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Mathematical Statistics

Stockholm University

How to predict crashes in financial

markets with the Log-Periodic Power Law

Emilie Jacobsson

Examensarbete 2009:7

Postal address:Mathematical StatisticsDept. of MathematicsStockholm UniversitySE-106 91 StockholmSwedenInternet:http://www.math.su.se/matstat

Mathematical StatisticsStockholm UniversityExamensarbete2009:7, http://www.math.su.se/matstat

How to predict crashes in financial markets

with the Log-Periodic Power Law

Emilie Jacobsson

September 2009

Abstract

Speculative bubbles seen in financial markets, show similarities in the way they evolve and grow. This particular oscillating movement can be captured by an equation called Log-Periodic Power Law. The ending crash of a speculative bubble is the climax of this Log-Periodic oscillation. The most probable time of a crash is given by a parameter in the equation. By fitting the Log-Periodic Power Law equation to a financial time series, it is possible to predict the event ofa crash. With a hybrid Genetic Algorithm it is possible to estimate the pa- rameters in the equation. Until now, the methodology of performing these predictions has been vague. The ambition is to investigate if the financial crisis of 2008, which rapidly spread through the world, could have been predicted by the Log-Periodic Power Law. Analysisof the SP500 and the DJIA showed the signs of the Log-Periodic PowerLaw prior to the financial crisis of 2008. Even though the analyzed indices started to decline slowly at first and the severe drops came much fur- ther, the equation could predict a turning point of the downtrend. The opposite of a speculative bubble is called an anti-bubble, moving as a speculative bubble, but with a negative slope. This log-periodic oscillation has been detected in most of the speculative bubbles that ended in a crash during the Twentieth century and also for some anti- bubbles, that have been discovered. Is it possible to predict the course of the downtrend during the financial crisis of 2008, by applying this equation? The equation has been applied to the Swedish OMXS30 index, during the current financial crisis of 2008, with the result of a predicted course of the index. ?Postal address: Mathematical Statistics, Stockholm University, SE-106 91, Sweden. E-mail: emjacobsson@hotmail.com. Supervisor: Ola Hammarlid. ?????t????? ??? y(t) =A+B(tct)z+C(tct)zcos(!log(tct) + )??? n1? ????n??????? ??? ?????? ?? ???? ??????? ??? ????? ??????? ??? ??? P

D=PminPmaxP

max??? f(x) =(z xz1 z expx z?for x0

0; for x <0???

F(x) = 1expn

x zo x zo =v uutN 1nX i=1(ri+1E[r])2??? r i+1= logp(ti+1)logp(ti)???

3= 0:0456?

N(D) =AexpfbjDjzg? ?????b=z???

logN(D) = logAbjDjz??? log(log(p)) =zlog(x)zlog()???? p i= 1i0:5N ??? ???i= 1;:::;N???? ?????N?? ??? ?????? ????? f(x) =( z(xc)z1 z expn (xc) zo ????xc;???z; >0

0; for x < c????

??? ??? ?? ??? ?????? ?? ??? ??????PN ????? ?????s??????? ??? ??? ?????? ??????? ?? ?????? ??? ???? ?????? ???? ??? ??? ??????? ??????? ?? ? ??????? ?? ? ?????? ?????? ?? ????? ??Pn j=1sj? s i(t+ 1) =sign0 KnX j=1s j(t) +"i1 A K

A(KcK)

h(t) =B(tct)???? 1t c t

0h(t)dt >0????

E t[p(t0)] =p(t)? ??? ???t0> t???? dp=(t)p(t)dtp(t)dj???? ?????j??????? ? ???? ??????? ?????? ??? ????? ? ?????? ??? ????? ??? ? E t[dp] =(t)p(t)dtp(t)h(t)dt= 0???? (t) =h(t)???? dp=h(t)p(t)dth(t)dj???? p

0(t) =h(t)p(t)

p

0(t)p(t)=h(t)????

log p(t)p(t0) =t t

0h(t0)dt0

log(p(t)) = log(p(t0)) +t t

0h(t0)dt0????

t t

0h(t0)dt0=Bt

t

0(tct0)dt=B(tct0)(+1)+ 1

t0=t t 0=t= =B2tctt0+ 1 =f???t0=tc???=+ 12(0;1)g=B (tct) log(p(t)) = log(p(to))B (tct)???? ????? ?????A(KcK) Reh A

0(KcK)

+A1(KcK) +i!+:::i

A00(KcK)

+A01(KcK) cos(!log(KcK) + ):::???? h(t)Reh B

0(tct)

+B1(tct) +i!+:::i

B00(tct)

+B01(tct) cos(!log(tct) + )???? log(p(t)) = log(p(to)) B

0(tct)+B01(tct)

cos(!log(tct) + ) y=A+B(tct)z+C(tct)zcos(!log(tct) + ) ??????? ???z= 0:330:18? ?? ??[0;2]? ??? ????? ???B(tct)z? ????cos(!log(tct) + )? y(t)A+Bf(t) +Cg(t)???? f(t) =( g(t) =( 0 B @P N iy(ti)PN iy(ti)f(ti)PN iy(ti)g(ti)1 C A=0 B @NPN if(ti)PN ig(ti)PN if(ti)PN if(ti)2PN if(ti)g(ti)PN ig(ti)PN if(ti)g(ti)PN ig(ti)21 C A0 @A B C1 A X

0y= (X0X)b????

?????X=0 B @1f(t1)g(t1)

1f(tn)g(tn)1

C

A???b=0

@A B C1 A b= (X0X)1X0y???? min

F() =NX

i(y(ti)^y(ti))2;?????= (tc;;!;z)????quotesdbs_dbs17.pdfusesText_23

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