We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter H∈(3/4,1) and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.
Mixed fractional Brownian motionrelative entropylarge financial marketentire asymptotic separationstrong asymptotic arbitrage60G2260G1591B2491B26Austrian Science FundJ3453-N25The third author gratefully acknowledges financial support from the Austrian Science Fund (FWF): J3453-N25.Introduction
Empirical studies of financial time series led to the conclusion that the log-return increments exhibit long-range dependence. This fact supports the idea of modelling the randomness of a risky asset using a fractional Brownian motion with Hurst parameter H>1/21/2$]]>. However, markets driven by a fractional Brownian motion have been extensively disputed, as this motion fails to be a semimartingale and, hence, they allow for a free lunch with vanishing risk (see [6]).
Many attempts were proposed to overcome this drawback of the fractional Brownian motion. In this work, we deal with the regularization method proposed by Cheridito in [2, 1] when H>3/43/4$]]>. This method consists in adding to the fractional Brownian motion a multiple of an independent Brownian motion, the resulting process, called mixed fractional Brownian motion, being Gaussian with the long-range dependence property. Moreover, as shown in [2, 1], when H>3/43/4$]]> the mixed fractional Brownian motion is equivalent to a multiple of a Brownian motion. Therefore, a Black–Scholes type model in which the randomness of the risky asset is driven by a mixed fractional Brownian motion is arbitrage free and complete. We call such a model a mixed fractional Black–Scholes model.
On one hand the fractional Black–Scholes model admits arbitrage. On the other hand, when the Hurst parameter H>3/43/4$]]>, adding a Brownian component (in the above explained way) makes the arbitrage disappear. In this paper we aim to go a step further and study the sensitivity to arbitrage of the mixed fractional Black–Scholes model when the Brownian component asymptotically vanishes. In [3, 4] it was argued that a good way of seeing the sensitivity to arbitrage of a market when one of its parameters converges to zero (or infinity), is to consider the family of markets indexed by the corresponding parameter and to use methods from large financial markets. To be precise, we study the asymptotic arbitrage opportunities in the sequence of mixed fractional Black–Scholes models when the scaling factor in front of the Brownian motion converges to zero. We focus on the notion of strong asymptotic arbitrage (SAA) introduced by Kabanov and Kramkov in [9] as the possibility of getting arbitrarily rich with probability arbitrarily close to one by taking a vanishing risk. Our model fits the standard framework of large financial markets, as each mixed fractional Black–Scholes model is arbitrage free (and even complete). We point out that the existence of arbitrage in the limiting market does not directly imply the existence of any kind of asymptotic arbitrage in the approximating sequence of mixed markets. In [9] the existence of strong asymptotic arbitrage was shown to be equivalent to the entire asymptotic separation of the sequence of objective probability measures and the sequence of equivalent martingale measures.
In order to show the existence of strong asymptotic arbitrage in the sequence of mixed fractional Black–Scholes models we use the result of [9] that was mentioned above. That means we show that the sequence of objective probability measures is entirely asymptotically separable from the sequence of equivalent martingale measures. Our main contribution is the proof of this entire asymptotic separability in the given model. We use the notion of relative entropy and a dichotomy result for sequences of Gaussian measures. Indeed, inspired by the work of Cheridito [2, 1], we first show, for each fixed market, that the entropy of the objective probability measure relative to the equivalent martingale measure, both restricted to a discrete partition, converges to infinity. Our proof then follows using tightness arguments for the sequence of Radon–Nikodym derivatives of the objective probability measures with respect to equivalent martingale measures and the fact that two sequences of Gaussian measures are either mutually contiguous or entirely separable. The latter is known in the literature as the equivalence/singularity dichotomy for sequences of Gaussian processes, see [5].
The paper is structured as follows. In Section 2, we set the mixed fractional Black–Scholes model and recall the framework of the large financial market. At the end of this part, we state the main result (Theorem 1). Section 3 is dedicated to the proof of Theorem 1, whereas Section 4 provides a discussion about the existence of strong asymptotic arbitrage using self-financing strategies constrained to jump only in a finite set of times. We end our work with Appendix A in which we recall the definition of relative entropy and an equivalent characterization in terms of the Radon–Nikodym derivative.
Preliminaries and main resultsSetting the model
Let (Ω,F,P) be a probability space.
A fractional Brownian motion ZH=(ZtH)t≥0 with Hurst parameter H∈(0,1) is a continuous centred Gaussian process with covariance function
Cov(ZtH,ZsH)=E(ZtHZsH)=12(t2H+s2H−|t−s|2H),s,t≥0.
In particular, Z12 is a standard Brownian motion.
A linear combination of different fractional Brownian motions is refered in the literature as a mixed fractional Brownian motion. In order to avoid localization arguments we only consider finite time horizon processes. In addition we focus on linear combinations of a standard Brownian motion (Bt)t∈[0,1] and an independent fractional Brownian motion (ZtH)t∈[0,1] with Hurst parameter H∈(3/4,1), both defined on (Ω,F,P). Cheridito shows in [2, 1] that, for each α∈R the mixed process MH,α:=(MtH,α)t∈[0,1] defined by
MtH,α:=αZtH+Bt,t∈[0,1],
is equivalent to a Brownian motion. By this we mean that the measure QH,α induced on C[0,1] by MH,α and the Wiener measure QW (induced by the Brownian motion on C[0,1]) are equivalent. As a consequence, the process MH,α is a (FtH,α)t∈[0,1]-semimartingale, where, for each t≥0, FtH,α:=σ((MsH,α)s∈[0,t])‾ is the right-continuous natural filtration augmented by the nullsets.
Now, for each α>00$]]>, we call by the α-mixed fractional Black–Scholes model the financial market consisting of a risk free asset normalized to one and a risky asset (StH,α)t∈[0,1] given by
StH,α:=S0H,αexp((μ−σ22α2)t+σ(ZtH+1αBt)),t∈[0,1],
where μ∈R and σ>00$]]> represent the drift and the volatility of the asset.1
SH,α is the solution of dStH,α=μStH,αdt+σStH,αd(ZH+1αB)t.
We denote by X:=(Xt)t∈[0,1] the coordinate process in C[0,1] and we define the process Sα:=(Stα)t∈[0,1] as
Stα:=S0αexp((μ−σ22α2)t+σαXt),t∈[0,1].
From the above discussion, we conclude that SH,α under P is equivalent to Sα under QW, which is a martingale when μ=0. For a general drift, we denote by Qμασ the measure induced on C[0,1] by the Brownian motion with drift −μασ (in particular Q0=QW). Thanks to the Girsanov theorem, the process SH,α under P is also equivalent to Sα under Qμασ, which is a martingale. Therefore, the α-mixed fractional Black–Scholes model with the filtration (FtH,α)t∈[0,1] has a unique equivalent martingale measure, and therefore is arbitrage-free and complete.
Asymptotic arbitrage
In this work, we treat the collection of α-mixed fractional Black–Scholes models with methods from large financial markets. This idea is formalized in the following definition.
(The large mixed fractional market).
We call by large mixed fractional market the family of α-mixed fractional Black–Scholes models, α>00$]]>, i.e. the family of markets
LH:=(Ω,F,(FtH,α)t∈[0,1],P,SH,α)α>0.0}.\]]]>
We aim to study the presence of asymptotic arbitrage in the large financial mixed fractional market when α tends to infinity, i.e. when the Brownian component asymptotically disappears. More precisely, we intend to investigate, using methods of [9], the presence of a so-called strong asymptotic arbitrage. The latter is an analogue concept of arbitrage but for sequences of markets rather than for a single market model. Intuitively, this kind of arbitrage for sequences of markets gives the possibility of getting arbitrarily rich with probability arbitrarily close to one while taking a vanishing risk. In order to make this idea precise we first specify the set of admissible trading strategies.
(Admissible trading strategy).
A trading strategy for SH,α is a real-valued predictable SH,α-integrable stochastic process Φ:=(Φt)t∈[0,1]. The trading strategy is said to be admissible if there is m∈R+ such that for all t∈[0,1]: (Φ·SH,α)t≥−m almost surely.
Now we proceed to recall the definition of strong asymptotic arbitrage of [9].
A strong asymptotic arbitrage (SAA) is said to exist in the large mixed fractional market as α tends to infinity if there exists a sequence (αℓ)ℓ≥1 converging to infinity and a sequence (Φℓ)ℓ≥1, where Φℓ is an admissible trading strategy for SH,αℓ, such that
(Φℓ·SH,αℓ)t≥−mℓ,t∈[0,1], ℓ≥1,
limℓ→∞P((Φℓ·SH,αℓ)1≥Mℓ)=1,
where mk and Mk are sequences of positive real numbers converging to zero and to infinity, respectively.
This definition is equivalent to the notion of strong asymptotic arbitrage of the first kind as given in [9]. This is trivially seen by taking V0ℓ(Φℓ)=mℓMℓ and Vtℓ(Φℓ)=mℓMℓ+1Mℓ(Φℓ·SH,αℓ)t that the SAA1 of [9] can be obtained from our SAA. It is equally trivial to get a SAA from the SAA1. The notion is further equivalent to the strong asymptotic arbitrage of the second kind from [9] as it is shown there that SAA1 and SAA2 are equivalent and hence can be subsumed under the name SAA.
Our approach to show the existence of arbitrage of this kind will be not constructive. Instead, we use an equivalent characterization of strong asymptotic arbitrage based on the notion of entire asymptotic separability of sequences of measures, which is defined as follows.
The sequences of probability measures (Pℓ)ℓ≥1 and (Qℓ)ℓ≥1 are said to be entirely asymptotically separable if there exists a subsequence ℓk and a sequence of sets Ak∈Fℓk such that limk→∞Pℓk(Ak)=1 and limk→∞Qℓk(Ak)=0. In this case we write (Pℓ)ℓ≥1△(Qℓ)ℓ≥1. In addition, two families of probability measures (Pα)α>00}}$]]> and (Qα)α>00}}$]]> are said to be entirely asymptotically separable, and we write (Pα)α>0△(Qα)α>00}}\bigtriangleup {({Q^{\alpha }})_{\alpha >0}}$]]>, if there is a sequence (αℓ)ℓ≥1 converging to infinity such that (Pαℓ)ℓ≥1△(Qαℓ)ℓ≥1.
The precise relation between this notion and the existence of SAA is given in [9, Proposition 4]. In the case of complete markets, this result takes the following simple form.
Consider a large financial market(Ωα,Fα,(Ftα)t∈[0,T],Pα)α>00}$]]>and assume that each small market is complete. For eachα>00$]]>, letQα∼Pαbe the unique equivalent martingale measure. Then the following conditions are equivalent
Therefore the study of SAA in LH reduces to determining whether (QH,α)α>00}}$]]> is entirely asymptotically separable from (Qμασ)α>00}$]]> or not.2
In the case when μ=0, the study of SAA reduces to showing that (QH,α)α>0△QW0}}\bigtriangleup {Q_{W}}$]]>.
Main result
We state now our main result.
There exists a strong asymptotic arbitrage in the large mixed fractional marketLHforα→∞.
As mentioned we will show that (QH,α)α>0△(Qμασ)α>00}}\bigtriangleup {({Q_{\frac{\mu \alpha }{\sigma }}})}_{\alpha >0}$]]>.
Proof of Theorem 1
In order to prove Theorem 1, we provide a series of lemmas from which the desired result is obtained as a direct consequence. Before proceeding, we introduce some notations.
Following the lines of [2, 1], we define, for all n∈N, Yn:C[0,1]→Rn by:
Yn(ω)=(ω(1n)−ω(0),ω(2n)−ω(1n),…,ω(1)−ω(n−1n))T
and denote by QH,α,n and Qμασn the restrictions of QH,α and Qμασ to the σ-algebra Fn:=σ(Yn). We fix the Hurst parameter H∈(3/4,1) and we avoid to mention the dependence on it by setting QH,α≡Qα and QH,α,n≡Qα,n.
We denote by Cn the covariance matrix of the increments of the fractional Brownian motion ZH:
Cn(i,j):=Cov(ZinH−Zi−1nH,ZjnH−Zj−1nH),1≤i,j≤n,
and by λ1n,…,λnn its eigenvalues. Since the matrix Cn is symmetric and positive semi-definite, all the λin, 1≤i≤n, are real and nonnegative.
We moreover set
Σ0:=1nIn+α2CnandΣ1:=1nIn+1n2μ2α2σ21n×n,
where In is the identity matrix and 1n×n is the n×n matrix with all coefficients equal to 1. Clearly, the matrices Σ0 and Σ1 are positive definite and therefore invertible.
The proof of Theorem 1 strongly relies on the concept of relative entropy (also called sometimes Kullback–Leibler divergence) of the probability measure Qα,n (respectively, Qα) relative to Qμασn (respectively, Qμασ), denoted by H(Qα,n|Qμασn) (respectively, H(Qα|Qμασ)), see [7, Section 6]. We recall the definition of relative entropy and some relevant results in the Appendix A.
For eachn≥1, we haveH(Qα,n|Qμασn)=12[tr(Σ1−1Σ0)−n+μ2α2σ2n21nTΣ1−11n+ln(det(Σ1)det(Σ0))],where1n∈Rnis the vector with all coordinates equal to 1, and, for each square matrix A,tr(A)anddet(A)denote the trace and the determinant of A, respectively.
Note first that
EQα,n[YnYnT]=Σ0andEQμασn[YnYnT]=Σ1.
Note also that
EQα,n[Yn]=0nandEQμασn[Yn]=−μασn1n,
where 0n∈Rn is the vector with all coordinates equal to 0. Since Yn is a Gaussian vector under the two measures, the result follows using Lemma 7 and performing a straightforward calculation. □
Using standard properties of the trace and the determinant, it is not difficult to see that
tr(Σ0)=∑i=1n(1n+α2λin)andln(det(Σ0))=∑i=1nln(1n+α2λin).
We set an:=1nμ2α2σ2 and note that Σ1=1n(In+an1n×n). The next lemma summarizes the properties of the matrix Σ1.
For eachn>11$]]>, the eigenvalues ofΣ1are1/nwith multiplicityn−1and1n+anwith multiplicity 1. In particular, we havedet(Σ1)=nan+1nn.The inverse ofΣ1is given byΣ1−1=n(In−annan+11n×n).
Denote dnλ:=det(Σ1−λIn)=det((1n−λ)In+ann1n×n). Subtracting the row i from the row i+1, for each 1≤i<n, in the matrix Σ1−λIn, we see that dnλ is equal to the determinant of the matrix
1n−λ+annannann⋯annλ−1n1n−λ0…00⋱⋱⋱0⋮⋱λ−1n1n−λ00⋯0λ−1n1n−λ.
Expanding the determinant by minors with respect to the last column we get
dnλ=(1n−λ)dn−1λ+ann(1n−λ)n−1,n>2.2.\]]]>
Iterating this identity, we obtain
dnλ=(1n−λ)n−2d2λ+(n−2)ann(1n−λ)n−1=(1n−λ)n−1(1n−λ+an).
The first two statements follow. For the last statement, one can easily check that
Σ1×n(In−annan+11n×n)=In.
This shows the desired result. □
For alln>11$]]>, we havelimα→∞H(Qα,n|Qμασn)=∞.
Our starting point is Lemma 1. Evaluating each term entering (3), we first obtain
tr(Σ1−1Σ0)=ntr(Σ0)−nannan+1tr(1n×nΣ0)=∑i=1n(1+nα2λin)−nannan+1(1+α2tr(1n×nCn)).
Note that
tr(1n×nCn)=∑i,j=1nCn(i,j)=E[(Z1H)2]=1.
Thus, taking an=1nμ2α2σ2, equation (6) becomes
tr(Σ1−1Σ0)=n+∑i=1nnα2λin−μ2α2μ2α2+σ2(1+α2).
For the third term in (3), using that 1nT1n×n1n=n2, one can easily derive that
1nTΣ1−11n=n2nan+1=n2σ2μ2α2+σ2.
For the last term in (3), we use (4) and Lemma 2 to obtain
ln(det(Σ1)det(Σ0))=ln(nan+1)−∑i=1nln(1+nα2λin)=ln(μ2α2+σ2σ2)−∑i=1nln(1+nα2λin).
Inserting (7), (8) and (9) in (3) yields
H(Qα,n|Qμασn)=12[∑i=1n(nα2λin−ln(1+nα2λin))−μ2α4μ2α2+σ2+ln(μ2α2+σ2σ2)].
Since the trace is similarity-invariant, we deduce that
∑i=1nλin=tr(Cn)=∑i=1nCn(i,i)=1n2H−1.
In addition, we have ln(μ2α2+σ2σ2)≥0. Therefore, (10) leads to
H(Qα,n|Qμασn)≥α22(n2−2H−μ2α2μ2α2+σ2)−n2ln(1+nα2λmaxn)≥α22(n2−2H−1)−n2ln(1+nα2λmaxn)=12ln(eθnα2(1+nα2λmaxn)n),
where θn:=n2−2H−1>00$]]> and λmaxn=maxi=1…nλin. The result follows taking the limit when α tends to infinity in the previous expression. □
If μ=0, using Lemma 1, the previous result extends directly to the case n=1.
The above proof also gives us the relation between the relative entropy of Qα,n relative to Qμασn, i.e. H(Qα,n|Qμασn), and the relative entropy of Qα,n relative to QWn, i.e. H(Qα,n|QWn). Indeed, using [1, Lemma 5.3] one can deduce from (10) that
H(Qα,n|Qμασn)=H(Qα,n|QWn)−12μ2α4μ2α2+σ2+12ln(μ2α2+σ2σ2).
We point out that we also have
limα→∞H(Qα|Qμασ)=∞.
Indeed, we know from [1, Lemma 5.3] that supnH(Qα,n|QWn)<∞, which directly implies that also supnH(Qα,n|Qμασn)<∞. Therefore, applying [7, Lemma 6.3] we obtain
H(Qα|QW)=supnH(Qα,n|QWn)andH(Qα|Qμασ)=supnH(Qα,n|Qμασn).
The statement then follows from the result for the restrictions.
For each n≥1, we denote the Radon–Nikodym derivative of Qα,n relative to Qμασn by
Lαn:=dQα,ndQμασn.
Using [7, Lemma 6.1] (see Lemma 7 in Appendix A), we see that
H(Qα,n|Qμασn)=EQα,n[ln(Lαn)]=EQμασn[Lαnln(Lαn)].
Moreover, let us recall the notion of (Qα,n)α>00}}$]]>-tightness: (Lαn)α>00}}$]]> is (Qα,n)α>00}}$]]>-tight if the following holds:
limN→∞lim supα→∞Qα,n(Lαn>N)=0.N\big)=0.\]]]>
For eachn>11$]]>, the family(Lαn)α>00}}$]]>is not(Qα,n)α>00}}$]]>-tight.
We know, by Lemma 3, that EQα,n[ln(Lαn)]=H(Qα,n|Qμασn) tends to infinity when α tends to ∞. Since the measures Qα,n and Qμασn are Gaussian, the result follows as a direct application of the remark on [5, p. 457] which says that tightness is equivalent to the boundedness of the following two families: EQα,n[ln(Lαn)], α>00$]]>, and VarQα,n[ln(Lαn)], α>00$]]>. □
Before we can state and prove the last lemma of this section, we recall now the definition of contiguity of sequences/families of probability measures.
A sequence of probability measures (Pℓ)ℓ≥1 is contiguous with respect to the sequence of probability measures (Qℓ)ℓ≥1, (Pℓ)ℓ≥1⊲(Qℓ)ℓ≥1, if for any sequence Aℓ∈Fℓ: limℓ→∞Qℓ(Aℓ)=0⇒limℓ→∞Pℓ(Aℓ)=0. We say that (Pℓ)ℓ≥1 and (Qℓ)ℓ≥1 are mutually contiguous if (Pℓ)ℓ≥1⊲(Qℓ)ℓ≥1 and (Qℓ)ℓ≥1⊲(Pℓ)ℓ≥1, in which case we write (Pℓ)ℓ≥1⊲⊳(Qℓ)ℓ≥1.
These notions extend to families of probability measures (Pα)α>00}}$]]> and (Qα)α>00}}$]]> as follows. We say that (Pα)α>00}}$]]> is contiguous (resp. mutually contiguous) to (Qα)α>00}}$]]> if for every sequence (αℓ)ℓ≥1 converging to infinity we have (Pαℓ)ℓ≥1⊲(Qαℓ)ℓ≥1 (resp. (Pαℓ)ℓ≥1⊲⊳(Qαℓ)ℓ≥1), in which case we write (Pα)α>0⊲(Qα)α>00}}\lhd {({Q^{\alpha }})_{\alpha >0}}$]]> (resp. (Pα)α>0⊲⊳(Qα)α>00}}\lhd \rhd {({Q^{\alpha }})_{\alpha >0}}$]]>).
For eachn>11$]]>, we have(Qα,n)α>0△(Qμασn)α>0.0}}\bigtriangleup {\big({Q_{\frac{\mu \alpha }{\sigma }}^{n}}\big)_{\alpha >0}}.\]]]>
Since, by Lemma 4, (Lαn)α>00}}$]]> is not tight with respect to (Qα,n)α>00}}$]]> we apply [8, Lemma V.1.6] and deduce that, for each n>11$]]>, (Qα,n)α⋪Qμασn. The dichotomy for sequences of Gaussian measures of [5, Corollary 4] says that two sequences of Gaussian measures on Rn are either mutually contiguous or entirely separable. So we conclude that, for each n>11$]]>, (Qα,n)α>0△(Qμασn)α>00}}\bigtriangleup {({Q_{\frac{\mu \alpha }{\sigma }}^{n}})_{\alpha >0}}$]]>. □
From Remark 2, when μ=0, the same arguments lead to the conclusion that Lemma (5) holds true for n=1.
From Proposition 1 (see also [9, Proposition 4]), we know that there is a SAA if and only if (Qα)α>0△(Qμασ)α>00}}\bigtriangleup {({Q_{\frac{\mu \alpha }{\sigma }}})}_{\alpha >0}$]]>.
Fix n>11$]]>. By Lemma 5, there exist a sequence (αℓ)ℓ≥1 converging to infinity and sets Aℓ∈Fn such that
limℓ→∞Qαℓ(Aℓ)=limℓ→∞Qαℓ,n(Aℓ)=0
and
limℓ→∞Qμαℓσ(Aℓ)=limℓ→∞Qμαℓσn(Aℓ)=1.
The result follows. □
Interpretation of the results in the restricted markets
Lemma 5 might suggest that, for each n>11$]]> (or following Remark 5, for each n≥1 if μ=0), there exists also some kind of asymptotic arbitrage in the large financial market consisting of the restrictions of the α-mixed fractional Black–Scholes models, α>00$]]>, to the grid En:={0,1n,…,n−1n,1}. However, we will show that this is impossible.
For simplicity, we only consider the case n=1 and μ=0. We also assume that S0H,α=1. Thus, for each α>00$]]>, the corresponding market is
StH,α=exp(σ(ZtH+1αBt)−σ22α2t),t=0,1.
In this case all possible strategies are constants and hence the value process V1α takes the following form
V1α=cα(S1H,α−1),
where cα∈R. Obviously, we cannot hope for admissibility (boundedness from below), see the discussion in the introduction of [10]. But even if we do not require any admissibility here, there is no way to choose a sequence of αℓ→∞ and corresponding value processes Vαℓ such that the following hold: there exists β>00$]]> and εℓ→0 with
(i)P(V1αℓ>β)>β,for allℓ,(ii)limℓ→∞P(V1αℓ≥−εℓ)=1.\beta \big)>\beta ,\hspace{2.5pt}\text{for all}\hspace{2.5pt}\ell ,\\ {} (ii)\hspace{1em}& \underset{\ell \to \infty }{\lim }P\big({V_{1}^{{\alpha _{\ell }}}}\ge -{\varepsilon _{\ell }}\big)=1.\end{aligned}\]]]>
This is not possible since Z1H as well as B1 are independent N(0,1) and hence are strictly positive as well as strictly negative, with positive P-probability (and here neither letting α→∞ nor multiplying S1H,α−1 by some constants, either positive or negative, will be of any help: whenever there will be a strictly positive part in the limit there will also be a strictly negative part in the limit with a non-disappearing probability). Hence there is no such thing as (14) which, in our discrete time t=0,1 situation, is the appropriate version of an asymptotic arbitrage.
The reason behind this apparent contradiction is that in contrast to the continuous time large financial market its discrete counterpart is not complete. Under the original measure P (which induces Qα on C[0,1]) we have that Z1H∼N(0,1) and B1∼N(0,1) and the two random variables are independent. We know that the Wiener measure QW is a martingale measure for Sα (understood on C[0,1]) for all α, hence QW|F1 is an equivalent martingale measure for (13). We will now construct a different martingale measure for the process (13) which is equivalent to P on (Ω,F).
Indeed, define a measure P˜ on (Ω,F) as follows: dP˜dP=g(X) where we have X:=exp(σZ1H) and g(x)=e−x1h(x) where h(x)=12πσxexp(−12(ln(x)σ)2) is the density of a lognormal distribution, i.e., the density of the law of X under P. Obviously this measure change has the purpose to make the distribution of X exponential with parameter 1. Recall that the measure Qα,1 is considered as a measure on R (the measure induced by M1H,α:=αZ1H+B1).
The measureP˜satisfies:
P˜∼P.
EP˜[S1H,α]=1=S0H,α, which means thatP˜is a martingale measure for (13), for each α.
LetQ˜α,1be the measure that is induced by(M1H,α,P˜)onR, for eachα>00$]]>. Then(Q˜α,1)α>0⊲⊳(Qα,1)α>00}}\lhd \rhd {({Q^{\alpha ,1}})_{\alpha >0}}$]]>.
To prove (1) observe that g(x)>00$]]> for all 0<x<∞ and 0<X<∞P-a.s. and hence dP˜αdP=g(X)>00$]]>P-a.s. Moreover
EP[g(X)]=∫0∞g(x)h(x)dx=∫0∞e−xdx=1,
and so P˜ is a probability equivalent to P.
For (2) we see that
EP˜[S1H,α]=EP[g(eσZ1H)exp(σ(Z1H+1αB1)−σ22α2)]=EP[g(eσZ1H)exp(σZ1H)]EP[exp(σαB1−σ22α2)]=EP[g(X)X],
where we used the independence of Z1H and B1 under P. Finally we have that
EP[g(X)X]=∫0∞g(x)xh(x)dx=∫0∞xe−xdx=1,
proving (2).
For (3) note that, for each A∈B(R), we have Qα,1(A)=P(M1H,α∈A) and Q˜α,1(A)=P˜(M1H,α∈A). Thus, using that P˜∼P, we infer, for a family of sets Aα, that Qα,1(Aα)=P(Dα)→0, for α→∞, if and only if Q˜α,1(Aα)=P˜(Dα)→0, where Dα={M1H,α∈Aα}∈F1. The result follows. □
In conclusion, Lemma 6 shows that there exists a family of equivalent martingale measures for the model (13) with good properties, in this case with the property of mutual contiguity. And this fact is reflected by the impossibility to find asymptotic arbitrage opportunities for the family of models (13), α>00$]]>.
Relative entropy
In this section, we recall the concept of relative entropy and some equivalent characterization. A more detailed presentation of the topic can be found in [7].
Let Q1 and Q2 be probability measures on a measurable space (Ω,F) and let P={Fi:i=1,…,n} be a finite partition of Ω, i.e. Ω=∪i=1nFi and Fi are pairwise disjoint. The entropy of the measure Q1 relative to Q2 is the quantity
H(Q1|Q2)=supP∑j=1nQ1(Fj)ln(Q1(Fj)Q2(Fj)),
where P is the class of all possible finite partitions P of Ω. In the above formula, we assume that 0ln0=0 and ln0=−∞.
([7, Lemma 6.1]).
If a probability measureQ1is absolutely continuous w.r.t. another probability measureQ2, then the relative entropyH(Q1|Q2)is related to the Radon–Nikodym derivativeφ=dQ1dQ2as follows:H(Q1|Q2)=EQ1[ln(φ)]=EQ2[φln(φ)].
Acknowledgement
We thank an anonymous referee whose comments and suggestions contributed to the quality of this version of the paper.
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