A. Shapoval 1,2, V. Gisin 1, V. Popov 1,3,4 1. Finance academy under the government of the RF 2. International institute of earthquake prediction thoory 3. Moscow State University 4. Space research institute

Super-exponential trends as the precursors of crashes

Are crises predictable?

Scheme of actions: 1. To detect the indicators of crises. 2. To construct the prediction algorithms involving these indicators.

Super-exponential growth Tulips mania in HollandTulips mania in Holland Demographical growth up to the middle of the previous centrury.Demographical growth up to the middle of the previous centrury. Boom in the 1920th on the American stock marketBoom in the 1920th on the American stock 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., Irrational agents –Weil, 1987, The bubbles because of the beliefs in them –Allen & Gorton, Groups with different information → the bubbles

Implicit detection of the bubbles West, 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. Super-exponential growth (speculative bubbles) preceeds the crashes Specification. Log-periodic oscillations

Evolutionary equation with a positive feedback Due to he special arrangements of the terms there exists the filter mapping the data into the normal sample! It gives a criterion of the model adequacy m>1, w(t) – the Wiener process, dj = 0 or 1 (Sornette, 02)

New model The solution is derived analytically!

Evaluation RegressionsRegressions Pattern recognitionPattern recognition Gel'fand, Guberman, Keilis-Borok, Knopoff, Press, Ranzman, Rotwain Sadovsky (1976) Gel'fand, Guberman, Keilis-Borok, Knopoff, Press, Ranzman, Rotwain Sadovsky (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 nature divides the time axis into the intervals of two sorts: (1) the alarm is announced (the event-to-predict is expected); (2) the alarm is not announced. Prediction efficiency

Error diagram Error diagram (Molchan, 1991) n and  are the rate of the failure-to-predict and the alarm rate n and  are the rate of the failure-to-predict and the alarm rate The complement startegy declares the alarm if A does not declareThe complement startegy declares the alarm if A does not declare A is better than B, A and C cannot be compared until the goal function is introducedA is better than B, A and C cannot be compared until the goal function is introduced The goal function:  = n + The goal function:  = 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

Precursor t  the collection of the sliding windows [t, t-w i ), i  I [t, t-w i ), i  I d i – the deviation of the solution from the data on [t, t-w i ), A(t) = #(d i (t) < d*) A(t) > A* bubbles

b A,N (t) – the trend of А on [t, t-N ) b X,N (t) – the trend of X on [t, t-N ) Either b A,N (t)<0, or b X,N (t)<0 the bubbles the alarm

A «calm period» thebubbles A(t) > A* b A,N (t)<0 or or b X,N (t)<0 the alarm Crash occurred or alarm was declared T days ago

HS: Dec 86 – Nov 08

HS: Dec 91 – Dec 97

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

Results The losses   [0.4, 0.5] are stable with respect to the parameters of the algorithm.The losses   [0.4, 0.5] are stable with respect to the parameters of the algorithm. The bubbles are usually identified directly before the end of the growth.The bubbles are usually identified directly before the end of the growth. Just a part of ascendent trends identified as the bubbles end with a crash.Just a part of ascendent trends identified as the bubbles end with a crash.

Conclusion The prediction efficiency is well estimated by the error diagram.The prediction efficiency is well estimated by the error diagram. The algorithm which predicts crashes following the booms is evaluatedThe algorithm which predicts crashes following the booms is evaluated The size of the fall following the boom has a significant random component.The size of the fall following the boom has a significant random component.

Thank you!

DJI: Oct 86 – Nov 87