Jinfeng LI

Abstract Based on a concept of asymptotic exponential arbitrage proposed by Föllmer-Schachermayer,the author introduces a new formulation of asymptotic arbitrage with two main differences from the previous one: Firstly,the realising strategy does not depend on the maturity time while the previous one does,and secondly,the probable maximum loss is allowed to be small constant instead of a decreasing function of time.The main result gives a sufficient condition on stock prices for the existence of such asymptotic arbitrage.As a consequence,she gives a new proof of a conjecture of Föllmer and Schachermayer.

Keywords Asymptotic arbitrage,Time-consistent,Small probable maximum loss

1 Formulations of Asymptotic Exponential Arbitrage

Let (Ω,F,F,P)be a filtered probability space where the filtration F=(Ft)t≥0satisfies the usual conditions,and the discounted price process S=(St)t≥0initially be any Rd-valued semimartingale.Based on a result of Schweizer [4],let us assume that the price process S has the form:

where M is a d-dimensional continuous local martingale with M0=0,λ is a d-dimensional predictable process,the market price of risk,satisfying

Let L(S)be the set of all predictable processes integrable with respect to S,and define for each T >0 the set

and in particular,

Clearly,H0⊂HTfor any T >0.

1.1 Föllmer-Schachermayer’s formulation

The following form of a long-term arbitrage was considered for the first time in Föllmer and Schachermayer[3].

Definition 1.1(Asymptotic Exponential Arbitrage)The process S=(St)t≥0allows asymptotic exponential arbitrage with exponentially decaying failure probability if there exist 0<<∞and constants C,κ1,κ2>0 such that for all T ≥,there is H ∈HTsatisfying

(a)(H ·S)T≥−e−κ1TP-a.s.;

(b)P[(H ·S)T≤eκ1T]≤Ce−κ2T.

We should note that the choice of the realising strategy H depends on the maturity T.Föllmer and Schachermayer[3]showed how such a notion is related to large deviation estimates for the market price of risk.They derived the results for some concrete models (the geometric Ornstein-Uhlenbeck process and the Black-Scholes model),and suggested the following general result which has been proved by Du and Neufeld [2]by means of a time-change argument.

Theorem 1.1(cf.[2])Let the filtration F be continuous in the sense that all local martingales are continuous.Assume that the market price of risk λ satisfies a large deviation estimate,i.e.,there are constant c1,c2>0 such that

Then S allows asymptotic exponential arbitrage.

1.2 Time-consistent asymptotic exponential arbitrage

In Föllmer-Schachermayer’s formulation,the realizing strategies for asymptotic exponential arbitrage depend on the horizon,that means the strategies may change as T varies.From the practical point of view,we expect that the arbitrage-realizing strategy can be independent of the horizon,in other words,has time-consistence.As a cost,a constant but small probable maximum loss is permitted.

Definition 1.2(Time-Consistent Asymptotic Exponential Arbitrage)The process S=(St)t≥0allows time-consistent asymptotic exponential arbitrage if there exist H ∈H0and constants T0,C,κ1,κ2>0 such that for all T ≥T0,

(a)(H ·S)T≥−1 P-a.s.;

(b)P[(H ·S)T≤eκ1T]≤Ce−κ2T.

This note aims to show that the condition (1.2)also suffices for time-consistent asymptotic exponential arbitrage.

Assumption 1.1The martingale M in (1.1)is continuous,and E(−αλ · M)is a true martingale for each α>0.

It is worth noting that here we do not require the continuity of filtration.The main result of this note is as follows.

Theorem 1.2Let Assumption 1.1 and the large deviation estimate (1.2)be satisfied.Then S allows time-consistent asymptotic exponential arbitrage.

We remark that a strategy realising time-consistent asymptotic exponential arbitrage naturally yields Föllmer–Schachermayer’s asymptotic exponential arbitrage.Indeed,if Htis a realising strategy for the former,thenis the one for the latter,where T is the maturity time.Moreover,Theorem 2.1 below gives a precise form of realising strategies.

2 Proofs

Since 1 ≪eκ1Twhen T is large,a tiny adjustment of κ1gives

(a′)1+(H ·S)T≥0 P-a.s.;

(b′)P[1+(H ·S)T≤eκ1T]≤Ce−κ2T.

Now let us denote

then we have the following lemma.

Lemma 2.1If there are κ1,κ2>0 such that

then S allows time-consistent asymptotic exponential arbitrage.

ProofTake 0<ε<κ2.There exist Hε∈H0and T0>0 such that for any T ≥T0,

that is

which concludes the result.

Let γ <0.Chebyshev’s inequality gives

thus

Taking limits and inferiors we have

Therefore S allows asymptotic exponential arbitrage provided the right-hand side is negative,and the original problem is converted to bound from above the value function of a long-term risk-sensitive control problem:

We consider an investor trading in the above market.More specifically,the wealth process,denoted by X=X(π),starting from 1,satisfies

where π denotes the strategy and E the stochastic exponential.

Define

and

where A denotes the admissible set containing strategy π such that π ∈L(M).Such an utilitybased optimal investment problem has been addressed in numerous literature.

Here we are going not to solve V(γ,T)explicitly,but to give an appropriate upper bound for it.

To this end,we select the following time-consistent strategy:

Then

where,for simplicity,we have denote

Thus from Hölder’s inequality,for

Thus

To get the exact infimum is not easy,so we take,for simplicity,

then Since the strategy π⋆is independent of T,we have

To sum up,we have proved the following proposition.

Proposition 2.1Under Assumption 1.1,the wealth process X=X(π⋆)realized by the strategy:

satisfies

for each T >0.Consequently,the function χ(·)define in (2.2)satisfies

By means of Varadhan’s integral lemma (cf.[1,Theorem 4.3.1]),we have the following lemma.

Lemma 2.2Suppose thatsatisfies a large deviation principle with a rate function I(x).Then

Recalling Lemma 2.1 and relation (2.1),we have actually proved the following theorem.

Theorem 2.1Let Assumption 1.1 be satisfied.Suppose thatsatisfies a large deviation principle with a rate function I(x).Then the wealth process X·=X·(H⋆)realized by

satisfies

Consequently,S allows time-consistent asymptotic exponential arbitrage provided the right-hand side of (2.8)is negative for some κ>0.

In fact,the condition (1.2)is sufficient to ensure time-consistent asymptotic exponential arbitrage of S.

Proof of Theorem 1.2Let X·=X·(H⋆)be the process defined in Theorem 2.1.For K >0,we have

thus by (1.2),

Recalling (2.4),we gain

which along with (2.1)yields

A proper choice of κ can ensure the negativeness of the right-hand side.The proof of Theorem 1.2 is complete.