# arXiv:1512.06582v1 n?u=RePEc:arx:papers:1512

Asymptotic pricing in large financial markets

Micha̷l Barski

Faculty of Mathematics and Computer Science, University of Leipzig, Germany

Faculty of Mathematics, Cardinal Stefan Wyszyński University in Warsaw, Poland

Michal.Barski@math.uni-leipzig.de

arXiv:1512.06582v1 [q-fin.MF] 21 Dec 2015

December 22, 2015

Abstract

The problem of hedging and pricing sequences of contingent claims in large financial

marketsisstudied. Connectionbetweenasymptoticarbitrageandbehavioroftheα-quantile

price 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

1 Introduction

A large financial market is a sequence of small arbitrage-free markets. Absence of arbitrage

opportunity on each element of the sequence does not guarantee that there is no arbitrage ”in

the limit”. Different concepts of asymptotic arbitrage were introduced in  and  and their

connections withsomepropertiesofmeasurefamilies: contiguity andasymptoticseparation were

shown. For other similar results in this field see also , . For other notions as asymptotic

free lunch and its relation with existence of a martingale measure for the whole market see ,

.

Another problem arises as a natural consequence of asymptotic arbitrage theory: how one can

calculate the price of a contingent claim and what is the connection between the price and

the no-arbitrage property of the market. We formulate the problem of pricing not for a single

random variable but for the sequence of random variables instead. Motivation for such problem

stating is presented in section 3. For such a sequence we define different types of sequences of

hedging strategies. The first of them hedges each element of the sequence, thus it carries no

risk at all. Basing on that property we define a strong price, which is strictly related to the

price known from the classical theory of financial markets. The other type hedges the sequence

with 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 price

which is called a weak price. These definitions are presented in section 3.

Insection 4weprovidecharacterization theoremsforthepricesmentioned aboveforgeneral

large financial markets. This general description uses the no-arbitrage property of each small

market only. The question arises how the prices are related to each other, in particular the

strong and the weak one, under different types of asymptotic arbitrage. Example 4.6 shows

that asymptotic arbitrage actually does affect this relation. We study this problem and show

a relevant theorem for the sequence of complete markets. Analogous theorem for incomplete

1

markets remains an open problem.

A significant part of the paper is section 5 devoted to the large Black-Scholes market with

constant coefficients. In these particular settings we improved previous results and established

more precise characterization theorems which includes widely used derivatives such as call and

put options. In this section we also provide an alternative proof of the theorem describing

different kinds of asymptotic arbitrage which comes from . The method of proving is less

general then in , but using Neyman-Pearson lemma provides more indirect insight into the

construction of relevant sets. Moreover, similar methods based on non-randomized tests are

successfully used in other proofs in this section.

The paper is organized as follows. In section 2 we present definitions of some properties

of measure families and known facts about asymptotic arbitrage. For a more comprehensive

exposition see  for the statistical part and , , ,  for the financial part. In section

3 we formulate precisely the problem of pricing. Section 4 provides characterization theorems

which 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 origin

in the paper on quantile hedging  . Thus the results presented here can be treated as an

extension or further development in this field.

2 Basic definitions and results

By a large financial market we mean a sequence of small markets. Let (Ω n ,F n ,(Ft n ),P n ),

where t ∈ [0,T n ] or t ∈ {0,1,...,T n } be a sequence of filtered probability spaces and (Sn(t)), i i =

1,2,...,d n asequenceof semimartingales describingevolution of d n stock prices. A large financial

market will be called stationary if Sn+1 i (t) = Si n for i = 1,2,...,d n . This means that each

subsequent small market contains the previous one. To shorten notation assume that all the

markets have the same time horizon, i.e. T n = T for n = 1,2,....

As a trading strategy on the n-th small market we admit a pair (x n ,ϕ n ), where x n ≥ 0 and ϕ n

is an R dn valued predictable process integrable with respect to (S n (t)). The value of x n is an

initial endowment and ϕ i n (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 xn,ϕn

t = ∑ d n

i=1

ϕ i n (t)Si n (t)

is assumed to satisfy a self-financing condition, that is:

V xn,ϕn

t = x n +

V xn,ϕn

t = x n +

t∑

d n ∑

s=1 i=1

∫ t

0

ϕ n (t)dS n (t) for continuous time models

ϕ i n(s)(S i n(s)−S i n(s−1)) for discrete time models.

Definition 2.1 A pair (0,ϕ n ) is an arbitrage strategy on the n-th small market if V 0,ϕn

t ≥ 0

a.s. for each t and

P n (V 0,ϕn

T

> 0) > 0.

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 i n(t)) is a local martingale on [0,T] with respect to Q for i =

1,2,...,d n

Theorem 2.3 If Q n ≠ ∅ then there is no arbitrage strategy on the n-th small market.

2

The proof can be found in  for discrete time and in  for continuous time settings. It turns

out that the inverse statement remains true for the discrete case, but is false for continuous time.

Throughout all the paper we assume that:

Q n ≠ ∅ for all n = 1,2,... .

The fact that there is no arbitrage on each small market does not guarantee that there is no

asymptotic arbitrage opportunity. For thelarge financial markets we have the following concepts

of asymptotic arbitrage which comes from .

Definition 2.4 A sequence of strategies (x n ,ϕ n ) realizes the asymptotic arbitrage of the first

kind (AA1) if:

V xn,ϕn

t

lim

n

x n = 0,

≥ 0 for all t ∈ [0,T]

lim

n

P n (V xn,ϕn

T

≥ 1) > 0.

Definition 2.5 A sequence of strategies (x n ,ϕ n ) realizes the asymptotic arbitrage of the second

kind (AA2) if:

V xn,ϕn

t

lim

n

x n > 0,

≤ 1 for all t ∈ [0,T]

lim

n

P n (V xn,ϕn

T

≥ ε) = 0 for any ε > 0.

Definition 2.6 A sequence of strategies (x n ,ϕ n ) realizes the strong asymptotic arbitrage of the

first kind (SAA1) if:

V xn,ϕn

t

lim

n

x n = 0,

≥ 0 for all t ∈ [0,T]

lim

n

P n (V xn,ϕn

T

≥ 1) = 1.

Definition 2.7 A sequence of strategies (x n ,ϕ n ) realizes the strong asymptotic arbitrage of the

second kind (SAA2) if:

V xn,ϕn

t

lim

n

x n = 1,

≤ 1 for all t ∈ [0,T]

P n (V xn,ϕn

T

≥ ε) = 0 for any ε > 0.

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 the

secondkind)anddenotethis propertybyNAA1, (NAA2,NSAA1, NSAA2) ifforany sequence

(n k ) there are no trading strategies (x nk ,ϕ nk ) realizing the corresponding kind of asymptotic

arbitrage.

For characterization of the asymptotic arbitrage and for later purposes we introduce some definitions

from mathematical statistics.

3

Definition 2.8 Let (Ω n ,F n ),n = 1,2,... 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 ) −→ 0 =⇒ G n (A n ) −→ 0

2) (G n ) is asymptotically separable from (H n ) (notation:(G n ) △ (H n )) if there exists a

sequence A n ∈ F n such that

H n (A n ) −→ 0 and G n (A n ) −→ 1

For the family 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 nQ(A), A ∈ Fn - the lower envelope of Q n .

The following result provides characterization of asymptotic arbitrage in terms of sequences of

sets. For the proofs see , , .

Theorem 2.9 The following conditions hold

1. (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 1 and Q 2 be two probability measures with

density dQ 1

dQ 2

on a measurable space(Ω,F). We areinterested in findingset Ã , which is a solution

of the problem

{

Q 1 (A) −→ max

A ∈ F :

Q 2 (A) ≤ γ

with γ ∈ [0,1]. Then the explicit solution is given by the following lemma.

Lemma 2.10 (Neyman-Pearson)

If there exists constant β such that Q 2 { dQ 1

dQ 2

≥ β} = γ then Q 1 { dQ 1

dQ 2

≥ β} ≥ Q 1 (B) for any set

B satisfying Q 2 (B) ≤ γ.

We recall also the pricing theorem, which has its origin in the theorem on optional decomposition

of the supermartingales. For more details see  and for later extensions , .

Theorem 2.11 (Price characterization) Let Q be a set of martingale measures for the semimartingale

(S t ) describing evolution of the stock prices. Let H be a non negative contingent

claim. Then there exists a trading strategy (˜x, ˜ϕ), where ˜x = sup Q∈Q E Q [H] s.t.

∫ t

˜x+ ˜ϕ(s)dS(s) ≥ esssup E Q [H | F t ].

0 Q∈Q

The pair (˜x, ˜ϕ) is thus a hedging strategy and ˜x is the price of H.

4

3 Problem formulation

Definition 3.1 A contingent claim H on a large financial market is a sequence of random

variables H 1 ,H 2 ,... satisfying the following conditions

1) For each n = 1,2... H n : Ω n −→ R + is an F n measurable, non negative random variable.

2) For each n = 1,2,... sup Q∈Q nE Q [H n ] < ∞ holds.

In classical market models we have always one random variable which we want to price and

hedge. The question arises for justification of considering a sequence of random variables. We

present 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 1 ,F 2 ,...). Then H n can be defined as projections of G on the spaces (Ω n ,F n ,P n ), i.e.

H n = E Pn [G | F n ]. Thus, we want to price a derivative which depends on infinitely many

assets 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 assets

as well) only. Then we can define H n = G for each n and consider opportunity arising from

the fact that the number of assets which can be traded is increasing. We examine how the

increasing 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 xn,ϕn

T

≥ H n ∀ n = 1,2,....

Such class of sequences we denote by H 1 . A strong price of H is defined as

v(H) =

inf lim x n .

(x n,ϕ n)∈H 1 n→∞

Throughout the whole paper we assume that α is any number from the interval [0,1].

Definition 3.3 A sequence (x n ,ϕ n ) n is a sequence of α-hedging strategies if

lim P n (V xn,ϕn

T

≥ H n ) ≥ α.

n→∞

Such class of sequences we denote by H α . An α-quantile price of H is defined as

v α (H) =

A weak price of H is the 1-quantile price, i.e.

inf lim x n .

(x n,ϕ n)∈H α n→∞

ṽ(H) := v 1 (H).

As follows from the definition above, we consider sequences of strategies which do not allow to

exceed 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 by

Definition 3.2, where there is no risk for any n = 1,2,....

At this stage it is clear that v α (H) ≤ v β (H) ≤ ṽ(H) ≤ v(H) for α < β since the following inclusions

hold : H α ⊇ H β ⊇ H 1 ⊇ H 1 . The main goal of the paper is to provide the characterization

of the prices and solve the problem of equality between the strong and the weak price.

5

4 Prices characterization

Using the price characterization Theorem 2.11 on a classical market it is simple to show the

following.

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 sup Q∈Q nE Q [H n ]. By Theorem 2.11, for any (x n ,ϕ n ) ∈ H 1 we get

n

x n ≥ sup Q∈Q nE Q [H n ] and thus v(H) ≥ g.

Taking ˜x n := sup Q∈Q nE Q [H n ], from Theorem 2.11 we know that there exists a sequence of

strategies (˜ϕ n ) s.t. (˜x n , ˜ϕ n ) ∈ H 1 and thus v(H) ≤ g.

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 = 1,2,... and lim

In particular (A n ) belongs to the class A 1 if P n (A n ) −→

n

1.

n→∞

P n (A n ) ≥ α.

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 Hα ⊆ A α )

Each element in H α indicates an element in A α . Indeed, for (x n ,ϕ n ) ∈ H α let us define

An

xn,ϕn := {V xn,ϕn

T

≥ H n }. By definition of H α we obtain that (An

xn,ϕn ) ∈ A α . Thus, if we

denote the sequences of sets above by A Hα the following inclusion holds : A Hα ⊆ A.

Remark 4.4 (H Aα ⊆ 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 A n,ϕ A n) satisfies : x A n = sup Q∈Q nE Q [H n 1 An ] and ϕ A n hedges the contingent claim

H n 1 An (on a small market with index n). It follows that (x A n ,ϕA n ) ∈ H α since (A n ) ∈ A α . If

we denote the sequences of strategies of the form above by H Aα the following inclusion holds:

H Aα ⊆ 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 1 An ].

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 E Q [H n 1 A

xn,ϕn]

Q∈Q n n

and therefore

lim

n

x n ≥ lim

n

sup

Q∈Q n E Q [H n 1 A

xn,ϕn

n

].

6

By the definition of the α-quantile price and by Remark 4.3 we obtain

v α (H) =

inf lim x n ≥

(x n,ϕ n)∈H α n

inf lim sup E Q [H n 1 A

xn,ϕn

(x n,ϕ n)∈H α n Q∈Q n

= inf

lim

(A n)∈A Hα n

≥ inf

lim

(A n)∈A α n

sup E Q [H n 1 An ]

Q∈Q n

sup E Q [H n 1 An ]

Q∈Q n

n

]

(≤) Consider an arbitrary element (A n ) ∈ A α and a correspondingstrategy described in Remark

4.4. Following the notation of Remark 4.4 we have

and therefore

By Remark 4.4 we obtain

lim

n

sup

Q∈Q n E Q [H n 1 An ] = x A n

sup E Q [H n 1 An ] = lim

Q∈Q n n

inf lim sup E Q [H n 1 An ] = inf

(A n)∈A α n Q∈Q n

x A n

lim x A n

(A n)∈A α n

= inf lim x n

(x n,ϕ n)∈H Aα n

inf lim x n

(x n,ϕ n)∈H α n

= v α (H)

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 = [0,1], S i n(1) = S i n(0)(1+ξ i ), i = 1,2,...,n, n = 1,2,...

where (ξ i ) is a sequence of random variables given by

{

−1 on [0,1−

1

] := E

2

ξ i =

i i

δ(2 i −1)

on (1− 1 ,1] := F

2 i −δ(2 i −1) 2 i i , δ ∈ (0,1).

Sigma fields are assumed to be generated by the sequence (ξ i ), i.e. F n = σ(ξ 1 ,ξ 2 ,...,ξ n ), and the

n-th objective probability measure P n is a restriction of the Lebesgue’s measure P on [0,1] to the

sigma-field F n , i.e. P n = P |F n. Each martingale measure Q n on the n-th market is described

by the property : E Qn [ξ 1 ] = 0,E Qn [ξ 2 ] = 0,...,E Qn [ξ n ] = 0. Thus Q n is indicated by its values

on the intervals E 1 ,E 2 ,...,E n and one can check that

(

Q n (E 1 ) = δ 1− 1 )

2

Q n (E 2 ) = δ

(1− 1 )

2 2

...

Q n (E n ) = δ

(

1− 1

2 n )

.

7

It 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 ≡ 1.

Proposition 4.7 In the model specified above we have:

v α (1) = δα.

Proof : We shall construct explicitly a sequence of sets (Ãn) ∈ A α satisfying:

lim Q n (Ãn) = inf

(A n)∈A α

lim Q n (A n ).

Let: G 1 := E 1 , G n := E n \E n−1 for n=2,3,....Then

Consider a series expansion of α:

P n (G n ) = P n (F n ) = 1

2 n and

δ

(

2 n = Qn (G n ) < Q n (F n ) = 1−δ 1− 1 )

2 n

α =

∞∑

i=1

γ i

2 i, where γ i ∈ {0,1}.

Define Ãn as follows

Ã n :=

n⋃

i=1

1 {γi =1}G i

and notice, that P n (Ãn) = ∑ n

i=1 γ iP n (G i ) = ∑ n γ i

i=1

and therefore lim

2 i n→∞ P n (Ãn) = ∑ ∞ γ i

i=1

=

2 i

α, so (Ãn) ∈ A α . For any (A n ) ∈ A α we have limP n (Ãn) ≤ limP n (A n ) and limQ n (Ãn) ≤

limQ n (A n ).

Thus

v α (1) = inf lim

(A n)∈A α n

∞∑

=

i=1

γ i

δ

2 i = δ ∞ ∑

n∑

E Qn [1 An ] = lim Q n [Ã n ] = lim 1 {γi n n

=1}Q n (G i )

i=1

γ i

2 i = δα. □

i=1

Notice that δ = ṽ(1) < v(1) = lim

between the strong and the weak price is possible.

n

E Qn  = 1, so this example shows that strict inequality

Notice also that this model admits AA2 and does not satisfy AA1. Indeed, taking the sequence

(F n ), we get: P n (F n ) = 1

2

−→ 0 and Q n (F n n ) = 1−δ ( 1− 1 )

2 −→ 1−δ > 0 and thus there is

n

AA2. Let (A n ) be a sequence s.t. Q n (A n ) −→ 0. This means that for any l > 0, Q n (A n ) < δ 2 l

holds for all large n and one can check, that this implies that A n ⊆ (1 − 1 ,1] for all large n.

2 l

As a consequence we obtain lim n P n (A n ) < 1 and letting l to ∞ we get lim

2 l n P n (A n ) = 0. This

means that NAA1 and also NSAA1, NSAA2 hold.

This example shows that NAA1, NSAA1, NSAA2 is insufficient for the equality of the strong

and the weak price.

8

Theorem 4.5 yields immediately two following conclusions.

Remark 4.8 If we require that ṽ(H) = v(H) even for H of simple structure then the market

must satisfy NAA2. Indeed, suppose that AA2 holds. It implies that for any (A n ) ∈ A 1 ,

¯Q n (A n ) ↛ 1 holds. Taking H ≡ 1 we obtain

ṽ(1) = inf

(A n)∈A 1

lim

n

¯Q n (A n ) < 1 = v(1).

Remark 4.9 If there is SAA1 or equivalently SAA2, then for any H bounded, i.e. H n ≤ K

for some constant K > 0, we have v α (H) = 0 for any α ∈ [0,1]. Indeed, by Theorem 2.9 there

exists a sequence (Ãn) s.t. P n (Ãn) −→ 1 and ¯Q n (Ãn) −→ 0. Then we have

v α (H) ≤ ṽ(H) ≤ lim

n

sup E Qn [H n 1Ãn ] ≤ lim K ¯Q n (Ãn) = 0.

Q∈Q n n

The next theorem provides some insight into the problem of asymptotic pricing for complete

models.

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) = ṽ(H).

Proof : First notice, that for any fixed (A n ) ∈ A 1 by NAA2 we obtain

Now consider two sequences:

The following holds:

and thus

P n (A n ) −→ 1 ⇐⇒ P n (A c n ) −→ 0 =⇒ Qn (A c n ) −→ 0.

x n := E Qn [H n ]

y n := E Qn [H n 1 An ].

x n −y n = E Qn [H n −H n 1 An ] = E Qn [H n 1 A c n

] ≤ K ·Q n (A c n) −→ 0

lim

n

x n = lim

n

Taking infimum over all (A n ) ∈ A 1 we obtain the required result.

y n .

v(H) = lim

n

x n = inf

(A n)∈A lim n

y n = ṽ(H)

9

Remark 4.11 Assume that NAA2 holds. For incomplete market we can define the analogous

sequences 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 1 An ]

and for these sequences we obtain analogous inequality

x n −y n ≤ sup

Q∈Q n E Q [H n −H n 1 An ] = sup

Q∈Q n E Q [H n 1 A c n

] ≤ K · ¯Q n (A c n).

However, we do not know if the last term goes to 0 as n −→ ∞. We know that Q n (A c n) −→ 0

only and this is insufficient to perform the above proof for incomplete markets.

5 The large Black-Scholes model

LetWt 1,W2

t ,...beasequenceofindependentstandardBrownianmotionsonafilteredprobability

space (Ω,F t ,F,P),t ∈ [0,T]. We will consider a stationary market, wherethe n-th small market

has its natural filtration i.e. Ft n = σ((Ws 1,...,Wn s ) s∈[0,t]) and F n = FT n . The n-th objective

measure is an adequate restriction of P i.e. P n = P | F n and the discounted price processes are

given by

dSt i = Si t (b idt+σ i dWt i ) i = 1,2,...,n, t ∈ [0,T]

where b i ∈ R,σ i > 0 for i = 1,2,...,n. Such sequence forms a complete large market with

martingale measures given by densities

dQ n

dP n = Z n = e −(θn ,W n T )−1 2 ‖θn ‖ 2 T

where θ n = ( b 1

σ 1

,..., bn

σ n

) and Wt n = (W1 t ,...,Wn t ). Recall, that W∗n t = Wn t +θn t is a Brownian

motion under Q n . In this setting we show more indirect proofs for the absence of asymptotic

arbitrage 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 coefficients

only. In this section we show also, that Theorem 4.10 and Remark 4.9 remain true for

random variables satisfying some integrability conditions, which are satisfied for widely used

derivatives.

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 > 0 let A n ε denote a solution of the problem

{

A ∈ F n P n (A) −→ max

:

Q n (A) ≤ ε.

Then the following conditions are equivalent

1) NAA1

2) (P n ) ⊳ (Q n )

10

3) For any sequence (ε n ) ∈ E, P n (A n ε n

) −→ 0 holds.

4) ∑ ∞

i=1 (b i

σ i

) 2 < ∞

Proof : Equivalence of (1) and (2) is proved in .

(2) =⇒ (3)Let(ε n )beanyelementofE. ThenQ n (A n ε n

) ≤ ε n −→ 0andthusby(2), P n (A n ε n

) −→

0 holds.

(3) =⇒ (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 n ε n

) −→ 0 holds.

(3) ⇐⇒ (4)Statistical methodsprovideanexplicitformofthesetA n ε. AccordingtotheNeyman-

Pearson Lemma 2.10 it is of the form A n ε = { dPn

dQ

≥ γ}, where γ is a constant s.t. Q n (A n ε) = ε.

n

This construction provides

A n ε =

{

e (θn ,WT n)+1 2 ‖θn ‖ 2T }

≥ γ

=

{

(θ n ,WT) n ≥ lnγ − 1 }

2 ‖ θn ‖ 2 T

{

= (θ n ,(WT ∗n −θT)) ≥ lnγ − 1 } {

2 ‖ θn ‖ 2 T = (θ n ,WT ∗n ) ≥ lnγ + 1 }

2 ‖ θn ‖ 2 T .

Solving the following equation:

{

Q n (A n ε) = Q n (θ n ,WT ∗n ) ≥ lnγ + 1 } (

lnγ +

2 ‖ θn ‖ 2 1

2

T = 1−Φ

‖ )

θn ‖ 2 T

‖ θ n ‖ √ = ε

T

we obtain

γ = e ‖θn ‖ √ TΦ −1 (1−ε)− 1 2 ‖θn ‖ 2T .

We calculate the value P n (A n ε ).

P n (A n ε) =P

((θ n n ,WT) n ≥ lnγ − 1 ) (

lnγ −

2 ‖ θn ‖ 2 1

2

T = 1−Φ

‖ )

θn ‖ 2 T

‖ θ n ‖ √ T

=1−Φ

(

‖ θ n ‖ √ T Φ −1 (1−ε)− ‖ θ n ‖ 2 T

‖ θ n ‖ √ T

Now observe that if ∑ ∞

i=1 (b i

σ i

) 2 < ∞ then for any (ε n ) ∈ E

(

1−Φ Φ −1 (1−ε n )− ‖ θ n ‖ √ )

T

)

(

= 1−Φ Φ −1 (1−ε)− ‖ θ n ‖ √ )

T

−→ 0.

If ∑ ∞

i=1 (b i

σ i

) 2 = ∞ then ε n := 1−Φ(1+ ‖ θ n ‖ √ T) −→ 0 and

(

1−Φ Φ −1 (1−ε n )− ‖ θ n ‖ √ )

T = 1−Φ(1) ↛ 0.

The next two theorems provide characterization of NAA2, SAA1 and SAA2. The proofs are

similar and therefore we sketch some parts of them only.

Theorem 5.2 For ε > 0 let A n ε

denote a solution of the problem

{

A ∈ F n Q n (A) −→ max

:

P n (A) ≤ ε.

Then the following conditions are equivalent

11

1) NAA2

2) (Q n ) ⊳ (P n )

3) For any sequence (ε n ) ∈ E, Q n (A n ε n

) −→ 0 holds.

4) ∑ ∞

i=1 (b i

σ i

) 2 < ∞

Proof : (3) ⇐⇒ (4) The set A n ε is of the form

A n ε = { dQ

n

dP n ≥ γ }

where γ is s.t. P n (A n ε ) = ε. This procedure yields

{ (

ln

1

A n ε = (θ n ,WT) n γ

≤ Φ

− 1 2 ‖ )}

θn ‖ 2 T

‖ θ n ‖ √ T

(

and Q n (A n ε) = Φ Φ −1 (ε)+ ‖ θ n ‖ √ )

T .

If ∑ ∞

i=1 (b i

σ i

) 2 < ∞ then for any (ε n ) ∈ E

γ = e −[Φ−1 (ε)‖θ n ‖ √ T+ 1 2 ‖θn ‖ 2 T]

(

Φ Φ −1 (ε n )+ ‖ θ n ‖ √ )

T −→ 0.

If ∑ ∞

i=1 (b i

σ i

) 2 = ∞ then taking ε n := Φ(1− ‖ θ n ‖ √ T) −→ 0 we obtain

(

Φ Φ −1 (ε n )+ ‖ θ n ‖ √ )

T = Φ(1) ↛ 0.

Theorem 5.3 For ε > 0 let A n ε denote a solution of the problem

{

A ∈ F n P n (A) −→ max

:

Q n (A) ≤ ε.

Then the following conditions are equivalent

1) SAA1

2) SAA2

3) P n △ Q n

4) Q n △ P n

5) There exists a sequence (ε n ) ∈ E s.t. P n (A n ε n

) −→ 1

6) ∑ ∞

i=1 (b i

σ i

) 2 = ∞.

12

Notice, that the conditions for the set A n ε are based on property P n △ Q n . One can base the

proof on the property Q n △ P n . This requires replacing measures P n and Q n in the conditions

for A n ε. The first four conditions are proved in  and are included in the formulation above

for the clarity of exposition only. Equivalence of (3) and (5) are easy to prove.

Proof : (5) ⇐⇒ (6) We use the construction of A n ε found in the proof of Th. 5.1

{

A n ε = (θ n ,WT ∗n ) ≥ lnγ + 1 }

2 ‖ θn ‖ 2 T

γ = e ‖θn ‖ √ TΦ −1 (1−ε)− 1 2 ‖θn ‖ 2 T

(

P n (A n ε) = 1−Φ Φ −1 (1−ε)− ‖ θ n ‖ √ )

T

If ∑ (

i=1 (b i

σ i

) 2 < ∞ then for any (ε n ) ∈ E, 1 − Φ Φ −1 (1−ε n )− ‖ θ n ‖ √ )

T −→ 0 holds. If

∑ ∞

i=1 (b i

σ i

) 2 = ∞ then ε n := 1−Φ( 1 2 ‖ θn ‖ √ (

T) −→ 0 and 1−Φ Φ −1 (1−ε n )− ‖ θ n ‖ √ )

T −→

1.

Inthe sequel we will characterize theweak price of H satisfying someintegrability conditions.

If H = H, where H is one fixed random variable measurable with respect to F 1 , then it is clear

that E Qn [H] does not depend on n and thus indicates the strong price. This means that the

investor doesn’t have any profits from the fact that the market is getting large and that he

can use greater and grater number of strategies. It turns out that he can not make any profits

unless he uses 1-quantile hedging strategies. In this case, but if ∑ ∞

i=1 (b i

σ i

) 2 = ∞, the initial

endowment can be reduced to 0, i.e. the weak price is equal to 0. The condition ∑ ∞

i=1 (b i

σ i

) 2 < ∞

guaranties that the investor is not able to make any profits at all, no matter what strategies he

uses, because then v(H) = ṽ(H).

Theorem 5.4 Let H be a contingent claim on a large Black-Scholes market with constant coefficients.

Then

1) if ∑ ∞

i=1 (b i

σ i

) 2 < ∞ and lim

n

E[H 1+δ

n ] < ∞ for some δ > 0 then ṽ(H) = v(H).

2) if ∑ ∞

i=1 (b i

σ i

) 2 = ∞ and lim

n

E[H 4+δ

n ] < ∞ for some δ > 0 then ṽ(H) = 0.

Proof : (1) For any sequence (A n ) ∈ A 1 define x n := E Qn [H n ], y n := E Qn [H n 1 An ]. Let

p,q,p ′ ,q ′ 1

> 1 be real numbers and s.t.

p + 1 q = 1, 1

+ 1 = 1. Using Hölder inequality twice to

p ′ q ′

the difference x n −y n we obtain:

( ( )1

x n −y n =E Qn [H n 1 A c n

] = E[Z n H n 1 A c n

] ≤ E(Z n H n ) p)1 p

P(A c q

n)

( ( ) 1

≤ E(Zn pp′ ) ′( ) 1

)1

p

E(Hn pq′ ) q

p(P(A ′ )1

c

n ) q

) 1

=

(E(Z n pp′ pp

)

′( ) 1

E(Hn pq′ pq

)

′( )1

P(A c q

n) .

Straightforward calculations yields

) 1 (E(Z n pp′ ) pp ′ = e 1 2 ‖θn ‖ 2 T(pp ′ −1)

(5.4.1)

13

( ) 1

and thus NAA2 guaranties that lim n→∞ E(Zn pp′ pp

)

′ < ∞. Taking p,p ′ s.t. pq ′ = 1 + δ

( )1

and using fact that lim n→∞ P(A c n ) q

= 0 we conclude that lim n→∞ (x n − y n ) = 0. Thus

lim x n = lim y n and taking infimum over all (A n ) ∈ A 1 we get ṽ(H) = v(H).

(2) For any (A n ) ∈ A 1 , p,p ′ > 1 and q,q ′ s.t.

1

p + 1 q = 1, 1

p ′ + 1 q ′ = 1 using Hölder inequalities

we obtain:

E Qn [H n 1 An ] ≤

( )1 ( )1

E Qn (Hn)

p p

Q n q

(A n )

( ( ) 1

EZn

p′

p ′( EH pq′

n

=

( )1 ( )1

E(Z n Hn)

p p

Q n q

(A n )

) 1

)1

p(

q ′ )1

Q n q

(A n )

=

) 1 (EZ n

p′

pp ′ ( EH pq′

n

) 1 (

pq ′ )1

Q n q

(A n )

Now, similarly to the previously used methods let us solve an auxiliary problem of finding set

A n ε s.t.

{

A ∈ F n Q n (A) −→ min

:

P n (A) ≥ 1−ε.

Analogous calculations provide:

A n ε =

{ dQ

n

dP n ≤ γ }

=

{

(θ n ,W n ) ≥ −lnγ 1 }

2 ‖ θn ‖ 2 T

γ = e −[Φ−1 (ε)‖θ n ‖ √ T+ 1 2 ‖θn ‖ 2 T]

(

Q n (A n ε) = Φ −Φ −1 (ε)− ‖ θ n ‖ √ )

T .

(

Taking p = 2+ 1 2 δ,p′ = 2,ε n = Φ −ln(‖ θ n ‖ √ )

T) (AA2 guaranties that ε n → 0) we get

lim E[Hn pq′ ] = lim E[H4+δ n ] < ∞

and

( ( ) 1 (

EZn

p′ pp ′ )1 q

q)

Q n (A n ε n

) = e 1 p ′ −1

2 p−1 ‖θn ‖ 2T (

Φ −Φ −1 (ε n )− ‖ θ n ‖ √ )

T

= e 1

2+δ ‖θn ‖ 2T (

Φ ln(‖ θ n ‖ √ T)− ‖ θ n ‖ √ )

T

(5.4.2)

Replacing ‖ θ n ‖ √ T by x for the sake of convenience, we calculate the following limit using

d’Hospital formula.

lim e 1

2+δ x2 Φ(lnx−x) = lim

x→∞ x→∞

1 √

e −1 2 (lnx−x)2 ( 1 x −1)

= lim

x→∞ −2+δ

2 √ 2π

e − 1

2+δ x2 (− 1

2+δ )2x

[e x2 ( 1

2+δ −1 2 )−1 2 ln2 x+xlnx

x 2 − ex2 ( 1

2+δ −1 2 )−1 2 ln2 x+xlnx

x

]

= 0

14

The limit is equal to 0 since: limx 2 ( 1

2+δ − 1 2 )− 1 2 ln2 x+xlnx = −∞.

( ) 1 (

Summarizing, we have shown that lim n→∞ EZ p′ pp ′ EH pq′) 1 ( pq ′ Q n (A n ε n

)

parameters and thus ṽ(H) = 0.

n

)1

q

Remark 5.5 The integrability conditions imposed on H in the second item of Theorem 5.4 can

be a little bit weakened. It follows from 5.4.2 that we have to find parameters p,p ′ > 1 s.t.

1 p ′ −1

2 p−1 = 1

2+δ . We can impose additional requirement: pq′ → min. Then it can be checked, that

2+δ

the solution is: p = 1+

2 , p′ = 2+2 2+δ

2 +δ √

2+δ

and pq ′ = 2

2 + 1 √

2 (4+δ)+ 2

2 2+δ. If δ → 0

then pq ′ is arbitrarily close to

2

2

+2+1 < 4. Thus, we can assume that

√ √

2

2 +1 2 (4+δ)+ 2 √ 2+δ

2

lim E[Hn ] < ∞ for some δ > 0.

n

The next theorem provides a more precise characterization of the α-quantile price. But first let

us 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 respect

to the measure P n .

By q n (α) we denote the α-quantile of H n Z n , i.e. q n (α) = {infx : P n (H n Z z ≤ x) ≥ α}.

Denote by B α a set of sequences satisfying

lim β n ≥ α.

n−→∞

Theorem 5.7 Let H be a contingent claim on a large Black-Scholes model with constant coefficients.

1) Under assumption 5.6 the α-quantile price is given by the formula

[ ]

v α (H) = inf lim E H n Z n 1 {HnZ n≤q n(β n)} .

(β n)∈B α n→∞

2) Let assumption 5.6 be satisfied. If lim E[H 1+δ

n

n ] < ∞ for some δ > 0 and ∑ ∞

i=1 (b i

σ i

) 2 < ∞

then

[ ]

v α (H) = lim E H n Z n 1 {HnZ n≤q n(α)} .

n→∞

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 4+δ

n ] < ∞ for some δ > 0 and ∑ ∞

i=1 (b i

σ i

) 2 = ∞ then v α (H) = 0 for each α ∈ [0,1].

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 1 An ].

15

Let us consider any (A n ) ∈ A α and define β n := P n (A n ). Denote by Ãn a solution of the

following problem: {

E Qn [H n 1 An ] −→ min

Ã n :

P n (A n ) ≥ β n .

If we introduce measure ˜Q n by the density d˜Q n

written in the equivalent form:

dQ n :=

H n

E Qn [H n]

, then the above problem can be

Ã n :

{˜Qn (A n ) −→ min

P n (A n ) ≥ β n .

Therefore by Lemma 2.10 we conclude that Ãn is of the form: {H n Z n ≤ γ}, where γ is a

constant s.t. P n (H n Z n ≤ γ) = β n . By Assumption 5.6 we know that there exists such γ and it

is equal to q n (β n ). Thus Ãn = {H n Z n ≤ q n (β n )} and

E Qn [H n 1 An ] ≥ E Qn [H n 1 {HnZ n≤q n(β n)}].

Letting n → ∞ and taking infimum over all (A n ) ∈ A α we obtain:

[ ]

v α (H) ≥ inf lim E H n Z n 1 {HnZ n≤q n(β n)} . (5.7.3)

(β n)∈B α n→∞

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 in

5.7.3.

(2) Let α,β ∈ [0,1] be two real numbers s.t. β < α. For p,q,p ′ ,q ′ > 1 s.t. 1 p + 1 q = 1, 1 p ′ + 1 q ′ = 1

we have the following inequality:

E[H n Z n 1 {HnZ n≤q n(α)}]−E[H n Z n 1 {HnZ n≤q n(β)}] = E[H n Z n 1 {qn(β)≤H nZ n≤q n(α)}]

) 1 (E(Z n pp′ pp

)

′( ) 1

E(Hn pq′ pq

)

′( )1

q

P(q n (β) ≤ H n Z n ≤ q n (α))

) 1 (

However, by 5.4.1 we have

(E(Z n pp′ pp

)

′ ≤ lim n→∞

( ) 1

1+δ and denoting K 1 := lim n→∞ E(Zn pp′ pp

)

′ and K 2 :=

=

) 1

E(Zn pp′ )

) 1 (E(Z n pp′ pp

)

′( ) 1

E(Hn pq′ pq

)

′ (α−β)

pp ′ < ∞. Taking p,q ′ s.t. pq ′ =

(E(H pq′

n )

) 1

pq ′ , we obtain

E[H n Z n 1 {HnZ n≤q n(α)}]−E[H n Z n 1 {HnZ n≤q n(β)}] ≤ K 1 K 2 (α−β).

and interchanging the role of α and β we obtain

| E[H n Z n 1 {HnZ n≤q n(α)}]−E[H n Z n 1 {HnZ n≤q n(β)}] |≤ K 1 K 2 | α−β | . (5.7.4)

Now consider (β n ) ∈ B α . If lim

n−→∞

β n > α then E[H n Z n 1 {HnZ n≤q n(β n)}] > E[H n Z n 1 {HnZ n≤q n(α)}]

andthuslimE[H n Z n 1 {HnZ n≤q n(β n)}] ≥ limE[H n Z n 1 {HnZ n≤q n(α)}]. If lim

n−→∞

β n = αthenby5.7.4

wehave|E[H n Z n 1 {HnZ n≤q n(α)}]−E[H n Z n 1 {HnZ n≤q n(β n)}] |≤ K 1 K 2 | α−β n |andlettingn → ∞

we obtain lim E[H n Z n 1 {HnZ n≤q n(β n)}] = lim E[H n Z n 1 {HnZ n≤q n(α)}]. The conclusion from these

two cases is that v α (H) ≥ lim E[H n Z n 1 {HnZ n≤q n(α)}]. However, {H n Z n ≤ q n (α)} ∈ A α and

therefore

[ ]

v α (H) = lim E H n Z n 1 {HnZ n≤q n(α)} . (5.7.5)

n→∞

16

Letting n → ∞ in 5.7.4 and using 5.7.5 we obtain:

| v α (H)−v β (H) |≤ K 1 K 2 | 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 0 (H) = 0. By Theorem 5.4 v 1 (H) = v(H).

(3) It is an immediate consequence of Theorem 5.4 (2), since v α (H) ≤ ṽ(H). □

Remark 5.8 Consider the prices of a call option, i.e. H ≡ (S 1 T − K)+ . The distribution of

(S 1 T −K)+ Z n is discontinuous in 0. Let α 0 := P n ((S 1 T −K)+ = 0). It is clear, that for α ≤ α 0 ,

v α (H) = 0 holds. On the interval (0,∞) the distribution function is continuous, thus for α > α 0

Theorem 5.7 can be applied.

Conclusion

In this paper we have introduced and characterized two types of asymptotic prices. They are

based on different treating of hedging risk which disappears in infinity. Relations between them

strictly depend on the asymptotic arbitrage on the market. In case of the large Black-Scholes

model with constant coefficients it was possible to find more indirect formula for the α-quantile

price 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 equal

2) there is asymptotic arbitrage of all kinds - then the weak price is equal to zero, while the

strong is not.

Acknowledgement This paper is a part of the author’s PhD thesis. The author would like to

express his thanks to Professor ̷Lukasz Stettner for help and support while writing this paper.

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