A. ShapovalA. Shapoval1,21,2,, V. Gisin V. Gisin11, , V V.. Popov Popov1,3,41,3,4

1. Finance 1. Finance academy under academy under the government the government

of the RFof the RF

2. International 2. International institute of institute of earthquake earthquake

prediction thooryprediction thoory

3. Moscow State 3. Moscow State UniversityUniversity

4. Space 4. Space research research instituteinstitute

Super-exponential trends as Super-exponential trends as the precursors of crashesthe precursors of crashes

Are crises predictable?

Scheme of actions:Scheme of actions:

1. To detect the indicators of crises.1. To detect the indicators of crises.

2. To construct the prediction algorithms 2. To construct the prediction algorithms involving these indicators.involving these indicators.

Super-exponential growthSuper-exponential growth

• Tulips mania in HollandTulips mania in Holland• Demographical growth up to the middle Demographical growth up to the middle

of the previous centrury. of the previous centrury. • Boom in the 1920th on the American Boom in the 1920th on the American

stock marketstock market

Theoretical background• The absence of the bubbles under the

restrictive assumtions about rationality of the agents (Tirole, 1982).

• The bubbles exist under weaker assumptions:– De Long B. et al., 1990. Irrational agents– Weil, 1987, The bubbles because of the

beliefs in them– Allen & Gorton, 1993. Groups with

different information → the bubbles

Implicit detection of the bubbles

• West, 1987. Two ways to calculate some characteristics of the data. They have to coincide if the bubbles are absent. The West procedure tests the standard present value model against an unspecified alternative which is interpreted as having arisen from a speculative bubble.

• Wu, 1997, estimates the bubbles using the Kalman filter

Explicit detection of the bubbles

• Idea: to formulate a model equation for the the bubbles

Hypothesis. Hypothesis. Super-exponential growth Super-exponential growth

(speculative bubbles) (speculative bubbles) preceeds the crashespreceeds the crashes

Specification.Specification. Log-periodic oscillationsLog-periodic oscillations

3

1 2

1 cos log( )

c

mc

t tC

TP t C Ct t

Evolutionary equation with a positive Evolutionary equation with a positive feedbackfeedback

Due to he special arrangements of the terms there exists the Due to he special arrangements of the terms there exists the filter mapping the data into the normal sample!filter mapping the data into the normal sample!

It gives a criterion of the model adequacyIt gives a criterion of the model adequacy

2 11 2

m m mdB B B dt B dw hdj B

m>1, w(t) – the Wiener processm>1, w(t) – the Wiener process, , dj =dj = 0 or 1 0 or 1

(Sornette, 02)

New modelNew model

The solution is derived analytically!The solution is derived analytically!

mdB B dt Bdw hdj B

2

2

0 5( 1) ( 1)

0 5( 1) ( 1)

0

1

( ) ( )

( 1)

0, ( ) ( )

s

m m w

m s m w

m

Y t Y t e

m e ds

t Y t B t

EvaluationEvaluation

• RegressionsRegressions• Pattern recognitionPattern recognition Gel'fandGel'fand, Guberman, Keilis-Borok, , Guberman, Keilis-Borok,

KnopoffKnopoff,, Press, Press, Ranzman, Rotwain Ranzman, Rotwain SadovskySadovsky (1976) (1976)

Pattern recognition. IDEA

• To find a pattern that preceeds the events-to-predict but rarely occurs during «ordinary intervals»

• To construct a prediction algorithm involving this pattern

Prediction algorithm of any Prediction algorithm of any nature divides the time axisnature divides the time axis

into the intervals of two sorts:into the intervals of two sorts:(1) the alarm is announced (the (1) the alarm is announced (the

event-to-predict is expected);event-to-predict is expected);(2) the alarm is not announced.(2) the alarm is not announced.

Prediction efficiency

Error diagramError diagram (Molchan, 1991)

• n and n and are the rate of the are the rate of the failure-to-predict and the failure-to-predict and the alarm ratealarm rate

• The complement startegy The complement startegy declares the alarm if A does declares the alarm if A does not declarenot declare

• A is better thanA is better than B, A andB, A and CC cannot be compared until cannot be compared until the goal function is the goal function is introducedintroduced

• The goal function: The goal function: = = n + n +

Prediction of the daily falls of DJI and HS

• The alarm of a fixed duration T is declared immediately after the crash

• The red markers are the real prediction

• The black markers correspond to changes of T

PrecursorPrecursort t the collection of the sliding windowsthe collection of the sliding windows [t, t-w[t, t-wii ), i), iII

ddii – the deviation of the solution from the – the deviation of the solution from the data on [t, t-wdata on [t, t-wii ), ),

A(t) = #(dA(t) = #(dii (t) < d*)(t) < d*)

A(t) > A*A(t) > A* bubbles bubbles

bbA,N A,N (t) – the trend of(t) – the trend of А on А on [t, t-N[t, t-N ))

bbX,N X,N (t) – the trend of(t) – the trend of XX on on [t, t-N[t, t-N ))

Either Either bbA,N A,N (t)<0, or(t)<0, or b bX,N X,N (t)<0(t)<0

the bubbles the bubbles the alarm the alarm

A «calm A «calm period»period»

the the bubblesbubbles

A(t) > A*A(t) > A*

bbA,N A,N (t)<0(t)<0 or or bbX,N X,N (t)<0(t)<0

the alarmthe alarm

Crash occurred or Crash occurred or alarm was declared alarm was declared T T days agodays ago

HS: Dec 86 – Nov 08HS: Dec 86 – Nov 08

HS: Dec 86 – Nov 08HS: Dec 86 – Nov 08

HS: Dec 91 – Dec 97HS: Dec 91 – Dec 97

DJI: Oct 28 – Dec 08DJI: Oct 28 – Dec 08 n + n + =0.41=0.41, the parameters are fixed, the parameters are fixed

ResultsResults

• The losses The losses [0.4, 0.5] are stable with [0.4, 0.5] are stable with respect to the parameters of the respect to the parameters of the algorithm.algorithm.

• The bubbles are usually identified directly The bubbles are usually identified directly before the end of the growth.before the end of the growth.

• Just a part of ascendent trends identified Just a part of ascendent trends identified as the bubbles end with a crash.as the bubbles end with a crash.

ConclusionConclusion

• The prediction efficiency is well estimated by The prediction efficiency is well estimated by the error diagram.the error diagram.

• The algorithm which predicts crashes The algorithm which predicts crashes following the booms is evaluated following the booms is evaluated

• The size of the fall following the boom has a The size of the fall following the boom has a significant random component.significant random component.

Thank you!Thank you!

DJI: Oct 86 – Nov 87DJI: Oct 86 – Nov 87