RISK LAB

Log-periodic power law models of asset prices Vyacheslav Arbuzov (PSNRU, Prognoz Risk Lab)

A situation in which prices for securities, especially stocks, rise far above their actual value. This trend continues until investors realize just how far prices have risen, usually, but not always, resulting in a sharp decline.

Thefreedictionary.com

About financial bubbles

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It was very difficult to definitively identify a bubbleuntil after the fact—that is, when it is bursting we confirm its existence.

Mr. Greenspan

An upward price movement over an extended range that then implodes.Charles Kindleberger, MIT

A speculative bubble exists when the price of something does not equal its market fundamentals for some period of time for reasons other than random shocks.

Professor J.Barley Rosser, James Madison University

Tulipomania

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• 1585 – 1650 Netherlands• Creating futures and options on the tulips• The fall is 100 times

Crash 1929. Dow Jones

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The crisis in October 1987. S&P 500

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The crisis in October 1997. Index Hang Seng

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The collapse of the RTS in 1997

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Crash of index NASDAQ in 2000

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Crash of index Dow Jones. 2007

Crash of index RTS in 2008

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What is common???

Log Periodic Power Law (LPPL)

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Authors

A.Johansen, O.Ledoit, D.Sornette (JLS)

First publicationLarge financial crashes (1997)

Famous bookDidier SornetteWhy Stock Markets Crash (2004)

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𝑙𝑛 [𝑝 (𝑡 ) ]=𝐴+𝐵(𝑡𝑐−𝑡)𝑚

Power law?

𝑡𝑐

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𝐶 (𝑡𝑐− 𝑡)𝑚𝑐𝑜𝑠 [𝜔𝑙𝑜𝑔 (𝑡𝑐− 𝑡 )−𝜑 ]

Log Periodic ?

LPPL = log periodic + power law

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+

What is m?

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m = 0.3m = 0.01

m = 0.9 m = 1.7

+𝑙𝑛 [𝑝 (𝑡)]

What is ?

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= 3 = 7

= 30 = 15

+

𝑡𝑐−𝑡

𝑡𝑐−𝑡

𝑡𝑐−𝑡 𝑡𝑐−𝑡

𝑡𝑐−𝑡

What is ?

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= 7 = 9.5

+

Critical time estimation

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For each log periodic curve we fixed:- start time of the bubble - critical time when bubble crash orchange to another regime

𝑡𝑐1 𝑡𝑐2

Sample of

First model

Second model

With four parameters I can fit an elephant, and with five I can make

him wiggle his trunk.John von Neumann

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Estimation of parameters

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1

2

2

ln

ln

ln

i i i

i i i i i i

i i i i i i

A N f g p

B f f g f p f

C g f g g p g

+

B f(t)+

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Splitting the tolerance values on the grid

Finding the grid parameters providing a minimum sum of squared residuals

Optimizing found on the grid parameters using the Newton-Gauss

Estimation of parameters

Various sections of the cost function

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New method for estimating the parameters

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V.Filimonov and D.SornetteA Stable and Robust Calibration Schemeof the Log-Periodic Power Law Model(29 aug 2011)

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+ +

B f(t)+

𝑚 , 𝑡𝑐 ,𝜔

New method for estimating the parameters

The most important results

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Dimensionality of the nonlinear optimization problem is reduced from a 4-dimensional space to a 3-dimensional space

The proposed modification eliminates the quasi-periodicity of the cost function due to subordination of the phase parameter as a part of and to angular log-frequency parameter .

Various sections of the cost function after transformation

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Estimation of parameters

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The procedure for estimation of parameters

0t

ct

m

ABln[ ( )]p t

Filter

+ +

𝐶1

𝐶2

Models selection

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Main filtration (0<m<1, B<0)

Residuals stationarity tests (ADF test, Phillips–Perron test)

Lomb spectral analysis (or )

0 10 20 30 40

05

01

00

15

0

LOMB PERIODOGRAM

omega

P(o

me

ga

)

m

Lomb spectral analysis

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𝜏= 12𝜔

arctan (∑𝑗

𝑠𝑖𝑛 2𝜔𝑡 𝑗

∑𝑗

𝑐𝑜𝑠2𝜔𝑡 𝑗

)

The evolution of the bubble …

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The Crash Lock-In Plot (CLIP)

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D.Fantazzini, P.Geraskin, Everything You Always Wanted to Know about Log Periodic Power Laws for Bubble Modelling but Were Afraid to Ask (2011)

The Crash Lock-In Plot (CLIP) for MICEX

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P.S. prediction of the date of avalanches

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The practical task № 7. Estimate LPPL model

Commands to help :

help(nsl)

TASK :a. Download Index Data(ticker: “MICEX”) from 2001 to 2009b. Estimate parameters of model LPPL

MODEL LPPL:

+

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Thank you for your attention!