A Simple Algorithm Based on Fluctuations to Play the Market

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A simple algorithm based on fluctuations to play the market

L. Gil

Institut Non Lineaire de Nice,UMR 6618 CNRS-UNSA, 1361 Route des lucioles,

06560 Valbonne - Sophia AntipolisFRANCE

(Dated: December 3, 2008)

In Biology, all motor enzymes operate on the same principle: they trap favourable brownianfluctuations in order to generate directed forces and to move. Whether it is possible or not tocopy one such strategy to play the market was the starting point of our investigations. We foundthe answer is yes. In this paper we describe one such strategy and appraise its performance withhistorical data from the European Monetary System (EMS), the US Dow Jones, the german Daxand the french Cac40.

PACS numbers: 89.65.Gh, 0 5.40.-a, 87.10.+e

INTRODUCTION

In the 1970s, the Efficient Market Hypothesis (EMH)assumed that the actions of all traders and external news

result collectively into so large fluctuations that any pre-dictability pattern is neither detectable nor exploitable[1, 2]. No predictions can be performed from the knowl-edge of the past prices and therefore to make profit, onehas to be informed (possibly illegaly) of the future evo-lutions of currencies or securities.

Now, even [3] the champions of EMH are forced torecognize that first, several disagreements between theefficient market theory and genuine experimental obser-vations have been exhibited [4, 5, 6, 7], and second, thatthe central assumption that every trader or companyacts in the same self-interested way rational, cool andomniscient is only a gross caricature, in strong contra-diction with the experimental psychological observations[8]. The EMH hypothesis must be viewed as a first or-der (and nevertheless useful) approximation describingan average idealized behavior.

Since financial markets are not in strictly perfect equi-librium, it should be possible to beat the market usingalgorithms based on fluctuations. In this paper we ex-hibit and describe one such an algorithm and appraiseits performance with the german mark(DEM), the en-glish pound (GBP) and french franc (FRF) currenciesduring the European Monetary System (EMS) from 1980to 1992, and with the securities which compose the US

Dow Jones, the german Dax and the french Cac40 be-tween 2000 and 2006. The results are quite impressive,especially with the EMS, where average costless yearlyreturn up to 50% are obtained.

BUILDING OF THE ALGORITHM

The present algorithm is sustained by two basic ideas:first, renounce to predict the future prices, second, take

full advantage of the stochastic characteristic of theirtemporal evolution (we will show that the absence of fluc-tuations leads to vanishing returns). The algorithm takesits origin from an analogy with the mechanism called

noise induced transport, putted into evidence in Physicsand which has been found to be deeply involved in Biol-ogy.

In Physics, the Maxwells demon as well as the Feyn-mans ratchet [9] are very famous illustrations of one ofthe main implication of the second law of thermodynam-ics, i.e. that useful work cannot be extracted from equi-librium fluctuations. In 1990s, it has been realized thatthis restriction does not hold anymore for out of equi-librium situations and a lot of theoretical [10, 11, 12],as well as numerical and experimental [13] studies, havereported onto the possibility for asymmetric potentialsto rectify non colored fluctuations and to lead to a co-herent response from unbiased forcing. The paradigm ofsuch situation is the flashing brownian ratchet. In thissystem, brownian particles are subjected to a spatial pe-riodic potential, which breaks the parity symmetry and isperiodically turned on and off with time. When off, theparticles symmetrically spread. When on, parity sym-metry is broken. As a result, a noise induced transporttakes place. Numerous applications of these ideas havebeen found in Biology where brownian fluctuations areubiquitous and unimaginably tumultuous [13, 14]. Forexample, the proteins (kinesin and dynein) which areinvolved in the transport between the nucleus and the

membrane of a cell, are just allowed to attach and detachthemselves from the periodic chiral pilings up of moleculscalled microtubules. However they move!

In the field of finance, spatial position of the proteinstands for the security price, transport means positivereturn, brownian fluctuations stand for random walk ofthe price returns and out of equilibrium situations corre-spond to non efficient market. Stationary attached posi-tion correspond to short situation while detached (andthen sensitive to fluctuations) to long one. The old

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maxim buy at low price and sell at hight will play therole of the asymmetry of the potential. Of course, at thisstage, we are still left with the crucial point to preciselydefine what low and hight means (we will do it in thefollowing). But what has to be learned from the biolog-ical analogy, is there is no need of a (intelligent) humandecision, no need of a deep and smart understand-ing of the market, and that a systematic, repetitive and

programmable trading is enought. After all, proteins arenot clever!

Now we built up step by step the algorithm base onfluctuations we will use to play the market. As a startingpoint, we restrict ourselves to a market composed of onlyone security, whose price is X1(t). Time is discret andthe interval between two successive times is constant andarbitrary set to one. We first assume that

X1(tn) = A1 + a11(tn) (1)

where 1 is an independent aleatory variable equal to 1with probability 12 . There are no longscale trends be-

cause A1 is constant with respect to time, only binomialfluctuations of amplitude a1 with a1 < A1. It is obvi-ous that, for such a system, the average return of thebuy and hold strategy is just 0. By analogy with motorenzymes which trap favourable brownian fluctuations inorder to progress, the aim of our Maxwells demon willbe to select the fluctuations of price which correspond toan increase of security number. His behavior (MD1) isdefined as

1. The demon is playing once at each time tn.

2. At each time tn, the demon is in possession of eithersecurities or money, but neither a mixing of both.

3. At time tn, the choice between security and moneyis ruled by

R1: If X1(tn) > A1 then the demon tries to getmoney. Therefore, if he was in possession of secu-rities, then he sells them.

R2: If X1(tn) < A1 then he tries to get securities.Therefore, if he was not in possession of securities,then he buys them.

Fig.1 shows both a typical price trajectory {X1(t1),X1(t2).... X1(tn)} and the associated demons behaviorwhile fig.2 describes the corresponding time evolution of

his wealth W and the number N1 of securities which arein his possession. Despite the dynamic is limited to the8 first days, one can already observe that, although thethird and sixth day price configurations are the same,the wealth has been increased by a multiplicative factor=A1+a1

A1a1greater than 1. Defining the daily return r(tk)

and the cumulative daily return R(tn) as

r(tk) = log

W(tk)

W(tk1)

R(tn) = log

W(tn)

W(t1)

(2)

FIG. 1: A typical trajectory showing the choice (black disk)between security (X1) and money versus time. The downwardarrows correspond to (R1) case while the upward ones to (R2).The lack of arrows correspond to situations where the choiceat tn already corresponds to the position at tn1.

FIG. 2: Evolution of the number of security (N1) and ofthe total wealth W for the given price trajectory displayed infig.1. Althought the time t=3, t=6 and t=8 configurations arestrictly identical, the wealth has been multiplied respectivelyby , 2 and 3.

and averaging over the 2n possible trajectories, we theneasily obtain R(tn) =

n8 log().

Of course, the binomial distribution of the price fluctu-ations (1) is not a necessary condition for the efficiency ofthe algorithm. It is just a convenient distribution whichallows some easy analytical computations. In fact, pro-

vided the knowledge of the future average price, the al-gorithm can be applied to any more complex distribu-tions, like the ones used in fig.3 or fig.5, and still leads toan increase of the non vanishing number of security ( N1)with time (fig.4).

Now the crucial question is : is something stands inour way and prevent us to use the previous algorithm(MD1) for playing the market? The answer is yes! Firsthindrance, the improbable knowledge, at the beginningof the process, of the average security price A1. Second


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FIG. 3: A typical trajectory showing the selected security(black disks) versus time for an arbitrary distribution.

FIG. 4: Evolution of the number of security (N1) and ofthe total wealth W for the 14 first days corresponding tothe trajectory displayed in fig.3. It is crucial to note theincreasing of the non vanishing N1 numbers. For example,

because X1(t2)X1(t3)

, X1(t5)X1(t7)

and X1(t9)X1(t14)

are all > 1, we are sure

that N1(t14) > N1(t7) > N1(t3) > N1(t1).

possible obstacle, the final price of the security (i.e. theprice of the security when one stops to play). Indeed forthe distributions used in fig.1 and fig.3, the final priceshave been arbitrary chosen in a close neighborhood ofthe average price. But, when it is no more the case as infig.5, the final price of the securities may be so low that itcan not be balanced by the increase of the non vanishingnumbers of securities (N1).

FIG. 5: The non vanishing number of security does increasewith time, however the last security price is so low that thealgorithm has lost money at the end of the process (t18).

In order to face up to these problems, we propose to

use, instead of the global average values A1, a local onedefined thanks to the m last known prices as

X1m (tn) =1

m

m1j=0

X1(tnj) (3)

such that, at time tn, the new demons behavior (MD2) isruled by the comparison between X1(tn) and X1m (tn)in place ofA1. It is worth noting that, with this modifica-tion, we are no more able to prove that the non vanishingnumbers of security increase with time, nor that there isgain. Also, the algorithm (MD2) can be optimized by asuitable choice of the m parameter. But, to what is thisoptimal value related to?

TEST WITH ARTIFICIAL DATA

Before testing the algorithm (MD2) with historicaldata, we first investigate its efficiency with numerous andwell controlled artificial aleatory data, satisfying the fol-lowing modified Geometric Ornstein-Uhlenbeck process[15, 16]

dX1 = 1 (A1 X1) dt + 1X1dW1dA1 = 1A1dt

(4)

where W1 is a Wieners process. The interpretation of themodel is straightforward: when 1 is vanishing, eq.(4)describes the well known random walk of the prices. Onthe contrary, 1 > 0 means that there exist some funda-mentals (liabilities, monopolies, patents or dividends...)which attracts the dynamics of the security price X1 inthe neighborhood of A1. Note that A1 is not forced tobe constant and may depend exponentially on time if1 = 0. We proceed in the following way. First we


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numerically integrate eq(4) over 1000 units of time us-ing the Milsteins scheme [17], a Box-Muller [18] normalnoise generator and X1(0) = A1(0) as an initial condi-tion. Then we apply the modified algorithm (MD2) andcompute the associated cumulative return. These lasttwo steps are repeated 30000 times (enough for conver-gence) and the final result is obtained by averaging.In a first step, we restrict ourselves to situations where

A1 is constant with time. The corresponding results aresummarized in fig.(6) which displays the average cumu-lative return versus m for various values of 1. Severalimportant observations are in order:

1. In absence of fluctuations (or when 1 >> 1), thecumulative return is just equal to 0 (fig.7).

2. In presence of fluctuations, but in absence of restor-ing force (i.e. 1 = 0), the average cumulative re-turn is always vanishing, whatever the value of m.

3. Therefore, a cumulative return distinct from 0 re-quires the presence of both fluctuations and restor-ing forces. In such a case, one observe that Rincreases with m up to an asymptotic value. Thisincrease has to be related to the fact that the morem increases, the more the local average X1m (tn)goes close to its asymptotic value A1.

4. A characteristic value mc can be defined as thevalue of m for which the average cumulative re-turn is equal to 90% of its asymptotic value. Asexpected, mc decreases with the restoring forces(fig.7).

FIG. 6: Application of the MD2 algorithm onto data gener-ated by (4) with A1=2, 1=0.01 and 1=0. The plot displaysthe average cumulative return R(t1000) after 1000 units oftime versus m for various values of 1. From bottom to top,1=0, 1=9.6, 1=0.3, 1=4.8, 1=0.6, 1=2.4 and 1=1.2.

When A1 is no more constant with time, the local averageX1m (tn) and A1(tn) can strongly move away one from

FIG. 7: Same regime of parameters as in fig.6. The blackdisks (left scale) correspond to mc versus 1 and the whitesquares (right scale) to the asymptotic values of the averagecumulative return R(t1000) versus 1.

the other as m is increased. In such a case, the algorithm(MD2) is certainly less efficient. Hence we expect anddo observe (fig.8) a decrease of the average cumulativereturn for sufficiently high value of m. The maximumis reached for a value of m close to mc and thereforedepends on 1. The characteristic time

1|1|

controls the

decrease of the cumulative return after the maximum aswell as the amplitude of the maximum.

FIG. 8: Application of the MD2 algorithm onto data gen-erated by (4) with A1=2, 1=0.01 and 1=1.2. The plotdisplays the average cumulative return R(1000) versus mfor various values of 1. From bottom to top, black squares,white disks, black disks, white squares, white stars, black starsand crosses correspond respectively to 1=0 10

3, +2 103,+4.0 103, +8.0 103, +12 103, +16 103 and 16 103.

Note that the arbitrary chosen exponential dependenceof A1 with time is not crucial and that similar resultswould have been obtained with other time dependences(like periodic or even random): only relevant is the ex-


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istence of two distinct characteristic time scales, one as-sociated with the price evolution, the other, much moreslower, associated with the temporal evolution of A1.

EUROPEAN MONETARY SYS TEM

We now apply our modified algorithm to genuine data

corresponding to the daily record of historical prices. Forthe sake of efficiency, we look for data for which we ex-pect the existence of a strong restoring force. It turns outthat there is no doubt about the existence of such a forcefor the time evolution of the currencies which belonged tothe European Monetary System (EMS). Indeed, the EMSwas an economic and politic arrangement where most na-tions of the European Economic Community linked theircurrencies to prevent large fluctuations relative to oneanother. In the early 1990s the European Monetary Sys-tem was strained by the differing economic policies andconditions of its members, especially the newly reunifiedGermany, and Britain permanently withdrew from thesystem (see the 1992 G. Soross speculation).

In order to map the situation with two currencies ontothe previous one, it is sufficient to consider one of the cur-rency as playing the role of the security, and the other asthe money in which the value of the security is expressed.The results of the application of the MD2 to the EMSsdata are then shown in fig.9. Obviously, the algorithmworks! Even if the transaction costs are taken into ac-count, average yearly returns of 24% (DEM/GBP) and44% (DEM/FRF) are obtained between 1980 and 1992(13 years), for a suitable choice of m. Second and impor-tant observation, the optimal value of m does not deeply

depend on the choice of the time interval (out of sample).Finally, the curves roughly look like those obtained with(4). Therefore if the theoretical framework (4) holds,we can in addition deduce that: (i) the restoring forcescould be about the same in UK and in France and the op-timal value mc 5 could correspond to a weekly actionof the central banks. (ii) The characteristic time asso-ciated with the decrease of the cumulative return afterits maximum is higher for the DEM/GBP than for theDEM/FRF pair. Also the maximum is smaller. Bothobservations could be related to the well known higherstability of the DEM/FRFs exchange rate (strongly po-litically motivated by the building of the euro) compared

to the DEM/GBPs one (reticence of the UK govern-ment).

Fig.(10) shows the time evolution of the cumulativedaily returns for the application of the MD2 with m = 5.The main features of the curves are the quite regulargrowth between 1980 and 1990 and its abrupt stop in theearly 1990s. In the framework of (4), this stop is under-stood as the vanishing of the restoring forces. Note thestop date is in perfect agreement with the acknowledgedbeginning of the EMSs trouble.

FIG. 9: Application of the MD2 to the historical EMS cur-rencies. The plots display the cumulative return R versusm. A cost of 3/10000 per transaction has been applied. Thedashed lines correspond to the 1980-1992 time interval. Forthe continuous lines, the time interval 1980-1990 has beensplitted into 3 equal parts (quoted I, II and III).

FIG. 10: Application of the MD2 with m=5 to the EMS cur-

rencies. The plots display the cumulative daily return versustime. The top curves correspond to a free cost transactionwhile for the lower one a cost of 3/10000 has been applied.

STOCK EXCHANGE

The next question is: what happens with the stock ex-change? In one hand, one can expect the existence ofliabilities, patents, companys reputation and monopolyto lead to the existence of a restoring forces. But in theother hand, rumors, non informed agents, mimicry andspeculation play an opposite role and can drive the se-

curity prices far from their fundamental values. In thefollowing, we investigate the efficiency of our algorithmonto the components of the Dow Jones, Dax and Cac40indexes, between 2000 and 2006. We proceed in the fol-lowing way. Among the components of each index, thesecurities which are not regularly quoted for the time in-terval between 2000/01/01 and 2006/05/12 ( 6.5 years)are thrown away. We are then left with Ns=30 securitiesfor the Dow Jones, Ns=29 for the Cac40 and Ns=24 forthe Dax. Then, for each index, we restrict our analysis


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to the days for which all the securities which remain af-ter the first selection are quoted. We then apply MD2separately to each component j of a single index. Sep-arately means that the money which is obtained witha given security is not used to speculate with an otherone. The same initial investment W(0) is used for eachcomponent. At time t, the cumulative return of a givenindex is then defined as

Rindex = log

Nsj=1 Wj(t)

NsW(0)

(5)

where Wj(t) stands for the wealth which is obtained play-ing MD2 with the security j. Transaction costs of 0.1%have been applied. The corresponding results are thendisplayed in fig.11 for the Cac40, fig.12 for the Dax andfig.13 for the Dow Jones.

FIG. 11: Application of MD2 algorithm to the Cac40 index.The 3 continuous lines correspond to the plot of the cumu-lative return (5) versus m, but the thick one deals with thehistorical records between 2000 and 2006, while the thin onesare associated respectively with the first (I) and second (II)half of the same time interval.

FIG. 12: Same caption as fig.11 but for the Dax index.

Several remarks are in order: (i) for each index, theshape of the curves associated with either the whole-ness or halves of the time interval, looks like the same,(ii) the european and US behaviors are deeply different.Roughly, for large values of m, we observe a decrease ofthe cumulative return for the Cac40 and Dax, and anincrease for the Dow Jones. In the framework of (4),this observation could suggest the existence of a stronger

FIG. 13: Same caption as fig.11 but for the Dow Jones index.

restoring force for the european indexes than for the USone, (iii) for the whole time interval, the best europeanresults are obtained for m 8. For the Cac40, the max-imum value is close to 0.7 and corresponds to a yearlyaverage return of 11%. For the Dax, the yearly returnis smaller and close to 3.5%. The smallest yearly re-turn ( 2%) is obtained for the Dow Jones. Although

small, the yearly returns provided by the MD2 algorithmover the whole time interval are always better than thebuy and hold strategy. Indeed, this last strategy leads,for the same time interval and the same selected securi-ties, to an average yearly return of 0.0% for the Cac40, 0.5% for both the Dax and the Dow Jones.

THE FABULOUS CASE OF THE CAC40

Concerning the financial market crashes, the french ex-ception has already been brought to light [19]. As an

other evidence of this strangeness, we now discuss a mod-ification of the MD2 algorithm which will reveal itselfas especially competitive for the Cac40 market, but ofmediocre interest for Dax and without interest for theDow Jones.

In the previous algorithm (MD2), each security wasconsidered independently from the other because theamount of money which is assigned to a particular secu-rity can not be used to buy an other security (eq. 5). Thisconstraint is now relaxed in the new algorithm (MD3),which is defined as:

1. The daemon is playing once at each time tn.

2. At a given time, the daemon is in possession ofonly one type of security, cash being considered asa special type of security.

3. At time tn, the choice of the new position is givenby the following rule: Assume that there are Nssecurities whose prices are X1(t), X2(t),....XNs(t).Assume also that at t = tn, the Maxwell daemonis in possession of Nj(tn) securities of type j (k =


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j = Nk = 0). Defining rj,k and j,k as

rj,k =1m

m1p=0

Xj(tnp)Xk(tnp)

j,k =

1m

m1p=0

Xj(tnp)Xk(tnp)

2 r2j,k

(6)

then the new position correspond to the k valuewhich maximizes

Xj(tn)Xk(tn)

rj,k

j,k(7)

4. As for MD2, m is a free parameter which standsfor the number of past data used to compute themobile averages.

5. The daemon starts with cash (W(0)) and the cu-

mulative return at time tn is defined as lnW(tn)W(0)

.

Fig.(14) displays the stupendous results of the appli-

cation of the MD3 algorithm to the components ofthe Cac40 between 2000/01/01 and 2006/05/12 ( 6.5years). First, the optimal value of m does not dependon the time interval (out of sample). Second this opti-mal value is found to be close to 25 days, so to say onemonth since only workdays are taken into account. Fi-nally but not the least, average yeraly return up tu 60%are obtained!

FIG. 14: Application of the MD3 algorithm to the Cac40index. 0.1% transaction cost have been applied. The picturedisplay the cumulative return versus m. The red points areassociated with the cumulatice return between 2000 and 2006,while the green and the blue ones are associated respectivelywith the first (I) and second (II) half of the same time interval.

CONCLUSION

By analogy with the way motor enzymes trapfavourable brownian fluctuations, we have built an algo-rithm which is able to make the best from out of equilib-rium price fluctuations and to play the market. Testingits efficiency with genuine historical data, positive cumu-lative returns have been measured even in presence of

a 0.1% transaction cost. Especially stupendous are theresults dealing with the application of the algorithm toEMS currencies or with the Cac40 components.

Several remarks are in order:

1. With no doubt, the birth of the EMS was first astrong political resoluteness. However, numerouseconomics experts, governors of the central banks

and members of the european monetary comityhave been lengthily consulted. How is it possiblethat such an opportunity to make money at theexpense of nations has not been identified?

2. The money which is captured by our algorithmscomes from the irrational behavior of uninformednoisy traders. Therefore we really expect thepresent algorithms will become unprofitable as soonas our paper will be published, either because irra-tional traders will be taught a lesson or because theprofitability of the algorithms will vanish with thenumber of users.

3. More than a way to make money, the algorithmscan be used as simple, sensitive and straightforwardtools to detect non efficient market, certainly lesscircuitous than the measure of the excess volatilityor the deviation from the normal distribution.

The author is grateful to J. Viting-Andersen and V.Planas-Bielsa for valuable discussions.

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A Simple Algorithm Based on Fluctuations to Play the Market