Generalized statistical arbitrage concepts and related gain strategies
GGENERALIZED STATISTICAL ARBITRAGE CONCEPTS ANDRELATED GAIN STRATEGIES
CHRISTIAN REIN, LUDGER R ¨USCHENDORF, AND THORSTEN SCHMIDT
Abstract.
Generalized statistical arbitrage concepts are introduced corre-sponding to trading strategies which yield positive gains on average in a classof scenarios rather than almost surely. The relevant scenarios or market statesare specified via an information system given by a σ -algebra and so this no-tion contains classical arbitrage as a special case. It also covers the notion of statistical arbitrage introduced in Bondarenko (2003).Relaxing these notions further we introduce generalized profitable strategieswhich include also static or semi-static strategies. Under standard no-arbitragethere may exist generalized gain strategies yielding positive gains on averageunder the specified scenarios.In the first part of the paper we characterize these generalized statisticalno-arbitrage notions. In the second part of the paper we construct severalprofitable generalized strategies with respect to various choices of the informationsystem. In particular, we consider several forms of embedded binomial strategiesand follow-the-trend strategies as well as partition-type strategies. We studyand compare their behaviour on simulated data. Additionally, we find goodperformance on market data of these simple strategies which makes themprofitable candidates for real applications. Introduction
Since the mid-1980s trading strategies which offer profits on average in comparisonto little remaining risk have been implemented and analyzed. The starting pointwere pairs trading strategies, see Gatev et al. (2006) for an historic account andfurther details. In this strategy one trades two stocks whose prices have a highhistoric correlation and whose spread widened recently by buying the looser andshorting the winner. Many variants of this simple strategy followed, see Krauss(2017) for a survey and a guide to the literature. This raised interest in a deepertheoretical understanding of these approaches.In this paper, we elaborate on the notion of statistical arbitrage (SA) introducedin Bondarenko (2003). The author considers a finite horizon market in order torestrict the class of admissible pricing rules. A trading strategy with zero initial costis called statistical arbitrage if(i) the expected payoff is positive and,(ii) the conditional expected payoff is non-negative in each final state of theeconomy.Unlike pure arbitrage strategies a statistical arbitrage can have negative payoffsprovided the average payoff in each final state is non-negative. This concept sup-plements previous forms of restrictions like ‘good deals’ or opportunities with highSharpe ratios or with high utility (see Hansen and Jagannathan (1991), Cochraneand Saa-Requejo (2000) and ˇCern`y and Hodges (2002)) or ‘approximate arbitrageopportunities’ and investment opportunities with a high gain-loss ratio (see Bernardoand Ledoit (2000)). All these restrictions lead to essential reductions of the pricingintervals.
Date : July 26, 2019. a r X i v : . [ q -f i n . M F ] J u l CHRISTIAN REIN, LUDGER R¨USCHENDORF, AND THORSTEN SCHMIDT
Bondarenko (2003) discusses the concept of statistical arbitrage in connectionwith various forms of risk preferences, w.r.t. the solution of the joint hypothesisproblem, for tests of the efficient market hypothesis (EMH) and the efficient learningmarket (ELM). The main economic assumption introduced by Bondarenko is theassumption that the pricing kernel is path independent, i.e. it is a function dependingonly on the final state of the underlying price model but not depending on the wholehistory. This assumption implies that the payoff process deflated by the conditionalrisk neutral density of the final state is a martingale, i.e. has no systematic trend.The main result in (Bondarenko, 2003, Proposition 1) states that the existence of apath-independent pricing kernel is equivalent to the absence of SA strategies.Following Hogan et al. (2004), another strand of literature considers tradingstrategies which achieve positive gains on average together with vanishing risk in anasymptotic sense, see for example Elliott et al. (2005); Avellaneda and Lee (2010).In Section 2 we generalize the concept of statistical arbitrage: starting froma general information system given by a σ -field G , a statistical G -arbitrage is atrading strategy with positive expected gain conditionally on G . The existenceof a pricing measure with G -measurable density implies absence of G -arbitrage.Investigating in Section 3 in detail a class of trinomial models we find that theconverse direction in Bondarenko’s equivalence theorem is not valid in general.For two-period binomial models we fully characterize SA and construct statisticalarbitrage strategies. In Section 4 we introduce generalized trading strategies includingalso static or semi-static strategies and derive various characterizations of thecorresponding SA concepts; in particular we give conditions which imply equivalenceresults with the existence of G -measurable pricing densities. In Section 5 we constructfor discrete and continuous time models various SA-strategies, test them in severalexamples and give an application to market data. A basic class of strategies isobtained by embedding binomial trading strategies into the continuous time modelsusing first-hitting times. Further classes are strategies induced by partitioning thepath space and strategies which follow some trend in the data.Several of theses strategies are examined and compared. As a result we obtainsome useful gain strategies and suggestions relevant for practical applications.2. Generalized gain strategies
Consider a filtered probability space (Ω , F , P ) with a filtration F = ( F t ) ≤ t ≤ T .The filtration is assumed to satisfy the usual conditions, i. e. it is right continuousand F contains all null sets of F : if B ⊂ A ∈ F and P ( A ) = 0 then B ∈ F . Wealso suppose that F = F T .Following the classical approach to financial markets as for example in Delbaenand Schachermayer (2006), we consider a finite time horizon T ∈ N . The marketitself is given by a R d +1 -valued locally bounded non-negative semi-martingale S =( S , . . . , S d ). The num´eraire S is set equal to one, such that the prices are consideredas already discounted.A dynamic trading strategy φ is an S -integrable and predictable process such thatthe associated value process V = V ( φ ) is given by V t ( φ ) = (cid:90) t φ s dS s , ≤ t ≤ T. (1)The trading strategy φ is called a -admissible if φ = 0 and V t ( φ ) ≥ − a for all t ≥ φ is called admissible if it is admissible for some a >
0. We further assume that themarket is free of arbitrage in the sense of no free lunch with vanishing risk (NFLVR),which is equivalent to the existence of an equivalent local martingale measure Q , seeDelbaen and Schachermayer (2006). Here, a measure Q which is equivalent to P , Q ∼ P , such that S is a F -(local) martingale with respect to Q is called equivalent ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 3 (local) martingale measure, EMM (ELMM). Let M e denote the set of all equivalentlocal martingale measures.A statistical arbitrage is a dynamic trading strategy which is on average profitable,conditional on the final state of the economy S T . More generally, we consider ageneral information system represented by a σ -field G ⊂ F T and consider strategieswhich are on average profitable conditional on G . For example, G could be generatedby the event { S T > K } , or the events S T ∈ K i , where ( K i ) i ∈I is a partition of R d , orby { max ≤ t ≤ T S t > K } . We call such strategies G -arbitrage strategies. Sometimeswe call a statistical G -arbitrage strategy also a G -profitable strategy or G -arbitrage,for short. By E we denote expectation with respect to the reference measure P . Definition 2.1.
Let G ⊆ F T be a σ -algebra. An admissible dynamic tradingstrategy φ is called a statistical G -arbitrage strategy , if V T ( φ ) ∈ L ( P ) andi) E [ V T ( φ ) | G ] ≥ , P -a.s. , ii) E [ V T ( φ )] > G ) := { φ : φ is a G -arbitrage } denote the set of all statistical G -arbitrage strategies. The market model satisfiesthe condition of no statistical G -arbitrage NSA ( G ) ifSA( G ) = ∅ . For G = F T , NSA( G ) is equivalent to the classical no-arbitrage condition (NA)since then E [ V T ( φ ) | G ] = V T ( φ ). Recall that NA is implied by NFLVR. If G = σ ( S T ),one recovers the notion of statistical arbitrage introduced in Bondarenko (2003).A further interesting type of examples is the case where G = σ ( { S T ∈ K i , i ∈ I} ), { K i } i ∈I being a partition of the state space, such that a statistical arbitrageoffers a gain in any { S T ∈ K i } on average , i.e. E [ V T ( φ ) | S T ∈ K i ] ≥ i ∈ I . Similarly one can also consider path-dependent strategies, like for example G = σ ( { max ≤ t ≤ T S t ∈ K i , i ∈ I} ). Remark 2.2 (Relation to good-deal bounds) . The general approach to good-dealbounds in ˇCern`y and Hodges (2002) allows to consider statistical arbitrages as aspecial case: indeed, if we define A = { Z : E [ Z | G ] ≥ E [ Z ] > } as set of good deals then a statistical G -arbitrage φ is a good deal strategy if V T ( φ ) ∈ A . The corresponding good-deal pricing bound is given by π ( X ) = inf { x : ∃ φ admissible s.t. X + x + V T ( φ ) ∈ A } . Remark 2.3.
We note some easy consequences of Definition 2.1.(i) The tower property of conditional expectations immediately yields that largerinformation systems G allow for less profitable G -arbitrage strategies i. e. G ⊂ G implies that SA( G ) ⊂ SA( G ). As a consequence we get that in thiscase NSA( G ) ⇒ NSA( G ) . (2)(ii) If G = {∅ , Ω } , then φ ∈ SA( G ) iff E P [ V T ( φ )] > On the statistical no-arbitrage notion
The notion of no statistical arbitrage is motivated by the question whether it ispossible to construct a trading strategy φ such that in any final state of the priceprocess S T the trader gets a gain on average (i. e. conditional on σ ( S T )). CHRISTIAN REIN, LUDGER R¨USCHENDORF, AND THORSTEN SCHMIDT
Proposition 1 in Bondarenko (2003) states that (in discrete time), NSA is equiva-lent to the existence of an equivalent martingale measure Q with path independentdensity Z , i. e. dQdP = Z ∈ σ ( S T ) , (3)where we use the notation Z ∈ σ ( S T ) for Z being σ ( S T )-measurable. We showin Section 3.2, that this equivalence needs additional assumptions which is onemotivation of our work. In Section 3.3 we explicitly construct statistical arbitrageswhose study is the second motivation of our work.On the other side, existence of an equivalent martingale measure with pathindependent density Z implies that NSA holds without further assumptions. Thisalso holds true for the generalized notion NSA( G ), as we now show. Proposition 3.1.
If there exists Q ∈ M e such that dQdP is G -measurable, thenNSA ( G ) holds.Proof. The proof follows from the Bayes-formula for conditional expectations. If Z = dQdP ∈ G , then for any X ∈ L ( P ) it holds that E P [ X | G ] = E Q [ XZ | G ] E Q [ Z | G ] = E Q [ X | G ] . (4)If there would be a statistical arbitrage strategy φ with E P [ X | G ] ≥ E P [ X ] > X = V T ( φ ) ∈ L ( P ), then, by (4), E Q [ X | G ] ≥ , Q -a.s.Moreover, since φ is admissible, V ( φ ) is a Q -supermartingale by Fatou’s lemma,and we obtain that E Q [ X ] = E Q [ V T ( φ )] ≤ V ( φ ) = 0 . (5)Hence, 0 = E Q [ X | G ] = E P [ X | G ]in contradiction to E P [ X ] > (cid:3) Remark 3.2 (Alternative admissible strategies) . An inspection of the proof, inparticular Equation (5), shows that the claim also holds when we consider asadmissible such strategies φ for which V ( φ ) is a Q -martingale.In the following we discuss whether also the converse direction in the Bondarenkoresult is true, i. e. the question if no statistical arbitrage implies the existence of anequivalent martingale measure with path-independent density. Moreover we studythe question how statistical G -arbitrage strategies can be constructed.3.1. Statistical arbitrage in trinomial models.
In this section we consider aspecial one-dimensional trinomial model of the following type which we will call thetrinomial model . While the first step is binomial, the second time-step is trinomial.In this regard, assume that d = 1, Ω = { ω , . . . , ω } and T = 2. Let S = s ∈ R ≥ and S take the two values s +1 and s − such that S ( ω ) = S ( ω ) = S ( ω ) = s +1 , S ( ω ) = S ( ω ) = S ( ω ) = s − . The existence of an equivalent martingale measure Q ∼ P is equivalent to ∆ S i = S i − S i − taking positive as well as negative values in each sub-tree. For the firsttime step we assume without loss of generality that s +1 − s > s − − s < s ++2 , s + − , s −− and the top state s ◦ with s ◦ > s ++2 > s + − > s −− >
0. While the + / − statesare reached by following a standard binomial, recombining two-period model, i.e. S ( ω ) = s ++2 , S ( ω ) = S ( ω ) = s + − , S ( ω ) = s −− , ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 5 S ( ω ) = S ( ω ) S ( ω ) S ( ω ) S ( ω ) S ( ω ) = S ( ω ) S ( ω ) S ( ω ) Figure 1.
The considered trinomial model with T = 2 time steps.The first step is binomial, the second step is also (recombining)binomial with an additional top state { ω , ω } .the top state is reached by S ( ω ) = S ( ω ) = s ◦ . We illustrate the scheme in Figure 1.To ensure absence of arbitrage we assume that s ++2 − s +1 > s − < s + − < s +1 , s −− − s − <
0. The gains from trading at time 2 with a self-financing strategy φ aregiven by V ( φ ) = φ ∆ S + φ ∆ S . (6)While φ is constant since F = {∅ , Ω } , φ can take two different values which wedenote by φ +2 and φ − (taken in the states { ω , ω , ω } and { ω , ω , ω } , respectively).Since G = F = σ ( { ω , ω } , { ω , ω } , { ω } , { ω } ) the strategy φ is a statisticalarbitrage if and only if φ ∆ S ( ω ) + φ +2 ∆ S ( ω ) ≥ , (7) φ ∆ S ( ω ) + φ − ∆ S ( ω ) ≥ , (8) φ ∆ S ( ω ) P ( ω ) + φ +2 ∆ S ( ω ) P ( ω )+ φ ∆ S ( ω ) P ( ω ) + φ − ∆ S ( ω ) P ( ω ) ≥ , (9) φ ∆ S ( ω ) P ( ω ) + φ +2 ∆ S ( ω ) P ( ω )+ φ ∆ S ( ω ) P ( ω ) + φ − ∆ S ( ω ) P ( ω ) ≥ , (10)and, in addition, at least one of the inequalities is strict.Moreover, if we consider an equivalent martingale measure Q then the density Z is path-independent if and only if Z ( ω ) = Z ( ω ) and Z ( ω ) = Z ( ω ). As a nextstep we establish a criterion for our model to be free of statistical arbitrage. DenoteΓ = − ∆ S ( ω ) + ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) , Γ = ∆ S ( ω )∆ S ( ω ) (∆ S ( ω ) + ∆ S ( ω )) − ∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) . CHRISTIAN REIN, LUDGER R¨USCHENDORF, AND THORSTEN SCHMIDT S ( ω ) = S ( ω ) = 14 S ( ω ) = 13 S ( ω ) = 12 S ( ω ) = 10 S ( ω ) = S ( ω ) = 10 S ( ω ) = 8 S ( ω ) = 6 Figure 2.
An explicit trinomial model with T = 2 time steps Lemma 3.3.
Let ν := P ( ω ) P ( ω ) and ν := P ( ω ) P ( ω ) . In the trinomial model there is nostatistical arbitrage if ν = − ∆ S ( ω )∆ S ( ω ) ν and if it holds that Γ < ν ≤ Γ . (11)The proof is relegated to the appendix.3.2. A counter example.
In the following we use Lemma 3.3 to show that Propo-sition 1 in Bondarenko (2003) is not valid without additional conditions. Considerthe (incomplete) trinomial model specified in Figure 2.It is easy to check that the equivalent martingale measures Q specified by q =( Q ( ω ) , . . . , Q ( ω )) are given by the set Q = (cid:110) q ∈ R (cid:12)(cid:12)(cid:12) q = − q + 14 , q = − q + 14 , q = q − , q = − q + 34 , where q ∈ (cid:16) , (cid:17) , q ∈ (cid:16) , (cid:17)(cid:111) . Furthermore consider the underlying measure P uniquely specified by the vector p = ( P ( ω ) , . . . , P ( ω )) given by p = (0 . , . , . , . , . , . . We compute ν = p p = 3 and ν = p p = 3. ThenΓ = ∆ S ( ω )∆ S ( ω ) (∆ S ( ω ) + ∆ S ( ω )) − ∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) = 3 = ν , Γ = − ∆ S ( ω ) + ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) = 23 < ν and ν = − ∆ S ( ω )∆ S ( ω ) ν = ν = 3 = p p . According to Lemma 3.3 there is no statistical arbitrage in the stated example. But,on the other hand, there is no path independent density in this case because if therewould be a path independent density, i. e. a density Z with Z ( ω ) = Z ( ω ) and ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 7 S ( ω ) S ( ω ) S ( ω ) S ( ω ) = S ( ω ) S ( ω ) S ( ω ) Figure 3.
The considered recombining binomial model with two periods. Z ( ω ) = Z ( ω ), there would exist an equivalent martingale measure Q fulfilling theconditions q q = p p = 3 and q q = p p = 3 . (12)But the only q ≥ q = ( , , , , , ) which is not an element of Q . This example shows that Proposition 1 in Bondarenko (2003) needs additionalassumptions: indeed, we have shown that there does not exist a statistical arbitrageand at the same time there is no path-independent density. In Section 4 we studythis topic in more detail.3.3. Statistical arbitrage strategies in binomial models.
In this section wepropose a method to construct statistical arbitrage strategies in binomial models.Consider the following recombining two-period binomial model: assume thatΩ = { ω , . . . , ω } and T = 2. Let S = s > S ( ω ) = S ( ω ) = s + ,and S ( ω ) = S ( ω ) = s − as well as s ++ = S ( ω ), s + − = S ( ω ) = S ( ω ), and s −− = S ( ω ). This model is illustrated in Figure 3.Absence of arbitrage is equivalent to ∆ S i , i = 1 , s + > s , s − < s , and s ++ > s + , s − < s + − < s + , and s −− < s − i. e. we consider binomial models aspresented in Figure 3. Gains from trading are again given by (6). Also φ is constantand φ can take the two values { φ +2 , φ − } . As in Equations (7) - (10), φ is a statisticalarbitrage, iff φ ∆ S ( ω ) + φ +2 ∆ S ( ω ) ≥ φ ∆ S ( ω ) + φ − ∆ S ( ω ) ≥ φ ∆ S ( ω ) P ( ω ) + φ +2 ∆ S ( ω ) P ( ω )+ φ ∆ S ( ω ) P ( ω ) + φ − ∆ S ( ω ) P ( ω ) ≥ Z is path-independent if and only if Z ( ω ) = Z ( ω ). Equations (13) - (15) are equivalent to A φ ≥ φ = ( φ , φ +2 , φ − ) (cid:62) with A = ∆ S ( ω ) ∆ S ( ω ) 0∆ S ( ω ) 0 ∆ S ( ω ) q ∆ S ( ω ) + ∆ S ( ω ) q ∆ S ( ω ) ∆ S ( ω ) , (16)where q = P ( ω ) P ( ω ) . CHRISTIAN REIN, LUDGER R¨USCHENDORF, AND THORSTEN SCHMIDT
Proposition 3.4.
In the recombining two-period binomial model NSA holds if andonly if det( A ) = 0 . The proof is relegated to the appendix.
Remark 3.5.
It turns out that in the binomial model above NSA is equivalent toexistence of a path-independent density: indeed, the unique equivalent martingalemeasure is given by the vector B − ( q , . . . , q ) with q = ∆ S ( ω ) (cid:0) ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) (cid:1) ,q = − ∆ S ( ω ) (cid:0) ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) (cid:1) , (17) q = − ∆ S ( ω ) (cid:0) ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) (cid:1) , (18) q = ∆ S ( ω ) (cid:0) ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) (cid:1) and B =∆ S ( ω ) (cid:16)(cid:0) ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) + (cid:0) ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:17) + ∆ S ( ω ) (cid:16)(cid:0) ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) + (cid:0) ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:17) . Proposition 3.4 yields that NSA holds iff det( A ) = 0, which is according to Equation(54) equivalent to P ( ω ) P ( ω ) = ∆ S ( ω )(∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ))∆ S ( ω )(∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω )) =: ˜ q. (19)Using (17) and (18) we obtain from det( A ) = 0 that dQ ( ω ) dQ ( ω ) = ˜ q = dP ( ω ) dP ( ω ) , which means that NSA is equivalent to the existence of a path-independent density.The question now is what path properties imply absence of statistical arbitrageopportunities. Lemma 3.6.
In the recombining two-period binomial model there exists a statisticalarbitrage if and only if P ( ω ) P ( ω ) (cid:54) = ˜ q. (20) Proof.
To have the possibility of statistical arbitrage we know from Proposition3.4 that we need det( A ) (cid:54) = 0 which is, according to Remark 3.5, equivalent to P ( ω ) P ( ω ) (cid:54) = ˜ q . (cid:3) The following Lemma explicitly describes the statistical arbitrages in terms ofthe vector φ = ( φ , φ +2 , φ − ) Lemma 3.7.
In the recombining two-period binomial model with statistical arbitrage, φ = D ( ξ , ξ , ξ ) with ξ = (cid:0) q ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) + ∆ S ( ω )∆ S ( ω ) ,ξ = − (cid:0) ∆ S ( ω ) + q ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) − (cid:0) ∆ S ( ω ) − ∆ S ( ω ) (cid:1) ∆ S ( ω ) ,ξ = − (cid:0) q ∆ S ( ω ) − q ∆ S ( ω ) (cid:1) ∆ S ( ω ) − (cid:0) − ∆ S ( ω ) + ∆ S ( ω ) + q ∆ S ( ω ) (cid:1) ∆ S ( ω ) ,q = P ( ω ) P ( ω ) , and D = (cid:16) q ∆ S ( ω )∆ S ( ω ) + (cid:0) − ∆ S ( ω ) − q ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:17) ∆ S ( ω )+ ∆ S ( ω )∆ S ( ω )∆ S ( ω ) is a statistical arbitrage. ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 9
Proof. If P ( ω ) P ( ω ) (cid:54) = ˜ q we have statistical arbitrage according to Lemma 3.6 and thedeterminant of the matrix A in (16) is not equal to zero according to Proposition 3.4.In this case the matrix A is invertible. Hence, φ = A − is a statistical arbitrageand it is easily verified that φ = D ( ξ , ξ , ξ ). (cid:3) In Section 5 we will use this information and propose a dynamic trading strategyexploiting statistical arbitrages with the results of this section.3.4.
Risk of statistical arbitrages.
The word arbitrage might be misleading onthe riskiness of statistical arbitrages, because in the classical sense, an arbitrage is astrategy without risk. This is of course not the case for statistical arbitrages (or thefollowing generalizations of this concept). Since we consider arbitrage-free markets,all gains come with a certain risk and, higher profits are associated with higher risk.This is confirmed by our simulation results in Section 5.As a simple example consider the case where ∆ i S ( ω j ) ∈ { , − } , i.e. the stockeither rises by 5 or falls by 5. In addition, assume that q = P ( ω ) /P ( ω ) = 1 .
2. Then,using Equation (16) it is not difficult to compute φ = A − = (1 . , − . , − . (cid:62) .From this strategy we obtain that the gains at time 2, given by G ( ω ) = φ ( ω )∆ S ( ω ) + φ ( ω )∆ S ( ω ) , yield G ( ω ) = G ( ω ) = 1, corresponding to (13) and (14). In addition, we obtainthat G ( ω ) = 15 and G ( ω ) = −
17. If we assume that P ( ω ) = 0 . { ω , ω } computes to P ( ω ) G ( ω ) + P ( ω ) G ( ω ) = 0 . ·
15 + 0 . · ( −
17) = 0 . ≥ , (21)such that the strategy is indeed a statistical arbitrage. While the (average) gains inthe three relevant scenarios are 1 , . ,
1, the possible loss in scenario ω is equal to −
17, which is attained with probability 0 .
25, clearly pointing out the riskiness ofthe strategy.To exploit the averaging property of statistical arbitrage, we repeat this strategyin the following until we first record a positive P&L. These considerations showclearly, that a risk analysis of the implemented strategy is very important.4.
Generalized G -arbitrage strategies In connection with improvement procedures for payoffs we consider any static orsemi-static payoff X ∈ L ( P ) as a generalized strategy. This leads to the followingnotion of generalized statistical G -arbitrage strategies and the corresponding notionof generalized statistical G -arbitrage. This concept was used in several papers dealingwith improvement procedures of financial contracts, see for example Kassbergerand Liebmann (2017). We denote by L ( P, Q ) := L ( P ) ∩ L ( Q ) the set of randomvariables which are integrable with respect to P and Q . Definition 4.1.
Let G ⊆ F be a σ -algebra. The set of generalized statistical G -arbitrage-strategies with respect to Q ∈ M e is defined asSA( Q, G ) := { X ∈ L ( P, Q ) : E Q [ X ] = 0 , E P [ X | G ] ≥ P -a.s. and E P [ X ] > } The market satisfies NSA( Q, G ), the condition of no generalized statistical G -arbitrage with respect to Q , if SA( Q, G ) = ∅ . We aim at studying under which conditions there exist generalized statistical G -arbitrages and to describe connections between NSA( Q, G ) and NSA( G ). Thefollowing result in Kassberger and Liebmann (2017), Proposition 6, characterizes thegeneralized NSA( Q, G )-condition by showing that in fact this notion is equivalentto G -measurability of dZ = dQdP . Proposition 4.2.
Let Q ∈ M e . Then NSA ( Q, G ) is equivalent to the existence of a G -measurable version of the Radon-Nikodym derivative Z = dQdP . The proof of this result is achieved by Jensen’s inequality and using as candidateof a generalized G -arbitrage X = E [ Z | G ] Z − ≥ − . (22)Equation (22) also shows that the statistical arbitrage, if it exists, may be chosenbounded from below.One consequence of this characterization result is the characterization of NSA( G )for the case of complete market models. Recall that the Radon-Nikodym derivative Z = dQdP is path-independent, iff Z is σ ( S T )-measurable.A financial market is called complete , if every contingent claim is attainable, i.e. forevery F -measurable random variable X bounded from below, we find an admissibleself-financing trading strategy φ , such that x + V T ( φ ) = X . This is implied by theassumption that M e = { Q } : indeed, under this assumption, Theorem 16 in Delbaenand Schachermayer (1995 a ) yields that any X ∈ L ( Q ), bounded from below, ishedgeable and hence attainable. Proposition 4.3.
Assume that M e = { Q } . Then NSA ( G ) holds if and only if dQdP is G -measurable.Proof. We first show that existence of a G -measurable Q ∈ M e implies NSA( G ):choose Q ∈ M e , such that Z = dQdP is G -measurable. Then NSA( G ) follows as inthe proof of Proposition 3.1.For the converse direction assume that Z is not G -measurable. By Proposition4.2 it follows that there exists a generalized G -arbitrage, i.e. an X ∈ L ( P, Q )with E Q [ X ] = 0, E P [ X | G ] ≥ E P [ X ] >
0. As remarked above, X can bechosen bounded from below. Hence, Theorem 16 in Delbaen and Schachermayer(1995 a ) yields existence of an admissible self-financing trading strategy φ , such that x + V T ( φ ) = X . Moreover, the superhedging duality, i.e. Theorem 9 in Delbaen andSchachermayer (1995 a ) implies that x = E Q [ X ] = 0, and hence φ is a G -arbitrage.This is a contradiction and the claim follows. (cid:3) In particular this result implies that Proposition 1 in Bondarenko (2003) gives acorrect characterization of NSA for complete markets.
Example 4.4 (Statistical arbitrage for diffusions) . This example discusses theconsequences of Proposition 4.2 and Proposition 4.3 in the case of a diffusion model.Let S be a one-dimensional diffusion process satisfying dS t = a t dt + b t dB t , ≤ t ≤ T, (23)where B t is a P -Brownian motion, a and b are progressively measurable such that P ( (cid:82) T | a s | ds < ∞ ) = 1 and P ( (cid:82) T b s ds < ∞ ) = 1. Assume further that b > dt -almost surely that the Novikov-condition is satisfied, i. e. E (cid:34) exp (cid:16) (cid:90) T a s b s ds (cid:17)(cid:35) < ∞ . Then this model is complete and by Girsanov’s theorem has a unique equivalentlocal martingale measure Q with Radon-Nikodym derivative Z T = exp (cid:32) − (cid:90) T a t b t dB t − (cid:90) T a t b t dt (cid:33) . (24)If a t /b t = c dt -almost surely, then we obtain from Proposition 4.3 that there are nostatistical arbitrage opportunities. This holds in particular when a t = a and b t = b ,0 ≤ t ≤ T, i. e. in the case of constant drift and volatility (the Black-Scholes model). ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 11
On the other side, the diffusion model allows for statistical arbitrage except for thecase that ( a t /b t ) is constant dt -almost surely. A comparable result was obtainedin G¨onc¨u (2015) when studying the concept of statistical arbitrage introduced inHogan et al. (2004) in the Black-Scholes model.The following definition introduces the generalized G -no-arbitrage conditionwithout dependence on a specific pricing measure Q . Definition 4.5.
Let G ⊆ F be a σ -algebra. The set of generalized statistical G -arbitrage-strategies is defined asSA( G ) := { X ∈ L ( P ) : sup Q ∈M e E Q [ X ] ≤ , E P [ X | G ] ≥ P -a.s. and E P [ X ] > } . The market satisfies NSA( G ), i.e. no generalized statistical G -arbitrage , ifSA( G ) = ∅ . Note that the definition defines a generalized statistical G -arbitrage as a randomvariable X ∈ L ( P ), such that sup Q ∈M e E Q [ X ] ≤ E P [ X | G ] ≥ P -almost surely,and E P [ X ] >
0. In this sense, the strategies in SA( G ) are generalized statistical G -arbitrage-strategies under any choice of the pricing measure Q . Our next step isto establish a relation between G -arbitrages and generalized G -arbitrages. Note thatthe connection to trading strategies in a continuous-time setting requires, as usual,to allow that sup Q ∈M e E Q [ X ] may be negative, while for the definition of SA( Q, G )we were able to consider E Q [ X ] = 0. The precise reasoning for this is becomingclear in the proof of the next proposition.We use the concept of No Free Lunch with Vanishing Risk (NFLVR), whichis a mild strengthening of the no-arbitrage concept, and refer to Delbaen andSchachermayer (1994) for definition and further reading. According to the results inthis article we require in the following that S is locally bounded, i.e. there exists asequence of stopping times ( T n ) n ≥ tending to ∞ a.s. and a sequence ( K n ) n ≥ ofpositive constants, such that | S (cid:74) ,T n (cid:75) | < K n , n ≥ G -arbitrage strategies restricted to claims bounded frombelow is denoted bySA b ( G ) := SA( G ) ∩ { X ∈ L ( P ) : ∃ a ∈ R such that X ≥ − a } . Proposition 4.6.
Assume that S satisfies (NFLVR). ThenNSA b ( G ) ⇔ NSA ( G ) . Proof.
We first show that every G -arbitrage strategy is a generalized G -arbitragestrategy: consider φ ∈ SA( G ), i. e. E [ V T ( φ ) | G ] ≥ E [ V T ( φ )] >
0. . By thesuperreplication duality, Theorem 9 in Delbaen and Schachermayer (1995 b ), it holdsthat sup Q ∈M e E Q [ V T ( φ )] = inf { x | ∃ admissible ˜ φ, x + V T ( ˜ φ ) ≥ V T ( φ ) } . Choosing ˜ φ = φ it follows sup Q ∈M e E Q V T ( φ ) ≤
0. Note that in addition, admissi-bility of φ implies that V T ( φ ) is bounded from below and so V T ( φ ) ∈ A b ( G ).For the reverse implication we have, again by the superreplication duality, for X ∈ A b ( G ) that0 ≥ sup Q ∈M e E Q X = inf { x ∈ R | ∃ admissible φ, x + V T ( φ ) ≥ X } . Since the infimum is finite, Theorem 9 in Delbaen and Schachermayer (1995 b ) yieldsthat it is indeed a minimum. Without loss of generality, we may chose x = 0 andobtain the existence of an admissible dynamic trading strategy φ with X ≤ V T ( φ ).As X ∈ A b ( G ) it holds further that E P [ X | G ] ≥ , P -a.s., which leads us to E P [ V T ( φ ) | G ] ≥ E P [ X | G ] ≥ P -a.s. Then, E P [ V T ( φ )] ≥ E P [ X ] >
0, such that V T ( φ ) ∈ SA( G ). So the existenceof generalized G -arbitrage strategies is equivalent to the existence of G -arbitragestrategies V T ( φ ) in SA( G )and the claim follows. (cid:3) Some classes of profitable strategies
In Section 4 we saw conditions and examples of statistical arbitrages in a varietyof models. Here we are considering several classes of simple statistical arbitragestrategies for several classes of information systems G . While these strategies areeasy to apply for general stochastic models we investigate them on the Black-Scholesmodel which will allow for analytic properties of the trading strategies. We will seein the following section that similar results can be expected in more general marketmodels.The Black-Scholes model is, according to Example 4.4, free of statistical arbitrage,and we show in the following how to construct dynamic trading strategies allowingstatistical G -arbitrage for various choices of G . To this end, assume that S isa geometric Brownian motion, i.e. the unique strong solution of the stochasticdifferential equation dS t = µS t dt + σS t dB t , ≤ t ≤ T (25)where B is a P -Brownian motion and σ >
0. In the simulation we will first chose µ = 0 . σ = 0 . S = 2186 according to estimated drift and volatilityfrom the S&P 500 (September 2016 to August 2017), and later consider smallperturbations.Motivated by our findings in Section 3.1, we begin by embedding binomial tradingstrategies into the diffusion setting by considering two limits (up / down) and takingactions at the first times these limits are reached. In Section 5.2 we will introducesome related follow-the-trend strategies.5.1. Embedded binomial trading strategies.
We introduce a recombination ofseveral two-step binomial models embedded in the continuous-time model as longas the final time T is reached. As information system we consider the σ -field G generated by the stopping times when the final states of each of the binomial modelare reached (or the trivial σ -field otherwise).As we repeatedly consider embedded binomial models it makes much sense to talkon the outcome of the trading strategy on average conditional on the final states ofeach binomial model, i.e. by averaging the outcome over many repeated applicationsof the trading strategy and hence we may apply the concept of statistical arbitrage here.Let i denote the current step of our iteration and consider a multiplicative stepsize c >
0. We initialize at time t = 0. Otherwise consider the initial time of ournext iteration given by the time where finished the last repetition and denote thistime by t i and the according level by s i = S t i . Then we define the following twostopping times denoting the first and second period of our binomial model by t i = inf (cid:8) t ∈ [ t i , T ] | S t ∈ { s i (1 − c ) , s i (1 + c ) } (cid:9) (26)and t i = inf (cid:8) t ∈ ( t i , T ] | S t ∈ { s i (1 − c ) , s i , s i (1 + 2 c ) } (cid:9) , (27)with the convention that inf ∅ = T . This induces a sequence of σ -fields G i := σ ( S t i ) . Since S is continuous, this scheme allows to embed repeated binomial models S t i , S t i , S t i , i = 1 , , . . . into continuous time. The considered trading strategy isto execute the statistical arbitrage strategy for binomial models computed in Lemma3.7 at the stopping times t i , t i , t i . At t i the position will be cleared and we start the ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 13
Index D i ff u s i on t =
136 t = x - x - h x x + h x + Figure 4.
The embedding of a binomial model: at the hittingtimes t and t of the diffusion the steps of the embedded binomialmodel take place. The hitting levels are given by s (1 ± . t i +10 = t i . Generally, we assume that the time horizon T is sufficiently large such that the (typically small) levels s i (1 − c ) , . . . , s i (1 + 2 c )are reached at least once. Example 5.1.
Figure 4 illustrates the embedding of the binomial model: theboundary s (1 − c ) is hit at stopping time t = t and the boundary s (1 − c ) atstopping time t = t . The trading strategy φ from Lemma 3.7 then implies tradingbuying (selling) φ entities of the underlying at time t = 0 and φ − entities at t = t .At time t = t we will close the position and start this procedure again with t = t and with the new starting point s = S t . This leads to a recombination of several2-period binomial models, as illustrated in Figure 5.The constant c and with it the barriers for the hitting times will be chosen independence of µ and σ to ensure that we do not loose the statistical arbitrageopportunity. To be more precise we use c = 0 . · µσ which showed a good performance in our simulations. According to Lemma 3.6 thereis a statistical arbitrage opportunity if P ( ω ) P ( ω ) (cid:54) = ˜ q . It is easy to check from Equation(19) that ˜ q = 1 in the case considered here.To guarantee existence of a statistical arbitrage we calculate the path probabilities P ( ω ) , P ( ω ). The first exit time τ = inf { t ≥ | S t / ∈ ( a, b ) } from the interval ( a, b )satisfies P ( S τ = a ) = (cid:18) as (cid:19) ν (cid:0) bs (cid:1) | ν | − (cid:0) s b (cid:1) | ν | (cid:0) ba (cid:1) | ν | − (cid:0) ab (cid:1) | ν | , a < b, (28) Index D i ff u s i on x - x - h x x + h x + x + x + t =
203 t =
372 t =
476 t = Figure 5.
The embedded multi-period binomial trading modelwith trading points t , t , t and t . The statistical arbitrage in thiscase corresponds to repeated trading strategies from Lemma 3.7:we buy φ entities at t = 0, change the position to φ +2 at t andequalize the position at t . With the new starting time t = t thestrategy will be started again and adjusted at the stopping times t and t where ν = µσ − , see Borodin and Salminen (2012), formula 3.0.4 in Section 9 ofPart II. This in turn yields that q = P ( ω ) P ( ω ) = P (cid:0) S t = s (1 + c ) (cid:1) P (cid:0) S t = s (cid:1) P (cid:0) S t = s (1 − c ) (cid:1) P (cid:0) S t = s (cid:1) = (cid:18) − (1 − c ) ν (1+ c ) | ν | − (1+ c ) −| ν | (cid:0) c − c (cid:1) | ν | − (cid:0) − c c (cid:1) | ν | (cid:19) (1 + c ) − ν (cid:0) c c (cid:1) | ν | − (cid:0) c c (cid:1) | ν | (1+2 c ) | ν | − (1+2 c ) −| ν | (cid:18) (1 − c ) ν (1+ c ) | ν | − (1+ c ) −| ν | (cid:0) c − c (cid:1) | ν | − (cid:0) − c c (cid:1) | ν | (cid:19)(cid:18) − (cid:16) − c − c (cid:17) ν (1 − c ) −| ν | − (1 − c ) | ν | (1 − c ) −| ν | − (1 − c ) | ν | (cid:19) . (29)Clearly, in general q (cid:54) = 1, such that in these cases statistical arbitrage exists, whichwe exploit in the following.From Lemma 3.7 we obtain with D = 2( q − c s i ) that the trading strategy φ = ( φ , φ +2 , φ − ) is given by φ = (2 + q )( c s i ) D − , (30) φ +2 = ( q − c s i ) D − , (31) φ − = − q ( c s i ) D − . (32)We call the trading strategy which results by repeated application of φ at therespective hitting times the embedded binomial trading strategy . Simulation results.
As already mentioned, we simulate a geometric Brownianmotion according to Equation (25) with µ = 0 . σ = 0 . S = 2186, T = 1(year), discretize by 1000 steps and embed the according binomial models repeatedly ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 15 gain p.a. median VaR(0.95) gain/trade losses (mean) ∅ N max. N Table 1.
Simulation results for the embedded binomial tradingstrategy for 1 mio runs. This example serves as benchmark. Gainp.a. denotes the overall average gain in the time period of one year,[0 , Gain/trade denotes the average gain per trade, losses denotes the fraction of simulations where the outcome of the tradingstrategy was negative, and we also show the average of the lossestitled mean . Finally, we also state the average number and maximalnumber of embedded binomial models. −10000 −8000 −6000 −4000 −2000 0 2000 + + + + + + Figure 6.
Histogram of the profits and losses from the embeddedbinomial trading strategy used in Table 1.in this time interval. In this case we have q = 1 . q (cid:54) = ˜ q , i. e. the embedded binomial strategy in thiscase is a G -arbitrage strategy. We denote by N the (random) number of binomialmodels that are necessary for each simulated diffusion to gain either a profit fromtrading or to reach T and by G i the gain or loss of the i -th binomial model. Henceeither (cid:80) Ni =1 G i > N = T .For 1 million runs, we obtain the results presented in Table 1. For each run werecord either a gain or a loss from trading. The average gain per simulation run isshown in column one, its median in column two. The distribution of the P&L isskewed to the left with potential large losses with small probability which is reflectedby a median of 206 in comparison to an average gain of 33. In column 3 we depictthe 95% Value-at-Risk which is of size 5,320. Column 4 denotes the average gainper trade which is obtained by dividing the average gain by the average numberof trades (i.e. repeated binomial models). In column 5 we show the (fraction of) losses , i.e. the fraction of simulated processes exhibiting no gain from trading beforereaching the final time T , followed by their mean. The average number of trading c gain pa median VaR . gain pt losses (mean) ∅ N (max)0.0025 8,890 48,700 -373 743 0.045 -57,900 12 1500.005 465 3,810 58,400 66 0.077 -6,210 7 630.01 41 206 5,250 11 0.132 -621 4 240.02 9 10 371 5 0.185 -50 2 90.04 3 2 24 3 0.109 -2 1 4 Table 2.
Simulations for the embedded binomial trading strategywith varying boundary levels; gain p.a. denotes the gain per year,gain p.t. denotes gain per trade. In the simulations for Table 1 weused c = 0 . µ / σ .repeats ∅ N is followed by the maximal number of trading repeats over all runs (max N ).As becomes clear from Table 1 we can record an overall profit for many cases.We have a negative outcome in 13 . Varying barrier levels.
The most interesting parameter turns out to be the parameter c . It decodes the varying the barrier level and the results may be found in Table 2.It turns out that this parameter allows to balance gains and risk very well.First, the smaller the parameter c is chosen, the higher are the gains in general.The additional gain does imply an increase of risk: most prominently, the meanof the losses decreases with c . On the other side, we observe a decrease in theprobability for losses to occur. The Value-at-Risk confirms the increase of risk withdecreasing c , except for the lowest c = 0 . .
95% does no longersee this risk (while it is of course still present).A high value of c corresponds intuitively to a larger step sizes, which leads to lesstrades on average. The largest value of c gives a statistical arbitrage with small gainand smallest risk. The role of drift and volatility.
For the investor it is of interest which drift andwhich volatility of an asset promises a good profit. To investigate this question wedefine the fraction η := µσ and show simulation results for different values of η . In Table 3 we fix the volatility σ and consider varying drift, while in Table 4 we fix the drift µ and consider varyingvolatility.Larger values of η point to a high drift relative to volatility situations which wewould expect to be very well exploitable. In fact, our simulations show quite thecontrary: we observe large gains when η is actually small, while for larger η weobserve only minor gains. More precisely, for fixed σ we obtain decreasing gains forincreasing drift, while for fixed µ we observe increasing gains for increasing volatility.This effect is much more pronounced for the latter case (increasing σ ). Alreadyfrom the results with varying step sizes in Table 2 such an effect was to be expected, ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 17 η gain pa median VaR . gain pt losses (mean) ∅ N (max)0.33 211 11,600 252,000 45 0.13 -29,400 5 300.50 170 4,360 94,500 36 0.13 -11,000 5 300.75 109 1,730 38,100 23 0.13 -4,400 5 301.00 64 913 20,400 14 0.12 -2,340 5 301.25 77 561 12,400 17 0.12 -1,400 5 302.00 42 197 4,430 9 0.11 -490 4 313.00 34 81 1,680 8 0.10 -182 4 31 Table 3.
Simulations for the embedded binomial trading strategywith different values of the drift µ (and hence η ), fixed σ = 0 . n = 250 ,
000 runs; gain p.a. denotes the gain per year, gainp.t. denotes gain per trade. η gain pa median VaR . gain pt losses (mean) ∅ N (max)0.50 74,500 222,000 -48,400 4,340 0.036 -2,770,000 17 2700.75 6,020 59,900 480,000 582 0.056 -79,400 10 1201.00 241 4,710 80,500 37 0.090 -8,520 7 511.25 67 541 12,700 16 0.124 -1,460 4 282.00 8 6 165 5 0.144 -22 2 9 Table 4.
Simulations for the embedded binomial trading strategywith different values of the volatility (and hence η ), fixed µ = 0 . η lead to larger step sizes here and to lower gains. Intuitively,larger volatility implies more repetitions and therefore a higher likelihood for thestatistical arbitrage to end up with gains. This is also reflected by increasing valuesof N in Table 4.5.2. Follow-the-trend strategy.
As we have seen in the previous section, embed-ding a binomial model into continuous time is not able to exploit a large drift. Thismotivates the introduction of a further step into the embedded model in order to exploit existing trends in the underlying. We focus on an upward trend , while thestrategy is easily adopted to the case for a downward trend. We consider two-stepbinomial embedding: first, we specify barriers (up/down) as previously. If we twiceobserved up movements, we expect an upward trend and exploit this in a furtherstep. Consequently, here we will consider four stopping times (for iteration i ): initialtime τ i , and stopping times τ i , τ i as previously and, in addition τ i . Most notably,this modelling implies a different choice of the filtration G , see Equation (36).The associated strategy is to trade in the following way: the first trading occursas previously at the first time when the barriers s (1 + c ) or s (1 − c ) are hit. Thenext trading takes place when the neighbouring barriers are hit, in the first case s or s (1 + 2 c ) and in the second case s or s (1 − c ), respectively. If a trend was detected(i.e. the upper barrier s (1 + 2 c ) was hit, as we consider the case of a positive drift),trading continues until a suitable stopping time.More formally, this leads to the following procedure: let i denote the current stepof our iteration. We initialize at time τ = 0. Otherwise consider the initial time ofour next iteration given by the the time where we finished the last repetition anddenote this time by τ i and the according level by s i = S τ i . Then, using again theproperty that S is continuous, we define the following successive stopping times: Index D i ff u s i on τ i σ i σ i σ i τ i τ i s s (1 − c ) s (1 − c ) s (1 + 2 c ) s (1 + 4 c ) Figure 7.
Illustration of the stopping times defined in (33), (34)resp. (35). The first stopping takes place when the process reacheseither the first upper or lower boundary s i (1 ± c ). Starting fromthe upper boundary the next stopping takes place if the processincreases to the level s i (1 + 2 c ), decreases to the level s i (1 − c ) orcrosses the level s . In case the process reached the upper level athird stopping occurs at τ i .first, analogously to t i from Equation (26), let τ i = inf (cid:8) t ∈ ( τ i , T ] | S t ≥ s i (1 + c ) or S t ≤ s i (1 − c ) (cid:9) . (33)In the same manner the second stopping occurs if either the upper level is reached,or the mid-level is crossed, or the bottom level is reached. The levels of course differdepending on whether S τ i = s i (1 + c ) or S τ i = s i (1 − c ). In this regard, we define(for the first case) σ i = inf (cid:8) t ∈ ( τ i , T ] | S t ≥ s i (1 + 2 c ) (cid:9) σ i = inf (cid:8) t ∈ ( τ i , T ] | S t ≤ s i (cid:9) . For the second case, we set σ i = inf (cid:8) t ∈ ( τ i , T ] | S t ≤ s i (1 − c ) (cid:9) σ i = inf (cid:8) t ∈ ( τ i , T ] | S t ≥ s i (cid:9) . Altogether we obtain that τ i = (cid:40) σ i ∧ σ i if S τ i = s i (1 + c ) ,σ i ∧ σ i otherwise . (34)Finally, we set τ i = (cid:40) inf (cid:8) t ∈ ( τ i , T ] | S t ≤ s or S t ≥ s i (1 + 4 c ) (cid:9) , if S τ i = s i (1 + 2 c ) ,τ i , otherwise. (35)Denote by τ max the last stopping time of τ , τ , . . . which lies before T . Thenthe statistical arbitrages traded on the partition of S τ max generated by the values s (1 + 2 kc ) , k = 0 , , , . . . which defines the G on the path space of the diffusion. ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 19 S ( ω ) = s +++ S ( { ω , ω } ) S ( { ω , ω , ω } ) S = s S ( { ω , ω } ) S ( ω ) = s ++ − S ( { ω , ω } ) S ( ω ) = s −− Figure 8.
The embedded binomial model for the follow-the-trendstrategy with positive drift. The filtration generated by the finalstates is generated by each { ω i } for i = 1 , , { ω , ω } . We alsodenote the resulting outcomes by s = s , s + , s − , . . . and indicatethis notation at some places.Trading will be executed at times τ i to τ i when the process reaches one of thepredefined boundaries (or trading time is over). At time τ i we check if a positivetrend persists and trade on this trend. Recall the trading strategy φ = ( φ , φ +2 , φ − )from Equations (30) to (32). First, trading at the first two times is executed aspreviously at times t i , t i , see Lemma 3.7: we hold on [ τ i , τ i ) the fraction φ sharesof S . After reaching s i (1 + c ) ( s i (1 − c ), respectively) at time τ i the trading strategychanges to holding φ +2 ( φ − ) shares of S until τ i . The next trading can be split intothe following three cases:(i) τ i = σ i : in this case we reached the upper level s i (1 + 2 c ) and follow the(upward) trend by holding φ ++3 shares of S . This position will be equalized at τ i or if the final time is reached.(ii) τ i equals σ i or σ i : from the state s i (1 + c ) resp. s i (1 − c ) we arrived back at s i (or below resp. above). No trend was detected and the embedded binomialtrading strategy ends by liquidating the position.(iii) τ i equals σ i : again, no (upward) trend was detected and the strategy ends byliquidation the position.Since Lemma 3.7 treats a related, but slightly different case we explicitly checkin the following that the embedded binomial model indeed allows for statisticalarbitrage. The embedded binomial follow-the-trend strategy.
We consider ˜Ω = { ω , . . . , ω } asdepicted in Figure 8. Let S = s ∈ R ≥ and S take the two values s + and s − suchthat S ( ω ) = S ( ω ) = S ( ω ) = s + , S ( ω ) = S ( ω ) = s − . At time 2 we have the three possibilities S ( ω ) = S ( ω ) = s ++ , S ( ω ) = S ( ω ) = s + − and S ( ω ) = s −− . In the cases of ω , . . . , ω the model stops. If, however, wesaw two up-movements, the model continues and ends up at time 3 in the states S ( ω ) = s +++ or S ( ω ) = s ++ − . We assume without loss of generality that s + > s , s − < s , and s ++ > s + , s − < s + − < s + , and s −− < s − as well as s ++ − < s ++ < s +++ , i. e. we consider binomial models as presented in Figure 8. The dynamic trading strategies can be described by V ( φ ) = φ ∆ S + φ ∆ S + φ ∆ S , with φ , φ +2 , φ − and φ ++3 being the respective values in the states ˜Ω, { ω , ω , ω } , { ω , ω } and { ω , ω } at times 1 , , and 3, respectively. Moreover, we choose˜ G = σ ( { ω } , { ω , ω } , { ω } , { ω } ) , (36)i.e. the σ -field generated by the final states of the embedded binomial model. Thefollowing lemma shows that there is always statistical arbitrage in the follow-the-trendstrategy if there is statistical arbitrage in the recombining two-period sub-modelconsisting only of the first two periods.Denote γ = 1 D q ∆ S ( ω )∆ S ( ω )∆ S ( ω )∆ S ( ω ) − (cid:0) q ∆ S ( ω ) + ∆ S ( ω ) (cid:1) ∆ S ( ω ) − q ∆ S ( ω )∆ S ( ω ) (37)with D given in Lemma 3.7. The following results shows, that in the follow-the-trendmodel there is statistical arbitrage, if (20) holds. Proposition 5.2. If φ is the strategy from Lemma 3.7, then for any α ≥ , ψ = ( ψ , ψ +2 , ψ − , ψ ++3 ) with ψ ++3 = 1 − α ∆ S ( ω ) − ∆ S ( ω ) and ψ ψ +2 ψ − = φ − ∆ S ( ω ) ψ ++3 γ is a ˜ G -arbitrage strategy, if (20) holds. Of course, the possible choice α = 1 leads to ψ ++3 = 0, such that in this case thestatistical arbitrage in the first two periods is exploited and the strategy coincideswith that of Lemma 3.7. Proof.
Following Definition 2.1 the strategy ψ is a statistical ˜ G -arbitrage strategy ifthe following holds ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) ≥ ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) ≥ ψ ∆ S ( ω ) P ( ω ) + ψ +2 ∆ S ( ω ) P ( ω )+ ψ ∆ S ( ω ) P ( ω ) + ψ − ∆ S ( ω ) P ( ω ) ≥ , (40) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) ≥ A = ∆ S ( ω ) ∆ S ( ω ) 0 ∆ S ( ω )∆ S ( ω ) 0 ∆ S ( ω ) 0 q ∆ S ( ω ) + ∆ S ( ω ) q ∆ S ( ω ) ∆ S ( ω ) 0∆ S ( ω ) ∆ S ( ω ) 0 ∆ S ( ω ) . Then Equations (38)–(41) are equivalent to ˜ A ψ ≥
0. Note that S i ( ω ) = S i ( ω ) for i = 1 , A ψ = ˜ x with ˜ x = ( x , . . . , x ) (cid:62) reveals ψ ++3 = x − x ∆ S ( ω ) − ∆ S ( ω ) . ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 21 gain pa mean VaR . gain pt losses (mean) ∅ N (max)27.8 164 4,180 9.17 0.171 -554 3 21 Table 5.
Simulations for the follow-the-trend strategy for 1 mioruns. In comparison to Table 1 (where the notation is explained)we find slightly smaller gains together with a smaller risk.As for Lemma 3.7, we will consider the case where ˜ A is invertible. Note that thethree times three submatrix (upper left) of ˜ A equals the matrix A from Equation(16). Then, denoting x = ( x , x , x ) (cid:62) , ψ ψ +2 ψ − = A − x − A − ∆ S ( ω ) ψ ++3 = A − x − ∆ S ( ω ) ψ ++3 γ with vector γ from Equation (37). Up to now we where free to choose any ˜ x ∈ R > .If we choose, as for Lemma 3.7, x = , then φ = A − is the strategy computedin Lemma 3.7 and the result follows. (cid:3) Simulation results.
We study the performance of the follow-the-trend strategy onbasis of various simulations and compare it to the results of the embedded binomialstrategies. As previously, we simulate a geometric Brownian motion according toEquation (25) with µ = 0 . σ = 0 . S = 2186, T = 1 (year), discretizeby 1000 steps and embed the according models repeatedly in this time interval. Inthis case, Proposition 5.2 grants the existence of statistical arbitrage which we willexploit in the following.Contrary to the intention of improving the average gain of the follow-the-trendstrategy, the simulations show that this goal is not achieved. But, in general, thefollow-the-trend strategy leads to a reduction of risk compared to the embedded-binomial trading strategy, visible through the reduced Value-at-Risk in Tables 5to 8. The reduction of the average gain and its mean can be explained from theobservations in Section 3.4: the follow-the-trend-strategy introduces additionalscenarios with smaller gains (compare Figure 8). This leads to a reduction of theaverage gain and, at the same time, to a reduction of risk.The results from Table 6 to 8 show a similar dependence on the choice of theparameters and of the barrier of the follow-the-trend strategy compared to theembedded binomial strategy. In general, we record smaller gains together withsmaller risk with one exception: the last line of Table 8 shows that a small σ allowsthe follow-the-trend strategy to exploit the existing (although small) positive trendin the data better. Of course, this comes with a higher risk, which is clearly visible.Summarizing, the follow-the-trend strategy shows (in general) smaller gainstogether with a smaller risk. The follow-the-trend strategy is, however, able toexploit a positive trend when σ is very small. c gain pa median VaR . gain pt losses (mean) ∅ N (max)0 . µ / σ
404 3,300 51,300 71.1 0.098 -5,590 6 440 . µ / σ
32 162 4,130 10.7 0.169 -548 3 180 . µ / σ . µ / σ Table 6.
Simulations for the follow-the-trend strategy with varyingbarrier levels c . In the simulations for Table 5 we used c = 0 . µ / σ . η gain pa median VaR . gain pt losses (mean) ∅ N (max)0.33 282 9,340 203,000 71 0.16 -26,100 4 240.50 122 3,500 76,200 31 0.16 -9,780 4 240.75 99 1,390 30,400 26 0.16 -3,890 4 221.00 78 734 16,200 20 0.15 -2,050 4 231.25 54 452 9,950 15 0.15 -1,260 4 232.00 34 162 3,570 10 0.14 -436 3 213.00 24 66 1,390 7 0.13 -165 3 21 Table 7.
Simulations for the follow-the-trend strategy with vary-ing values of the drift (and hence η = µ / σ ) with fixed σ = 0 . η gain pa median VaR . gain pt losses (mean) ∅ N (max)0.33 65,600 2,030,000 22,700,000 6,640 0.06 -2,770,000 10 1000.50 2,010 40,700 586,000 284 0.09 -62,500 7 580.75 292 3,930 69,200 60 0.12 -7,940 5 341.00 44 732 16,400 11 0.15 -2,080 4 241.25 27 200 5,330 9 0.18 -729 3 172.00 10 15 469 5 0.20 -68 2 9 Table 8.
Simulations for the follow-the-trend strategy with vary-ing values of the volatility σ and fixed µ = 0 . Partition strategies on the final value.
In this section we study statisticalarbitrage with respect to the information system G fin defined by { S T ≥ s } = { ω , ω , ω } , and { S T < s } = { ω , ω } . (42)This information system corresponds to the two scenarios that the value of the assetincreased or decreased at time T . The statistical G fin -arbitrage corresponds to astrategy which yields an average profit in both of these scenarios.As an example, we continue in the setting of the follow-the-trend model consideredin the previous Section 5.2, although other settings are clearly possible. Recall thatthis means we are focusing on an upward trend. We add the assumption that s ++ − < s such that also the third period allows for interesting outcomes (below or above s , compare Figure 8). The new information system will lead to a differenttrading strategy as we detail in the following. ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 23
Proposition 5.3.
In the follow-the-trend model with s ++ − < s there is G fin -arbitrage if (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) (cid:17) + (cid:16) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) (cid:17) P ( ω ) P ( ω )+ (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) (cid:17) P ( ω ) P ( ω ) ≥ , (43) (cid:16) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) (cid:17) + (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) (cid:17) P ( ω ) P ( ω ) ≥ and, in addition, at least one of the inequalities is strict. The proof is immediate. Note that here there is a lot of freedom in choosingsuch strategies. Indeed, we will pursue choosing a strategy matching our previousstrategies for better comparability.
Example 5.4.
We consider a special case of (43), (44): we additionally assumethat the first line of Equation (43) and the first line of Equation (44) is non-negative.Then, the strategy ψ is a G fin -arbitrage if ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) ≥ , (45) ψ ∆ S ( ω ) + ψ − ∆ S ( ω )+ (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) (cid:17) P ( ω ) P ( ω ) ≥ , (46) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) ≥ ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) + ψ ++3 ∆ S ( ω ) ≥ , (48)and at least one inequality is strict. Note that we used ∆ S ( ω ) = ∆ S ( ω ),∆ S ( ω ) = ∆ S ( ω ) = ∆ S ( ω ) and ∆ S ( ω ) = ∆ S ( ω ) from Section 5.2. Thischoice is similar to the previously studied partition strategies and we compute astrategy explicitly. In this regard, define the matrix A by A = ∆ S ( ω ) ∆ S ( ω ) 0 ∆ S ( ω )∆ S ( ω ) + r ∆ S ( ω ) r ∆ S ( ω ) ∆ S ( ω ) 0∆ S ( ω ) 0 ∆ S ( ω ) 0∆ S ( ω ) ∆ S ( ω ) 0 ∆ S ( ω ) with r = P ( ω ) P ( ω ) . If A is invertible, for any α ≥
0, the strategy ψ given by ψ ++3 = 1 − α ∆ S ( ω ) − ∆ S ( ω )and ψ ψ +2 ψ − = φ − ∆ S ( ω ) ψ ++3 γ is a G fin -arbitrage. Here, φ = D ( ξ , ξ , ξ ) with ξ = (cid:16) r ∆ S ( ω ) − ∆ S ( ω ) (cid:17) ∆ S ( ω ) + ∆ S ( ω )∆ S ( ω ) ,ξ = (cid:16) ∆ S ( ω ) − ∆ S ( ω ) (cid:17) ∆ S ( ω ) + (cid:16) ∆ S ( ω ) − ∆ S ( ω ) − r ∆ S ( ω ) (cid:17) ∆ S ( ω ) ,ξ = r ∆ S ( ω ) (cid:16) ∆ S ( ω ) − ∆ S ( ω ) (cid:17) − r ∆ S ( ω )∆ S ( ω ) , and D = (cid:16) r ∆ S ( ω )∆ S ( ω ) − (cid:0) ∆ S ( ω ) + r ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:17) ∆ S ( ω )+ ∆ S ( ω )∆ S ( ω )∆ S ( ω ) , computed analogously to Lemma 3.7. In addition, γ = 1 D r ∆ S ( ω )∆ S ( ω )∆ S ( ω )∆ S ( ω ) − (cid:0) r ∆ S ( ω ) + ∆ S ( ω ) (cid:1) ∆ S ( ω ) − r ∆ S ( ω )∆ S ( ω ) , and the computation of the strategy is finished. (cid:5) Remark 5.5.
Under the same assumptions as in the previous example we aimto find a G fin -arbitrage strategy fulfilling equations (45) - (48). In that case thestrategy (Φ , ψ ++ ) with Φ = ( ξ , ξ , ξ ) as in Lemma 3.7 and − S ( ω ) ≤ ψ ++ ≤ − S ( ω )is a G fin -arbitrage strategy. To see this remind that ξ ∆ S ( ω ) + ξ ∆ S ( ω ) ≥ ,ξ ∆ S ( ω ) + ξ ∆ S ( ω )+ (cid:16) ξ ∆ S ( ω ) + ξ ∆ S ( ω ) (cid:17) P ( ω ) P ( ω ) ≥ ,ξ ∆ S ( ω ) + ξ ∆ S ( ω ) ≥ ξ ∆ S ( ω ) + ξ ∆ S ( ω ) ≥ , where ξ ∆ S ( ω ) + ξ ∆ S ( ω ) = ξ ∆ S ( ω ) + ξ ∆ S ( ω ). We are looking for ψ ++ with B + ψ ++ ∆ S ( ω ) ≥ ,B + ψ ++ ∆ S ( ω ) ≥ , where B := ξ ∆ S ( ω ) + ξ ∆ S ( ω ). This results in B ∆ S ( ω ) ≥ − ψ ++ ,B ∆ S ( ω ) ≤ − ψ ++ . Note that B ≥
0, as Φ is a statistical arbitrage strategy. Besides that we have∆ S ( ω ) > S ( ω ) < − B ∆ S ( ω ) ≤ ψ ++ ≤ − B ∆ S ( ω ) . As B was set equal to 1 in Lemma 3.7 we gain in this setting the special condition − S ( ω ) ≤ ψ ++ ≤ − S ( ω ) , but of course strategies can be derived for any B ≥ ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 25 gain pa median VaR . gain pt losses (mean) ∅ N (max)28.6 167 4,290 8.76 0.158 -544 3 20 Table 9.
Statistical G fin -arbitrage trading strategy simulationresults for 1 mio simulations with c i = 0 . η S σ i . In comparison toTable 1 (the embedded binomial strategy) we find slightly smallergains together with smaller losses, while the gains are larger thanin Table 5 (the follow-the-trend strategy). c gain pa median VaR . gain pt losses (mean) ∅ N (max)0 . µ / σ
356 3,280 51,500 58 0.09 -5,510 6 490 . µ / σ
28 166 4,290 9 0.15 -543 3 190 . µ / σ . µ / σ Table 10.
Simulation results for the statistical G fin -arbitrage trad-ing strategy with varying boundaries of the embedded binomialmodel. Gain p.t. is gain per trade and ¯ N equals the maximal N inthe simulations. η gain pa median VaR . gain pt losses (mean) ∅ N (max)0.33 192 9,600 207,000 45.2 0.15 -25,700 4 260.50 112 3,560 77,700 26.7 0.15 -9,600 4 250.75 97 1,430 31,200 23.7 0.14 -3,830 4 261.00 73 751 16,600 18.3 0.14 -2,020 4 261.25 55 458 10,100 13.9 0.14 -1,230 4 242.00 34 163 3,600 9.15 0.13 -428 3 253.00 24 67 1,410 6.82 0.12 -162 3 24 Table 11.
Statistical G fin -arbitrage trading strategy for varying µ but with fixed σ = 0 .
01. Gain p.t. is gain per trade and ¯ N equalsthe maximal N in the simulations. Simulation results.
Again, we study the performance of the strategy, this timethe strategy derived in Example 5.4 with a partition (above/below) on the finalvalue of the stock. We perform various simulations. As previously, we simulate ageometric Brownian motion according to Equation (25) with µ = 0 . σ = 0 . S = 2186, T = 1 (year), discretize by 1000 steps and embed the according modelsrepeatedly in this time interval. The properties for existence of a v in this settingare confirmed numerically.As pointed out before, the statistical arbitrages are with respect to differentinformation fields. By our variant of G fin -arbitrage chosen in Example 5.4 we indeedfind very similar results to the follow-the-trend strategy as one can see in Table 9 to12. η gain pa median VaR . gain pt losses (losses) ∅ N (max)0.75 203 3,890 69,800 38 0.11 -7,810 5 371.00 71 752 16,600 18 0.14 -2,020 4 251.25 28 205 5,500 9 0.17 -715 3 182.00 10 15 494 5 0.19 -67 2 113.00 4 3 51 3 0.09 -5 1 6 Table 12.
Statistical G fin -arbitrage trading strategy for varying σ but fixed µ = 0 . Summary on the different strategies.
The previous results confirm statis-tical G -arbitrage for all three introduced strategies with respect to the correspondingchoices of G . Although we observe similar patterns through all strategies like highergains for smaller boundaries or an decreasing average profit for increasing η thereare significant differences between the strategies:(i) the average profit achieved is best for the embedded binomial strategy.(ii) The follow-the-trend strategy and the G fin -arbitrage strategy show similarbehaviour: while showing smaller gains on average, these two strategies havesmaller risk. 6. Application to market data
In this section we apply the previously studied approaches to real stock data. Itis quite remarkable that the positive impression from the simulated data persists onmarket data. We study data from the Kellogg Company and from Deutsche Bankand study the performance of the G fin -arbitrage from Chapter 5.3.Before we can start with that we have to do some preparations. As we determinedthe strategies above assuming a positive drift we have to calculate the correspondingstrategy for negative drift at first. This is because we will determine the drift in thefollowing examples using real market data and in this case of course there will beboth, sections with positive and negative drift as well.We consider ˜Ω = { ω , . . . , ω } as depicted in Figure 9. Let S = s ∈ R ≥ and S take the two values s + and s − such that S ( ω ) = S ( ω ) = s + , S ( ω ) = S ( ω ) = S ( ω ) = s − . At time 2 we have the three possibilities S ( ω ) = s ++ , S ( ω ) = S ( ω ) = s + − and S ( ω ) = S ( ω ) = s −− . In the cases of ω , . . . , ω the model stops. If, however,we saw two down-movements, the model continues and ends up at time 3 in thestates S ( ω ) = s −−− or S ( ω ) = s −− + . We assume without loss of generalitythat s + > s , s − < s , and s ++ > s + , s − < s + − < s + , and s −− < s − as well as s −−− < s −− < s −− + , i. e. we consider binomial models as presented in Figure 9.We have a look at statistical arbitrage with respect to the information system G fin defined by { S T > s } = { ω , ω } , and { S T ≤ s } = { ω , ω , ω } . (49)Analogously to the case with positive drift we add the assumption that s −− + > s .This will lead to a different trading strategy as we detail in the following. ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 27 S ( ω ) = s ++ S ( { ω , ω } ) S = s S ( { ω , ω } ) S ( ω ) = s −− + S ( { ω , ω , ω } ) S ( { ω , ω } ) S ( ω ) = s −−− Figure 9.
The embedded binomial model for the follow-the-trendstrategy with negative drift. The filtration generated by the finalstates is generated by each { ω i } for i = 1 , , { ω , ω } . We alsodenote the resulting outcomes by s = s , s + , s − , . . . and indicatethis notation at some places. Proposition 6.1.
In the follow-the-trend model with s −− + > s there is G fin -arbitrage if (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) (cid:17) P ( ω )+ (cid:16) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) + ψ −− ∆ S ( ω ) (cid:17) P ( ω ) ≥ , (50) (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) (cid:17) P ( ω )+ (cid:16) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) (cid:17) P ( ω )+ (cid:16) ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) + ψ −− ∆ S ( ω ) (cid:17) P ( ω ) ≥ and, in addition, at least one of the inequalities is strict. Example 6.2.
We consider a special case of (50), (51): we additionally assumethat the first line of Equation (50) and the last line of Equation (51) is non-negative.Then, the strategy ψ = ( ψ, ψ + , ψ − , ψ −− ) is a G fin -arbitrage if ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) ≥ , (cid:16) ψ ∆ S ( ω ) + ψ +2 ∆ S ( ω ) (cid:17) P ( ω ) P ( ω )+ ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) ≥ ,ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) + ψ −− ∆ S ( ω ) ≥ ψ ∆ S ( ω ) + ψ − ∆ S ( ω ) + ψ −− ∆ S ( ω ) ≥ , year D o ll a r s t o ck p r i c e Figure 10.
Daily closing prices of the shares of the Kellogg Com-pany during January 1, 2000 and December 31, 2017. Prices arepresented in US-Dollar.and at least one inequality is strict. In this regard, define the matrix A by A = ∆ S ( ω ) ∆ S ( ω ) 0 0∆ S ( ω ) + r ∆ S ( ω ) r ∆ S ( ω ) ∆ S ( ω ) 0∆ S ( ω ) 0 ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) 0 ∆ S ( ω ) ∆ S ( ω ) with r = P ( ω ) P ( ω ) . If A is invertible, for any α ≥
0, the strategy ψ given by ψ −− = 1 − α ∆ S ( ω ) − ∆ S ( ω )and ψ ψ +2 ψ − = φ − ∆ S ( ω ) ψ −− γ is a G fin -arbitrage. Here φ is the strategy from Lemma 3.7 and γ = 1 D ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω ) (cid:0) r ∆ S ( ω ) + ∆ S ( ω ) (cid:1) + r ∆ S ( ω )∆ S ( ω ) , with D = (cid:16) r ∆ S ( ω )∆ S ( ω ) − (cid:0) ∆ S ( ω ) + r ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:17) ∆ S ( ω )+ ∆ S ( ω )∆ S ( ω )∆ S ( ω ) . The approach now is to simulate the trading with a dynamic strategy, i. e.whenever the data leads to a positive drift we will use the strategy from Example5.4 while for a negative drift we will use the strategy described above.
Example 6.3 (Kellogg Company) . In Figure 10 we depict historical stock pricesof the Kellogg Company from January 1, 2000 to December 31, 2017. Tradingstrategies are used by implementing the strategies from Example 5.4 and 6.2 wherethe parameters of the geometric Brownian motion are estimated by the maximum-likelihood estimates from three years directly before the trading period (which is
ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 29 boundary GPTA: Kellog Deutsche Bank0 . S σ i . S σ i . S σ i . S σ i . S σ i Table 13.
Gains per traded assets (GPTA) for the G fin -arbitrage,applied to historical stock data of the Kellogg Company andDeutsche Bank AG from the year 2000 to 2017. Drift and volatil-ity were estimated by maximum-likelihood methods with a rollingwindow of length 3 years. year E u r o s t o ck p r i c e Figure 11.
Daily closing prices of the shares of the Deutsche BankAG during January 1, 2000 and December 31, 2017. Prices arepresented in Euro.a sliding-window approach with a window length of 3 years). Table 13 shows theachieved gains for different boundary values. The gains are normalized to one tradedasset to improve comparability. The results confirm the findings from the previoussection in the sense that we see gains for all chosen boundaries. If the boundaryis chosen too small or too large the trading strategy does, however, not performoptimally.
Example 6.4 (Deutsche Bank) . As a second example, we apply our methodologyto stock prices of Deutsche Bank from January 1, 2000 to December 31, 2017. Incontrast to the previous example, we observe higher volatility and also large lossesin the observation period. We proceed as for the Kellogg’s example and the resultsare shown in Table 13. Due to the present downward trend in the stock evolutionthe G fin -strategy is expected to perform as the embedded binomial strategy. Werecognize positive gains through all boundaries. Conclusion
We introduce the concept of statistical G -arbitrage and give a characterizationof it. Moreover, we examine various profitable strategies both on simulated and onmarket data. The choice of the information system G is either motivated naturallyby the aim to generate profitable strategies in average over certain pre-determinedscenarios or, alternatively, it can be used as a technical tool to generate profitablestrategies.Our data experiments show that the analysed strategies show a good performanceboth on simulated data and on market data. Appendix A. Proofs
Proof of Lemma 3.3.
Note that equations (7) - (10) reads Aξ ≥ A = ∆ S ( ω ) ∆ S ( ω ) 0∆ S ( ω ) 0 ∆ S ( ω )∆ S ( ω ) ν + ∆ S ( ω ) ∆ S ( ω ) ν ∆ S ( ω )∆ S ( ω ) ν + ∆ S ( ω ) ∆ S ( ω ) ν ∆ S ( ω ) . We do a change of basis for the mapping A and substitute the vector in the firstcolumn. This leads to a matrix ˜ A ,˜ A = S ( ω ) 00 0 ∆ S ( ω ) B ∆ S ( ω ) ν ∆ S ( ω ) B ∆ S ( ω ) ν ∆ S ( ω ) where B = ν (cid:18) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) (cid:19) + ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) B = ν (cid:18) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) (cid:19) + ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) . We denote by (cid:61) ( ˜ A ) the image of a mapping ˜ A . There exists statistical arbitrage if (cid:61) ( ˜ A ) ∩ R > (cid:54) = ∅ . The linear subspace spanned by ˜ A is given by α B B + β ∆ S ( ω )0∆ S ( ω ) ν ∆ S ( ω ) ν + γ S ( ω )∆ S ( ω )∆ S ( ω ) , (52)with α, β, γ ∈ R . Assume this space meets R ≥ . Then it follows from the condition β ∆ S ( ω ) = β ( s ++2 − s +1 ) ≥ β ≥
0. Similarily, γ ≤ S ( ω ) = s −− − s − <
0. Summing up the third and fourth coordinate from (52) we get α (cid:18) ν (cid:16) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) (cid:17) + ν (cid:16) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) (cid:17) + ∆ S ( ω )∆ S ( ω ) (cid:16) − ∆ S ( ω ) − ∆ S ( ω ) (cid:17) + ∆ S ( ω ) + ∆ S ( ω ) (cid:19) (53)+ γ (∆ S ( ω ) + ∆ S ( ω ))+ β (∆ S ( ω ) ν + ∆ S ( ω ) ν ) . Choosing ν = − ∆ S ( ω )∆ S ( ω ) ν , β (∆ S ( ω ) ν + ∆ S ( ω ) ν ) = 0 ENERALIZED STATISTICAL ARBITRAGE CONCEPTS 31 such that the last term in the above equation vanishes. As we assumed that thespace spanned by (52) meets R ≥ it must also hold true that (53) ≥
0. For ν < ∆ S ( ω )∆ S ( ω ) (∆ S ( ω ) + ∆ S ( ω )) − ∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) = Γ the coefficient of α in (53) is negative. Together with γ ≤ S ( ω ) , ∆ S ( ω ) > ν results in α ≤ ≥
0. Onthe other hand, if we claim ν > − ∆ S ( ω ) + ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) ∆ S ( ω ) − ∆ S ( ω ) ∆ S ( ω )∆ S ( ω ) = Γ it follows that B > αB + β ∆ S ( ω ) ν + γ ∆ S ( ω ) ≤ . Hence (cid:61) ( ˜ A ) ∩ R > = ∅ . It remains to prove that(i) Γ < Γ and(ii) there is no statistical arbitrage for ν = Γ .The statements (i) and (ii) are verified by analogous calculations which concludesthe proof. (cid:3) Proof of Proposition 3.4. “ ⇒ ” If det( A ) (cid:54) = 0 we choose for example ξ := A − ⇐ ” On the other hand, if det( A ) = 0 there still might be an arbitrage opportunityif the image of A intersects with the positive subspace of R , i.e. if (cid:61) ( A ) ∩ R > (cid:54) = ∅ .To show that this is not the case we change the basis for the mapping A andsubstitute the vector in the first column. This leads to a matrix ˜ A ,˜ A = S ( ω ) 00 0 ∆ S ( ω ) B ∆ S ( ω ) q ∆ S ( ω ) , where B = q (cid:0) ∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) ∆ S ( ω ) (cid:1) + ∆ S ( ω ) − ∆ S ( ω )∆ S ( ω ) ∆ S ( ω ) . Calculating det( A ) we see that det( A ) = 0 is equivalent to0 = − ∆ S ( ω ) (cid:0) ∆ S ( ω )∆ S ( ω ) − (cid:0) ∆ S ( ω ) + q ∆ S ( ω ) (cid:1) ∆ S ( ω ) (cid:1) − q ∆ S ( ω )∆ S ( ω )∆ S ( ω ) . (54)In the recombining binomial model this reduces to0 = q ∆ S ( ω ) (cid:16) − ∆ S ( ω )∆ S ( ω ) (cid:17) + ∆ S ( ω ) (cid:16) − ∆ S ( ω )∆ S ( ω ) (cid:17) which is equivalent to B = 0. In this case the linear subspace spanned by ˜ A is givenby α ∆ S ( ω )0 q ∆ S ( ω ) + β S ( ω )∆ S ( ω ) , (55)with α, β ∈ R . Because ∆ S ( ω ) > α ≥ β ≤ S ( ω ) < S ( ω ) < S ( ω ) >
0, we obtain for the third coordinate that αq ∆ S ( ω ) + β ∆ S ( ω ) ≤ (cid:61) ( A ) ∩ R > = ∅ , which concludes the proof. (cid:3) References
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