the strong and the weak price are equal2) there is asymptotic arbitrage of all kinds - then the weak price is equal to zero, while thestrong is not. Acknowledgement
This paper is a part of the author’s PhD thesis. The author would like toexpress his thanks to Professor Lukasz Stettner for help and support while writing this paper.
References [1] Delbaen F., Schachermayer W.
The fundamental Theorem of Asset Pricing for unboundedstochastic processes , Mathematische Annalen 312, 215-250, (1998)[2] F¨ollmer H., Kabanov Yu.M.
Optional decomposition and Lagrange multipliers
FinanceStoch.- 2, 69-81 (1998)[3] F¨ollmer H., Kramkov D.O.
Optional decompositions under constraints
Probab. TheoryRelat. Fields 109, 1-25 (1997)[4] F¨ollmer H., Leukert P.
Quantile Hedging , Finance and Stochastics 3, 251-273, (1999)[5] Jacod J., Shiryaev A.N.
Limit Theorems for Stochastic Processes , Berlin Heidelberg NewYork : Springer (1987)[6] Jacod J., Shiryaev A.N.
Local martingales and the fundamental asset pricing theorems inthe discrete-time case , Finance and Stochastics 2, 259-273, (1998)[7] Kabanov Yu.M., Kramkov D.O.
Large financial markets: asymptotic arbitrage and conti-guity , Prob. Theory Appl. 39(1), 182-187 (1994)[8] Kabanov Yu.M., Kramkov D.O.
Asymptotic arbitrage in large financial markets , FinanceStoch. 2(2), 143-172 (1998)[9] Klein I.,
A fundamental theorem of asset pricing for large financial markets , MathematicalFinance 10(4), 443-458 (2000)10] Klein I., Schachermayer W.
Asymptotic arbitrage in non-complete large financial markets ,Probab. Theory Appl. 41(4), 780-788 (1996)[11] Kramkov D.O.
Optional decomposition of supermartingales and hedging contingent claimsin incomplete security markets , Probab. Theory Relat. Fields 105, 459-479 (1996)[12] R´asonyi M.