Empirical Formulas A Practice-Oriented . . . Scale Invariance: A . - PowerPoint PPT Presentation

It Is Important to Take . . . Gaussian Random . . . Mandelbrot’s Fractal . . . Empirical Analysis of . . . Empirical Formulas A Practice-Oriented . . . Scale Invariance: A . . . for Individual Stock: . . . Economic Fluctuations: Probabilistic Approach Fuzzy Approach Towards Discussion Conclusion a New Justification Title Page Tanja Magoˇ c and Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ Department of Computer Science University of Texas at El Paso Page 1 of 16 El Paso, Texas 79968, USA emails t.magoc@gmail.com, vladik@utep.edu Go Back Full Screen Close Quit

It Is Important to Take . . . Gaussian Random . . . 1. It Is Important to Take into Account Economic Mandelbrot’s Fractal . . . Fluctuations Empirical Analysis of . . . A Practice-Oriented . . . • Fact: stock prices (and other related economic indices) Scale Invariance: A . . . fluctuate in an unpredictable (“random”) way. Individual Stock: . . . • Usually: these fluctuations are small. Probabilistic Approach • Sometimes: the fluctuations become large. Fuzzy Approach Discussion • Crises: large negative fluctuations bring havoc to the Conclusion economy and finance, lead to crisis situations. Title Page • Consequence: it is therefore important to correctly take ◭◭ ◮◮ such fluctuations into account. ◭ ◮ • Problem: in particular, it is extremely important to Page 2 of 16 correctly predict the probability of large fluctuations. Go Back Full Screen Close Quit

It Is Important to Take . . . Gaussian Random . . . 2. Gaussian Random Walk Model Mandelbrot’s Fractal . . . Empirical Analysis of . . . • Pioneering work: L. Bachelier’s PhD (1900). A Practice-Oriented . . . • Formula: fluctuations of different sizes x are normally Scale Invariance: A . . . − x 2 1 � � distributed, with pdf ρ ( x ) = 2 π · σ · exp . Individual Stock: . . . √ 2 σ 2 Probabilistic Approach • Fact: The random walk model indeed describes small Fuzzy Approach fluctuations reasonably well. Discussion • Problem: this model drastically underestimates the prob- Conclusion Title Page abilities of large fluctuations: ◭◭ ◮◮ – in the normal distribution, fluctuations larger than 6 σ have a negligible probability ≈ 10 − 8 , while ◭ ◮ – in real economic systems, even larger fluctuations Page 3 of 16 occur every decade (and even more frequently). Go Back • Consequences: we thus underestimate risk – and be- Full Screen come unprepared when large fluctuations occur. Close Quit

It Is Important to Take . . . Gaussian Random . . . 3. Mandelbrot’s Fractal Model Mandelbrot’s Fractal . . . Empirical Analysis of . . . • Pioneer: Benoit Mandelbrot (1960s), the father of frac- A Practice-Oriented . . . tals. Scale Invariance: A . . . • Empirical results: medium-scale fluctuations follow the Individual Stock: . . . power-law distribution ρ ( x ) = A · x − α for α ≈ 2 . 7. Probabilistic Approach • Problem: this model drastically over estimates the prob- Fuzzy Approach ability of large-scale fluctuations. Discussion � 1 . 7 1 , hence P ( x > x 0 ) � X 0 Conclusion • Indeed: P ( x > x 0 ) ∼ . P ( x > X 0 ) ≈ Title Page x 1 . 7 x 0 0 ◭◭ ◮◮ • Daily fluctuations of ≈ x 0 = 1% are normal, with prob- ability P ≈ 1. ◭ ◮ • Thus, the probability of a crisis is Page 4 of 16 1 1 Go Back P ( x > X 0 ) ≈ 300 . 30 1 . 7 ≈ Full Screen • Consequence: we should have crises every year. Close Quit

It Is Important to Take . . . Gaussian Random . . . 4. Empirical Analysis of Economic Fluctuations Mandelbrot’s Fractal . . . (Econophysics) Empirical Analysis of . . . A Practice-Oriented . . . • Empirical result: large fluctuations are distributed with ρ ( x ) = A · x − 4 , so P ( x > x 0 ) ∼ 1 Scale Invariance: A . . . (“cubic law”). x 3 Individual Stock: . . . 0 Probabilistic Approach • Fact: in the practical financial engineering applica- Fuzzy Approach tions, this cubic law is rarely used. Discussion • Main reason: the cubic law lacks a clear theoretical Conclusion justification. Title Page • Clarification: existing explanations depend on complex ◭◭ ◮◮ math assumptions – and are not clear to economists. ◭ ◮ • Consequence: prevailing economic models mis-estimate Page 5 of 16 probabilities of large fluctuations. Go Back • Our objective: provide simpler – and hopefully more Full Screen convincing – explanations for the cubic law. Close Quit

It Is Important to Take . . . Gaussian Random . . . 5. A Practice-Oriented Temporal Reformulation of the Mandelbrot’s Fractal . . . Probabilities Empirical Analysis of . . . A Practice-Oriented . . . • Ideally: we should be able to predict when the fluctu- Scale Invariance: A . . . ations will reach a given size x 0 . Individual Stock: . . . • In reality: economic fluctuations are random (unpre- Probabilistic Approach dictable). Fuzzy Approach • Conclusion: we can only predict the average time t ( x 0 ) Discussion before such a fluctuation occurs. Conclusion Title Page • Relation to ρ ( x 0 ) : during the time period t , we have ∆ t time quanta, hence t t def ◭◭ ◮◮ N = ∆ t · ( ρ ( x 0 ) · h ) fluctuations. ◭ ◮ • Conclusion: t ( x 0 ) is when we have one fluctuation: ∆ t Page 6 of 16 t ( x 0 ) ≈ ρ ( x 0 ) · h. Go Back • Vice versa: once we find t ( x ), we get ρ ( x ) ≈ const Full Screen t ( x ) . Close Quit

It Is Important to Take . . . Gaussian Random . . . 6. Scale Invariance: A Natural Requirement Mandelbrot’s Fractal . . . Empirical Analysis of . . . • Fact: the numerical value of the fluctuation size x de- A Practice-Oriented . . . pends on the choice of a measuring unit. Scale Invariance: A . . . • Example: when European countries switched to Euros, Individual Stock: . . . all the stock prices were re-scaled x → x ′ = λ · x . Probabilistic Approach • Reasonable requirement: t ( x ) should not depend on the Fuzzy Approach choice of the unit. Discussion Conclusion • Clarification: the fluctuation of 0.1 Euros will happen Title Page faster than a fluctuation of 1 Euro. ◭◭ ◮◮ • Clarified requirement: we have to also re-scale time. ◭ ◮ • Resulting requirement: for every λ > 0, there exists a Page 7 of 16 value r ( λ ) for which, for all x and for all λ , we have Go Back t ( λ · x ) = r ( λ ) · t ( x ) . Full Screen Close Quit

It Is Important to Take . . . Gaussian Random . . . 7. Scale-Invariance Implies Power Law Mandelbrot’s Fractal . . . Empirical Analysis of . . . • Scale invariance (reminder): t ( λ · x ) = r ( λ ) · t ( x ) . A Practice-Oriented . . . • Step 1: differentiate w.r.t. λ and take λ = 1: Scale Invariance: A . . . x · dt Individual Stock: . . . def dx = α · t, where α = r ′ (1) . Probabilistic Approach Fuzzy Approach • Step 2: separate variables: Discussion dt t = α · dx x . Conclusion Title Page • Step 3: integrate: ◭◭ ◮◮ ln( t ) = α · ln( x ) + c. ◭ ◮ Page 8 of 16 • Step 4: exponentiate: Go Back t = C · x α , hence ρ ( x ) ∼ const t ( x ) ∼ x − α . Full Screen Close Quit

It Is Important to Take . . . Gaussian Random . . . 8. Individual Stock: Idealized Case Mandelbrot’s Fractal . . . Empirical Analysis of . . . • Remaining question: what is α ? A Practice-Oriented . . . • Up to now: we considered fluctuations which occur Scale Invariance: A . . . within a single time quantum ∆ t . Individual Stock: . . . • Possibility: we can consider different time quanta: e.g., Probabilistic Approach a time quantum ∆ t ′ = k · ∆ t for some integer k . Fuzzy Approach • Reminder: price fluctuations are reasonably accurately Discussion described by a random walk. Conclusion Title Page • Scaling for a random walk: fluctuation over ∆ t ′ = k · ∆ t √ ◭◭ ◮◮ is k times larger than for ∆ t . √ ◭ ◮ • When t → k · t , we get x → k · x . √ Page 9 of 16 • In terms of λ and r ( λ ), this means that when λ = k , we have r ( λ ) = k , i.e., r ( λ ) = λ 2 . Go Back • For t ( x ) = C · x α , the requirement t ( λ · x ) = r ( λ ) · t ( x ) Full Screen leads to α = 2 and ρ ( x ) ∼ x − 2 , i.e., P ( x > x 0 ) ∼ x − 1 0 . Close Quit

It Is Important to Take . . . Gaussian Random . . . 9. From the Idealized Case of an Individual Stock to Mandelbrot’s Fractal . . . Stock Market: Two Approaches Empirical Analysis of . . . A Practice-Oriented . . . • We considered the case of an individual stock which is Scale Invariance: A . . . not interacting with other stock prices. Individual Stock: . . . • In reality, stocks are inter-related: a change in one Probabilistic Approach stock price causes a change in prices of other stocks. Fuzzy Approach • How can we take this dependence into account? Discussion Conclusion • In this talk, we describe two approaches for taking this Title Page dependence into account: ◭◭ ◮◮ – a probabilistic approach, and ◭ ◮ – a fuzzy approach. Page 10 of 16 • We will show that both approaches lead to the same distribution. Go Back • This makes us even more confident that this is indeed Full Screen a correct distribution. Close Quit