Formulation of the . . . The Emergence of . . . Applications to . . . Financial Applications Log-Periodic Power Law as a Towards a General . . . Predictor of Catastrophic Analysis of the Problem Let Us Use Natural . . . Events: A New First Result and . . . Main Result Mathematical Justification Home Page Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , and Title Page Songsak Sriboonchitta 3 ◭◭ ◮◮ 1 Department of Computer Science, University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA, vladik@utep.edu Page 1 of 24 2 Department of Mathematical Sciences, New Mexico State University Las Cruces, New Mexico 88003, USA, hunguyen@nmsu.edu Go Back 3 Faculty of Economics, Chiang Mai University Chiang Mai, Thailand, songsakecon@gmail.com Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 1. Outline Applications to . . . • To decrease the damage caused by meteorological dis- Financial Applications asters, it is important to predict these disasters. Towards a General . . . Analysis of the Problem • In the vicinity of a catastrophic event, many parame- Let Us Use Natural . . . ters exhibit log-periodic power behavior. First Result and . . . • By fitting the formula to the observations, it is possible Main Result to predict the event. Home Page • Log-periodic power behavior is observed in ruptures of Title Page fuel tanks, earthquakes, stock market disruptions, etc. ◭◭ ◮◮ • In this talk, we provide a general system-based expla- ◭ ◮ nation of this law. Page 2 of 24 • This makes us confident that this law can be also used Go Back to predict meteorological disasters. Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 2. Formulation of the Problem Applications to . . . • To decrease the damage caused by meteorological dis- Financial Applications asters, it is important to predict these disasters. Towards a General . . . Analysis of the Problem • A natural idea is to see how similar disaster prediction Let Us Use Natural . . . problems are solved in other application areas. First Result and . . . • We need to predict mechanical disasters, earthquakes, Main Result financial disasters, etc. Home Page • Some predictions comes from the observation that: Title Page – in the vicinity of a catastrophic event, ◭◭ ◮◮ – many parameters exhibit so-called log-periodic power ◭ ◮ behavior, Page 3 of 24 – with oscillations of increasing frequency. Go Back • Let us therefore describe this behavior in detail. Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 3. The Emergence of Log-Periodic Power Law in Applications to . . . Disaster Prediction Financial Applications • The history of log-periodic power law applications started Towards a General . . . with space exploration. Analysis of the Problem Let Us Use Natural . . . • To be able to safely return home, a spaceship needs to store fuel. First Result and . . . Main Result • A satellite is moving at a speed of 8 km/sec, much Home Page faster than the speediest bullet. Title Page • At such a speed, a micro-meteorite or a piece of space ◭◭ ◮◮ debris can easily cause a catastrophic leak. ◭ ◮ • To avoid such a bullet-type penetration, engineers use Kevlar, bulletproof material. Tests showed that: Page 4 of 24 – while in general, Kevlar-coated tanks performed re- Go Back ally well, Full Screen – on a few occasions, the Kevlar tanks catastrophi- Close cally exploded. Quit
Formulation of the . . . The Emergence of . . . 4. The Emergence of Log-Periodic Power Law in Applications to . . . Disaster Prediction (cont-d) Financial Applications • D. Sornette noticed that: Towards a General . . . Analysis of the Problem – an explosion is usually preceded by oscillations; Let Us Use Natural . . . – their frequency increases as we approach the critical First Result and . . . moment of time t c . Main Result • He observed that the dependence of each corresponding Home Page parameter x on time t has the form Title Page x ( t ) = A + B · ( t c − t ) z + C · ( t c − t ) z · cos( ω · ln( t c − t )+ ϕ ) . ◭◭ ◮◮ ◭ ◮ • By fitting this model to the observations, we can pre- dict the moment t c of the catastrophic event. Page 5 of 24 • Sornette called the dependence (1) Log-Periodic Power Go Back Law (LPPL, for short). Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 5. Applications to Earthquake Prediction Applications to . . . • D. Sornette’s wife, A. Sauron-Sornette, is also a scien- Financial Applications tist: she is a geophysicist. Towards a General . . . Analysis of the Problem • Naturally, the two scientist spouses talk about their Let Us Use Natural . . . research. First Result and . . . • From the mechanical viewpoint, an earthquake is sim- Main Result ply a mechanical rupture. Home Page • So, they decided to check whether the log-periodic power Title Page law occurs in earthquakes. ◭◭ ◮◮ • In many cases, they observed the log-periodic power ◭ ◮ law behavior in the period preceding an earthquake. Page 6 of 24 • This technique is not a panacea: not all earthquakes can be this predicted. Go Back Full Screen • However, some can be predicted, and the ability to predict an earthquake decreases the damage. Close Quit
Formulation of the . . . The Emergence of . . . 6. Financial Applications Applications to . . . • With colleagues, D. Sornette observed similar log-periodic Financial Applications fluctuations before financial crashes. Towards a General . . . Analysis of the Problem • A similar observation was independently made by Feigen- Let Us Use Natural . . . baum and Freund. First Result and . . . • Both papers appeared in 1996 in physics journals, and Main Result were not widely understood by economists. Home Page • In Summer 1997, D. Sornette and O. Ledoit used their Title Page techniques to predict the October 1997 market crash. ◭◭ ◮◮ • By investing in put options, made a well-documented ◭ ◮ (and well-publicized) 400% profit on their investment. Page 7 of 24 • This caused attention of economists. Go Back • Now log-periodic power law predictions are important part of the econometric toolbox. Full Screen • Not all financial crashes are predictable, but some are. Close Quit
Formulation of the . . . The Emergence of . . . 7. Towards a General Explanation for Log-Periodic Applications to . . . Power Law Financial Applications • Log-periodic power law is observed in different systems. Towards a General . . . Analysis of the Problem • This seems to indicate that the this law is caused by Let Us Use Natural . . . general properties of system. First Result and . . . • Some theoretical explanations have been published. Main Result Home Page • However, these explanations are based on a very spe- cific model of a system. Title Page • It is desirable to come up with a more general expla- ◭◭ ◮◮ nation. ◭ ◮ • This would make us confident that this law can also be Page 8 of 24 used to predict meteorological disasters. Go Back Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 8. Analysis of the Problem Applications to . . . • We are interested in the dependence of quantities de- Financial Applications scribing the system on time t : x = x ( t ). Towards a General . . . Analysis of the Problem • In principle, we can have arbitrary functions x ( t ). Let Us Use Natural . . . • However, our objective is to make predictions by using First Result and . . . appropriate computer models. Main Result Home Page • In the computer, at any given moment of time, we can only represent finitely many parameters. Title Page • It is therefore reasonable to consider finite-parametric ◭◭ ◮◮ families of functions x ( t ) = f ( c 1 , . . . , c n , t ). ◭ ◮ • Usually, we know the approximate values c (0) 1 , . . . , c (0) n Page 9 of 24 of the parameters c i . Go Back Full Screen Close Quit
Formulation of the . . . The Emergence of . . . 9. Analysis of the Problem (cont-d) Applications to . . . def = c i − c (0) Financial Applications • In this case, the differences ∆ c i are small, so i Towards a General . . . we can keep only linear terms in the Taylor expansion Analysis of the Problem x ( t ) = f ( c 1 , . . . , c n , t ) = f ( c (0) 1 + ∆ c 1 , . . . , c (0) n + ∆ c n , t ) . Let Us Use Natural . . . First Result and . . . • So, x ( t ) = f 0 ( t ) + ∆ c 1 · e 1 ( t ) + . . . + ∆ c n · e n ( t ) , where = ∂f Main Result def def = f ( c (0) 1 , . . . , c (0) f 0 ( t ) n , t ) and e i ( t ) . Home Page ∂c i Title Page • Substituting ∆ c i = c i − c (0) into this formula, we get i ◭◭ ◮◮ x ( t ) = e 0 ( t ) + c 1 · e 1 ( t ) + . . . + c n · e n ( t ) , ◭ ◮ def c (0) � where e 0 ( t ) = f 0 ( t ) − · e i ( t ) . i Page 10 of 24 • In other words, the desired dependencies x ( t ) are linear Go Back combinations of the appropriate functions e i ( t ). Full Screen Close Quit
Recommend
More recommend
