Long-time fluctuations in a dynamical model of stock market indices
Bibliographic record
Abstract
Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. Recent empirical studies of stock market indices examined whether the distribution $P(r)$ of returns $r(\ensuremath{\tau})$ after some time $\ensuremath{\tau}$ can be described by a (truncated) L\'evy-stable distribution ${L}_{\ensuremath{\alpha}}(r)$ with some index $0<\ensuremath{\alpha}<~2.$ While the L\'evy distribution cannot be expressed in a closed form, one can identify its parameters by testing the dependence of the central peak height on $\ensuremath{\tau}$ as well as the power-law decay of the tails. In an earlier study [R. N. Mantegna and H. E. Stanley, Nature (London) $376,$ 46 (1995)] it was found that the behavior of the central peak of $P(r)$ for the Standard Poor 500 index is consistent with the L\'evy distribution with $\ensuremath{\alpha}=1.4.$ In a more recent study [P. Gopikrishnan et al., Phys. Rev. E $60,$ 5305 (1999)] it was found that the tails of $P(r)$ exhibit a power-law decay, with an exponent $\ensuremath{\alpha}\ensuremath{\cong}3,$ thus deviating from the L\'evy distribution. In this paper we study the distribution of returns in a generic model that describes the dynamics of stock market indices. For the distributions $P(r)$ generated by this model, we observe that the scaling of the central peak is consistent with a L\'evy distribution while the tails exhibit a power-law distribution with an exponent $\ensuremath{\alpha}2,$ namely, beyond the range of L\'evy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".