Time Series Technical Analysis via New Fast Estimation Methods: A Preliminary Study in Mathematical Finance

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Time Series Technical Analysis via New Fast Estimation Methods: A Preliminary Study in Mathematical Finance

Michel Fliess, Cédric Join

To cite this version:

Michel Fliess, Cédric Join. Time Series Technical Analysis via New Fast Estimation Methods: A

Preliminary Study in Mathematical Finance. IAR-ACD08 (23rd IAR Workshop on Advanced Control

and Diagnosis), Nov 2008, Coventry, United Kingdom. �inria-00338099v2�

estimation methods: a preliminary study in

mathematial nane

Mihel FLIESS

∗,∗∗

Cédri JOIN

∗,∗∗∗

∗

INRIA-ALIEN

∗∗

LIX (CNRS,UMR7161), Éole polytehnique

91128 Palaiseau, Frane

Mihel.Fliesspolytehnique.edu

∗∗∗

CRAN (CNRS,UMR 7039),Nany-Université

BP239, 54506 Vand÷uvre-lès-Nany, Frane

Cedri.Joinran.uhp-nany.fr

Abstrat: Newfast estimation methods stemming from ontrol theory lead to afresh look at

time series, whih bears someresemblaneto tehnial analysis. The results are applied to

atypialobjet of nanial engineering, namely the foreast of foreignexhange rates,via a

model-free setting,i.e., viarepeatedidentiationsofloworderlineardiereneequationson

slidingshort time windows. Severalonviningomputersimulations,inludingthepredition

ofthe position and of thevolatilitywith respet tothe foreastedtrendline, areprovided. Z^-

transformanddierentialalgebraarethemainmathematialtools.

Keywords: Timeseries,identiation,estimation, trends,noises,model-freeforeasting,

mathematialnane,tehnialanalysis,heterosedastiity,volatility,foreignexhangerates,

lineardiereneequations,Z-transform,algebra.

1. INTRODUCTION

1.1 Motivations

Reent advanes in estimation and identiation (see,

e.g.,(Fliess &Sira-Ramírez,2003;Fliess&Sira-Ramírez,

2004; Fliess, Join & Sira-Ramírez, 2004; Fliess & Sira-

Ramírez,2008;Fliess,Join&Sira-Ramírez,2008)andthe

referenes therein) stemming from mathematial ontrol

theorymaybesummarizedbythetwofollowingfats:

• ^{Their algebrainature permits to derive exatnon-}

asymptotiformulaeforobtainingtheunknownquan-

titiesin realtime.

• ^{Thereisnoneedtoknowthestatistialpropertiesof}

theorrupting noises.

Those tehniques have already been applied in many

onrete situations, inluding signal proessing (see the

referenes in (Fliess, 2008)). Their reent and suessful

extension to disrete-time linear ontrol systems (Fliess,

Fuhshumer, Shöberl, Shlaher & Sira-Ramírez, 2008)

haspromptedustostudytheirrelevanetonanialtime

series.

Remark 1.1. Therelationshipbetweentimeseriesanalysis

and ontrol theory is well doumented (see, e.g., (Box,

Jenkins & Reinsel, 1994; Gouriéroux & Monfort, 1995;

Hamilton, 1994) and the referenes therein). Our view-

point seemsneverthelesstobequitenewwhenompared

to theexisting literature.

Remark 1.2. The title of this ommuniation is due to

analysis, or harting (see, e.g., (Aronson, 2007; Béhu,

Bertrand&Nebenzahl,2008;Kaufman,2005;Kirkpatrik

& Dahlquist, 2006; Murphy, 1999) and the referenes

therein),whihiswidelyusedamongtradersandnanial

professionals.

1

1.2 Lineardiereneequations

Consider the univariate time series {x(t) | t ∈ N}^: x(t)

is not regarded here as a stohasti proess like in the

familiar ARMA and ARIMA models but is supposed to

satisfyapproximatively alineardiereneequation

x(t+n)−a1x(t+n−1)− · · · −anx(t) = 0 ⁽¹⁾

where a1, . . . , an ∈ R^{. Introdue as in digital signal}

proessingtheadditivedeomposition

x(t) =xtrendline(t) +ν(t) ⁽²⁾

where

• xtrendline(t) ^{is the trendline2 whih satises Eq. (1)}

exatly;

• ^{theadditivenoise}ν(t)^{isthemismathbetweenthe}

realdata andthetrendline.

Thus

x(t+n)−a1x(t+n−1)− · · · −anx(t) =ǫ(t) ⁽³⁾

where

ǫ(t) =ν(t+n)−a1ν(t+n−1)− · · · −anν(t) ⁽⁴⁾

1

Tehnial analysis is often severely ritiized in the aademi

worldandamongthepratitionersofmathematialnane(see,e.g.,

(Paulos,2003)).

2

Weonlyassumethat theergodimean ofν(t)^is0^,i.e.,

N→+∞lim

ν(0) +ν(1) +· · ·+ν(N)

N+ 1 = 0 ⁽⁵⁾

Itmeansthat,∀t∈N^{, themovingaverage}

MAν,N(t) = ν(t) +ν(t+ 1) +· · ·+ν(t+N)

N+ 1 ⁽⁶⁾

is loseto 0 ^if N ^{is largeenough.It follows from Eq. (4)}

that ǫ(t)^{also satisestheproperties(5)and (6).Most of}

thestohasti proesses,likenitelinearombinations of

i.i.d. zero-mean proesses, whih are assoiated to time

series modeling, do satisfy almost surely suh a weak

assumption.Ouranalysis

• ^{doesnotmakeanydierenebetween}non-stationary andstationarytimeseries,

• ^{does not need the often tedious and umbersome}

trend and seasonalitydeomposition (our trendlines

inludetheseasonalities,iftheyexist).

1.3 Amodel-freesetting

Itshould belearthat

• ^{aonretetimeseriesannotbewell} approximated in generalbyasolutionofaparsimonious Eq.(1),

i.e., alineardiereneequationofloworder;

• ^{the useof large order lineardierene equations, or}

ofnonlinearones,mightleadtoaformidableompu-

tationalburden fortheir identiations withoutany

lear-utforeastingbenet.

We adopt therefore the quite promising viewpoint of

(Fliess &Join,2008)where theontrol of omplex sys-

temsisahievedwithouttryingaglobalidentiationbut

thanksto elementarymodels whih areonly validduring

ashorttime intervalandareontinuouslyupdated.

3

We

utilize here low order dierene equations.

4

Then the

windowsize for themovingaverage(6) doesnot needto

belarge.

1.4 Content

Set. 2, whih onsiders the identiability of unknown

parameters, extendsto the disrete-time ase aresult in

(Fliess, 2008). The onvining omputer simulations in

Set.3arebasedontheexhangeratesbetweenUSDollars

anduros.Besidesforeastingthetrendline,wepredit

• ^{the position of the future rate w.r.t. the foreasted}

trendline,

• ^{thestandarddeviationw.r.t.theforeastedtrendline.}

Thoseresultsmightleadtoanewunderstandingofvolatil-

ityandriskmanagement.

5

Set.4onludeswithashort

disussiononthenotionoftrend.

3

Seethe numerousexamplesandthe referenesin(Fliess&Join,

2008)foronreteillustrations.

4

Compare with (Markovsky, Willems, van Huel, de Moor &

Pintelon,2005).

5

See(Taleb,1997)foraritialappraisaloftheexistingliterature

onthissubjet,whihisofutmostimportaneinnanialengineer-

ing.(Extreme)risksaredisussedin(Bouhaud&Potters,1997;Da-

orogna,Gençay,Müller,Olsen&Pitet,2001;Mandelbrot&Hud-

son,2004;Sornette,2003)fromquitedierentperspetives.Itisthe

trendlinewhihwouldexhibitabrupthangesinoursetting(ompare

2. PARAMETERIDENTIFICATION

2.1 Rationalgenerating funtions

Consideragain Eq.(1). TheZ^-transformX ^ofx^satises

(see,e.g.,(Doetsh,1967;Jury,1964))

zⁿ[X−x(0)−x(1)z⁻¹− · · · −x(n−1)z⁻⁽ⁿ⁻¹⁾]

− · · · −an−1z[X−x(0)]−anX= 0 ⁽⁷⁾

It showsthatX^{, whih is alled thegenerating funtion}

ofx^{, isarationalfuntionof}z^,i.e., X∈R(z)^: X =P(z)

Q(z) ⁽⁸⁾

where

P(z) =b0zⁿ⁻¹+b1zⁿ⁻²+· · ·+bn−1∈R[z]

Q(z) =zⁿ− · · · −an−1z−an∈R[z]

Hene

Proposition2.1. x(t)^, t ≥ 0^{, satises a linear dierene}

equation(1)if,andonlyif,itsgeneratingfuntion X ^isa

rationalfuntion.

It is obvious that the knowledge of P ^and Q ^{permits to}

determinetheinitialonditionsx(0), . . . , x(n−1)^.

Remark2.2. Considertheinhomogeneouslineardierene

equation

x(t+n)−a1x(t+n−1)− · · · −anx(t)

= X

nite

̟(t)α^t+ X

nite

̟^′(t) sin(ωt+ϕ)

where ̟(t), ̟^′(t) ∈ R[t]^, α, ω, ϕ ∈ R^{. Then the} Z^-

transformX ∈R(z) ^ofx(t)^{is againrational. It isequiv-}

alentsayingthatx(t)^,t≥0^{, stillsatisesa}homogeneous diereneequation.

2.2 Parameteridentiability

Generalities Let

K=Q(a1, . . . , an, b0, . . . , bn−1)

be the eld generated over the eld Q ^{of rational num-}

bers by a1, . . . , an, b0, . . . , bn−1^{, whih are onsidered as}

unknownparametersandthereforeinouralgebraisetting

as independent indeterminates (Fliess & Sira-Ramírez,

2003; Fliess, Join & Sira-Ramírez, 2004; Fliess & Sira-

Ramírez, 2008). Write K¯ the algebrailosure of K (see, e.g.,(Lang,2002;Chambert-Loir,2005)).ThenX ∈K¯(z)^,

i.e., X ^{is arational funtion over} K¯. MoreoverK¯(z)^{is a}

dierential eld (see, e.g., (Chambert-Loir, 2005)) with

respet to the derivation

d

dz^{. Its subeld of onstants is}

thealgebraiallylosedeld K¯.

IntroduethesquareWronskianmatrixM^oforder2n+ 1

(Chambert-Loir,2005)whereitsχ^th-row,0≤χ≤2n^,is d^χ

dz^χzⁿX, . . . , d^χ dz^χX, d^χ

dz^χzⁿ⁻¹, . . . , d^χ

dz^χ1 ⁽⁹⁾

It followsfrom Eq. (7) that the rank of M ^is 2n ^{if, and}

only if, x ^{does not satisfy a linear dierene relation of}

orderstritlylessthann^{. Hene}

referenes therein). Our estimation tehniques permit an eient

hange-pointdetetion (Mboup,Join &Fliess,2008),whih needs

Theorem2.3. If x ^{does not satisfy a linear dierene}

equationoforderstritlylessthann^{,thentheparameters} a1, . . . , an, b0, . . . , bn−1

arelinearly identiable.

6

Identiability of the dynamis For identifying the dy-

namis, i.e., a1, . . . , an^{, without having to determine the}

initialonditionsonsiderthe(n+ 1)×(n+ 1)^Wronskian

matrixN^{,where its}µ^th-row,0≤µ≤n+ 1^,is d^n+µ

dz^n+µzⁿX, . . . , d^n+µ dz^n+µX

It is obtained by taking the X^{-dependent entries in the} n+ 1^{lastrowsoftype(9),i.e., in}disregardingtheentries dependingonb0, . . . , bn−1^.TherankofN ^isagainn^.Hene

Corollary 2.4. a1, . . . , an ^arelinearlyidentiable.

Identiability of the numerator Assume now that the

dynamis is known but not the numerator P ^{in Eq. (8).}

Weobtainb0, . . . , bn−1 ^fromtherstn^rows(9).Hene

Corollary 2.5. b0, . . . , bn−1 ^arelinearlyidentiable.

2.3 Somehintson theomputerimplementation

We proeed as in (Fliess &Sira-Ramírez, 2003; Fliess &

Sira-Ramírez,2008)andin(Fliess,Fuhshumer,Shöberl,

Shlaher & Sira-Ramírez, 2008). The unknown linearly

identiable parameters are solutions of a matrix linear

equation, the oeients of whih depend on x^{. Let}

us emphasize that we substitute to x ^{its ltered value}

thanks to a disrete-time version of (Mboup, Join &

Fliess,2009).

7

3. EXAMPLE:FORECASTING 5DAYS AHEADTHE

$-EXCHANGERATES

We are utilizing data from the European Central Bank,

depited by the blue lines in theFigures 1and 2, whih

summarizethe2400^{lastdailyexhangeratesbetweenthe}

US Dollarsandtheuros.

8

3.1 Foreastingthe trendline

In order to foreast the exhange rate 5 ^{days ahead we}

applytherulesskethedinSet.2.3andweutilizealinear

diereneequation(1)oforder3^{(thelteredvaluesofthe}

exhangeratesaregivenbytheblaklinesintheFigures1,

2). Fig.3providestheestimated valuesoftheoeients

of the dierene equation. The results on the foreasted

valuesoftheexhangeratesaredepitedbytheredlinesin

theFigures1and2,whihshouldbeviewedasapredited

trendline.

6

It means following the terminology of (Fliess & Sira-Ramírez,

2003; Fliess & Sira-Ramírez, 2008) that a1, . . . , an, b0, . . . , bn−1

are uniquely determined by a system of 2n ^{linear equations, the}

oeientsofwhihdependon

d^χ

dz^χX ^and _dz^dχχz^m,0≤m≤n−1^.

7

See,e.g.,(Gençay,Selçuk&Hafner,2002)foranexellentpresen-

tationofvariouslteringtehniquesineonomisandnane.

8

Theauthors areperfetly aware thatonlyomputationsdealing

withhighfrequenydata mightbeofpratialvalue.Thistypeof

resultswillbepresentedelsewhere.

0 500 1000 1500 2000 2500

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Samples

Figure 1.Exhange rates(blue ),ltered signal(blak-

-),foreastedsignal(5 ^{daysahead)(red)}

0 500 1000 1500 2000 2500

−3

−2

−1 0 1 2 3

Samples

Figure 3. Parameter estimations a1 ^(red −.−^), a2 ^(blue

−−^)anda3^(blak−−⁾⁽5 ^daysahead)

3.2 Above orunder the preditedtrendline?

Consideragaintheerror ν(t)^{in Eq. (2)andits moving}

averageMAν,N(t)^{in Eq. (6). ForeastingMA}ν,N(t) ^asin

Set.3.1tellsusanexpetedposition withrespettothe

foreastedtrendline. The blue line of Fig. 4 displaysthe

resultfor the window size N = 100^{.The meaningof the}

indiators△^and∇^{is lear.}

Table1omparesforvariouswindowsizesthesignsofthe

preditedvaluesofMAν,N(t)^{,whih tellsusifoneshould}

expet to beaboveorunder thetrendline, with the true

positionsofx(t)^{withrespettothetrendline.Theresults}

areexpressedviaperentages.

3.3 Preditedvolatility w.r.t.the trendline

Introduethemoving standarddeviation

MSTDν,N(t) =

s PN

τ=0(ν(t+τ)−^MA_ν,N(t−N+τ))² N+ 1

1050 1100 1150 1200 1250 1300 1350 1400 1450 1.00

1.05 1.10 1.15 1.20 1.25 1.30 1.35

Samples

Figure2.ZoomofFigure1

0 500 1000 1500 2000 2500

−0.03

−0.02

−0.01 0.00 0.01 0.02 0.03

Samples

Figure 4. Predited position w.r.t. the trendline (5 ^days

ahead)∇^:above,△^:under

Window'ssize Perentage

50 65.6%

100 88.3%

200 62.3%

300 67.1%

Table 1. Comparison between the sign of the

preditedvalueofMAν,N(t)^{andthetrueposi-}

tionofx(t)^{w.r.t.thetrendline(}5^daysahead).

and foreast it as in Set. 3.2. The results, whih are

displayedforawindowsize N = 100 ^{in Table 2and Fig.}

5viathefamiliarondeneintervals,

9

onrmthetime-

dependeneofthevariane,i.e.,theheterosedastiity.

9

There is of ourse no need for the underlying statistis to be

Condeneintervals Predition Real

mean-3×^std,mean+3×^std 99% 98.7%

mean-2×^std,mean+2×^std 95% 92.2%

mean-std,mean+std 68% 64.4%

Table2.Condeneintervalvalidations(5^days

ahead)

0 500 1000 1500 2000 2500

−0.20

−0.15

−0.10

−0.05 0.00 0.05 0.10 0.15 0.20

Samples

Figure5.Condeneinterval(95%⁾⁽5^daysahead)

3.4 Forasting10 days ahead

Figures 6, 7, 8, 9 display the same type of results as

in Setions 3.1, 3.2, 3.3 via similar omputations for a

foreasting 10 ^{days ahead. The quality of the omputer}

simulationsonlyslightlydeteriorates.

quantitieslikeskewnessand kurtosis,whihwouldbeobtained by

1050 1100 1150 1200 1250 1300 1350 1400 1450 1.00

1.05 1.10 1.15 1.20 1.25 1.30 1.35

Samples

Figure7.ZoomofFigure6

0 500 1000 1500 2000 2500

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Samples

Figure 6.Exhange rates(blue ), lteredsignal(blak -

-),foreastedsignal(red) (10^daysahead)

4. CONCLUSION

Theexisteneoftrends,whihis

• ^{thekeyassumptionin tehnialanalysis,10}

• ^{quite foreign, to the best of our knowledge, to the}

aademimathematialnane,wheretheparadigm

of random walks is prevalent (see, e.g., (Wilmott,

2006)),

10

Trendsintehnialanalysisshouldnotbeonfusedwithwhatare

alledtrendsinthetimeseriesliterature (see,e.g.,(Gouriéroux &

0 500 1000 1500 2000 2500

−0.03

−0.02

−0.01 0.00 0.01 0.02 0.03

Samples

Figure 8.Predited position w.r.t. thetrendline (10 ^days

ahead)∇^:above,△^{: under}

isfundamentalinourapproah.Atheoretialjustiation

willappearsoon(Fliess &Join,2009).

11

Wehopeitwill

leadto asoundfoundation oftehnialanalysis,

12

whih

will bring asa byprodut easily implementable real-time

omputerprograms.

13

11

Theexisteneoftrendsdoesnotneessarilyontraditarandom

harater(see(Fliess&Join,2009)fordetails).

12

Seealso(Daorogna,Gençay,Müller,Olsen&Pitet,2001)fora

mostexiting studywhihemployshighfrequenydata.Thereare

alsoothertypesofattemptstoputtehnialanalysisonarmbasis

(see,e.g.,(Lo,Mamaysky&Wang, 2000)).See(Blanhet-Salliet,

Diop, Gibson, Talay & Tanré, 2007) for a omparison between

tehnial analysis and model-based approahes with parametri

unertainties.

13

Ourtehnisalreadyleadtosuhomputerprogramsinautomati

0 500 1000 1500 2000 2500

−0.20

−0.15

−0.10

−0.05 0.00 0.05 0.10 0.15 0.20

Samples

Figure9.Condeneinterval(95%⁾⁽10^daysahead)

Aknowledgement.TheauthorswishtothankG.Daval-Lelerq

(Soiété Générale - Corporate & Investment Banking) for helpful

disussionsandomments.

REFERENCES

Aronson D. (2007). Evidene-Based Tehnial Analysis.

Wiley.

Béhu T., Bertrand E., Nebenzahl J. (2008). L'analyse

tehnique(6

e

éd.).Eonomia.

Blanhet-SallietC.,DiopA.,GibsonR.,TalayD.,Tanré

E.(2007).Tehnialanalysisomparedtomathemat-

ial models based methods under parameters mis-

speiation.J. BankingFinane,31, 13511373.

BouhaudJ.-P.,PottersM.(1997).Théoriedesrisques-

naniers.Eyrolles.Englishtranslation(2000):Theory

ofFinanial Risks.CambridgeUniversityPress.

BoxG.E.P.,JenkinsG.M.,Reinsel,G.(1994).TimeSeries

Analysis: Foreastingand Control (3

rd

ed.). Prentie

Hall.

Chambert-Loir A. (2005). Algèbre orporelle. Édit. Éole

Polytehnique. English translation (2005): A Field

GuidetoAlgebra.Springer.

Daorogna M.M., Gençay R., Müller U., Olsen R.B.,

PitetO.V.(2001).AnIntrodutiontoHighFrequeny

Finane.Aademi Press.

Doetsh G. (1967). Funktionaltransformationen. In R.

Sauer,L.Szabó(Eds):MathematisheHilfsmitteldes

Ingenieurs,1.Teil,pp.232484.Springer.

DurlaufS.N.,PhilipsP.C.B.(1988).Trendsversusrandom

walksintimeseriesanalysis.Eonometria,56,1333

1354.

Fliess M. (2008). Critique du rapport signal à bruit en

ommuniationsnumériques.ARIMA,toappear.On-

linehttp://hal.inria.fr/inria- 003 1171 9/en /.

Fliess M., Fuhshumer S., Shöberl M., Shlaher K.,

Sira-RamírezH.(2008).Anintrodutiontoalgebrai

disrete-time linear parametri identiation with a

onrete appliation. J. europ. syst. automat., 42,

211232.

Fliess M., Join C. (2008). Commande sans modèle et

ommande à modèle restreint, e-STA, 5. Online

http://hal.inria.fr/inria -00 2881 07/e n/.

Fliess M., Join C. (2009). A mathematial proof of the

Fes,Maroo.Soononlinehttp://hal.inria.fr/INRIA.

FliessM.,JoinC.,Sira-RamírezH.(2004).Robustresidual

generation for linear fault diagnosis: an algebrai

settingwithexamples.Int.J.Control,77,12231242.

FliessM.,JoinC.,Sira-RamírezH. (2008).Non-lineares-

timationiseasy.Int.J.ModellingIdentiationCon-

trol,4,1227.

FliessM.,Sira-RamírezH.(2003).Analgebraiframework

forlinearidentiation.ESAIMControlOptimiz.Cal-

ulus Variat.,9,151168.

Fliess, M., Sira-Ramírez, H. (2004). Reonstruteurs

d'état. C.R.Aad.Si. ParisSer. I,338,9196.

FliessM.,Sira-RamírezH.(2008).Closed-loopparametri

identiation for ontinuous-time linear systems via

new algebrai tehniques. In H. Garnier, L. Wang

(Eds):Identiation ofContinuous-timeModels from

SampledData,pp.363391,Springer.

GençayR.,SelçukF.,WhitherB.(2002).AnIntrodution

to Wavelets and Other Filtering Methods in Finane

andEonomis.AademiPress.

Gouriéroux C., Monfort A. (1995). Séries temporelles

et modèles dynamiques (2

e

éd.). Eonomia. English

translation(1996):TimeSeriesandDynamiModels.

CambridgeUniversityPress.

Hamilton J.D. (1994). Time Series Analysis. Prineton

UniversityPress.

KaufmanP.J.(2005).New Trading Systemsand Methods

(4

th

ed.).Wiley.

Jury E.I. (1964). Theory and Appliation of the z^-

Transform Method.Wiley.

Kirkpatrik C.D.,Dahlquist J.R.(2006). Tehnial Anal-

ysis: The Complete Resoure for Finanial Market

Tehniians.FTPress.

LangS. (2002).Algebra(3

rd

rev.ed.).Springer.

LoA.W., MamayskyH., Wang J. (2000).Foundationsof

tehnial analysis: omputational algorithms, statis-

tial inferene, and empirial implementation. J. Fi-

nane,55,1705-1765.

MandelbrotB.B.,HudsonR.L.(2004).The(Mis) Behav-

ior of Markets.BasiBooks.

Markovsky I., Willems J.C., van Huel S., de Moor B.,

Pintelon R. (2005). Appliation of strutured total

least squares for systemidentiation and model re-

dution. IEEE Trans. Automat. Control, 50, 1490-

1500.

Mboup M., Join C., Fliess M. (2008). A delay estima-

tion approah to hange-point detetion. Pro. 16

th

Medit. Conf. Control Automation, Ajaio. Online

http://hal.inria.fr/in ria- 0017 977 5/en /.

MboupM.,JoinC.,FliessM.(2009).Numerialdierenti-

ationwithannihilatorsinnoisyenvironment.Numer.

Algor.,DOI:10.1007/s11075-008-9236-1.

Murphy J.J. (1999). Tehnial Analysis of the Finanial

Markets(3

rd

rev.ed.).NewYorkInstituteofFinane.

Paulos J.A. (2003). A Mathematiian Plays the Stok

Market. BasiBooks.

Sornette D. (2003). Why Stok Markets Crash: Critial

EventsinComplexFinanialSystems.PrinetonUni-

versityPress.

TalebN.(1997).DynamiHedging.Wiley.

WilmottP.(2006).PaulWilmottonQuantitativeFinane

(2

nd

ed.,3volumes).Wiley.