# Economics and Finance: q-Statistical Stylized Features Galore

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## Abstract

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## 1. Introduction

## 2. Applications in Economics and Finance

#### 2.1. Prices and Volumes

#### 2.2. Volatilities

#### 2.3. Inter-Occurrence Times

#### 2.4. Wealth

## 3. Conclusions and Perspectives

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Empirical return densities (points) and q-Gaussians (solid lines) for normalized returns of the 10 top-volume stocks in the NYSE and in NASDAQ in 2001. The dotted line is a (visibly inadequate) Gaussian distribution. The 2- and 3-min curves are moved vertically for display purposes. From [103]. There exist in the literature quite a few other such examples, for other stocks and other years, with similar values of q.

**Figure 2.**Empirical distributions (points) and q-exponential-like curves (solid lines) for normalized volumes of the 10 top-volume stocks in the NYSE and in the NASDAQ in 2001. The solid lines are fittings with a q-exponential multiplied by a power-law (analogous to the density of states prefactor that typically emerges for the distributions of quasi-particles in, say, condensed matter physics); from [103]. There exist in the literature quite a few other such examples, for other stocks and other years, with similar values of q and of the rest of the fitting indices.

**Figure 3.**Implied volatilities as a function of the strike price for call options on JY currency futures, traded on 16 May 2002, with 147 days left to expiration. In this typical example, the current price of a contract on Japanese futures is $79, and the risk-free rate of return is $5.5\%$. Circles correspond to volatilities implied by the market, whereas triangles correspond to volatilities implied by our model with $q=1.4$ and $\sigma =10.2\%$. The dotted line is a guide to the eye. From [103].

**Figure 4.**Illustrations of the q-log-normal density for $\mu =0$ and $\sigma =1$: blue $q=5/4$, red $q=1$ and green $q=4/5$. From [108].

**Figure 5.**Probability density function of a five-day volatility vs. $\mathcal{B}$. The symbols are obtained from the data, and the lines are the best fits with the Gamma distribution (dashed green) and the double-sided q-log-normal (red) with $\mu =0.391$, $\sigma =1.15$ and $q=1.22$. For further details, see [108].

**Figure 6.**Illustration of the relative daily price returns ${X}_{i}$ of the IBM stock between (

**a**) January 2000 and June 2010 and (

**b**) 27 August and 23 October 2002. The red line shows the threshold $Q\simeq -0.037$, which corresponds to an average inter-occurrence time of ${R}_{Q}$ = 70. In (b), the inter-occurrence times are indicated by arrows. From [106].

**Figure 7.**The mean inter-occurrence time ${\mathrm{R}}_{Q}$ vs. the absolute value of the loss threshold −Q. The continuous curves are fittings with ${\mathrm{R}}_{Q}=A{e}_{{q}_{inter}}^{{B}_{inter}\left|Q\right|}=A[1+(1-{q}_{inter}){B}_{inter}{\left|Q\right|]}^{1/(1-{q}_{inter})}$. Top left: For the exchange rate of the U.S. Dollar against the British Pound, the index S&P500, the IBM stock and crude oil (West Texas Intermediate (WTI)), from left to right in the plot; the corresponding values for ${q}_{inter}$ are 0.95, 0.92, 0.97, 0.927 (with $A=2,\phantom{\rule{0.166667em}{0ex}}2.04,\phantom{\rule{0.166667em}{0ex}}1.95,\phantom{\rule{0.166667em}{0ex}}2.02$ and ${B}_{inter}=240,\phantom{\rule{0.166667em}{0ex}}175,\phantom{\rule{0.166667em}{0ex}}95,\phantom{\rule{0.166667em}{0ex}}60$). Similarly for the top right, bottom left and bottom right plots. From [107].

**Figure 8.**The mean inter-occurrence time ${\mathrm{R}}_{Q}$ versus the absolute value of the loss threshold −Q: ${ln}_{{q}_{inter}}$$({\mathrm{R}}_{Q}/A)$ versus the ${\mathrm{B}}_{inter}|-Q|$ representation of the same data of Figure 7. The continuous curve is a fitting with ${\mathrm{R}}_{Q}=A{e}_{{q}_{inter}}^{{B}_{inter}|-Q|}$. From [107].

**Figure 9.**(

**a**) The distribution function of the inter-occurrence times for the relative daily price returns ${X}_{i}$ of IBM in the period 1962–2010. The data points belong to ${R}_{Q}$ = 2, 5, 10, 30 and 70 (in units of days), from bottom to top. The full lines show the fitted q-exponentials ${p}_{Q}\left(r\right)\propto {e}_{{q}_{threshold}}^{-{\beta}_{threshold}\phantom{\rule{0.166667em}{0ex}}r}$ for typical values of ${R}_{Q}$. (

**b**) The dependence of the parameters ${\beta}_{threshold}$ (squares, lower curve) and ${q}_{threshold}$ (circles, upper curve) on ${R}_{Q}$ in the ${q}_{threshold}$-exponential. (

**c**) Confirmation that, for ${R}_{Q}=2$, the distribution function is a simple exponential (i.e., ${q}_{threshold}=1$). The straight line is proportional to ${2}^{-r}$. From [106].

**Figure 10.**The distribution function of the inter-occurrence times (as in Figure 9a) for the relative daily price returns of 16 examples of financial data, taken from different asset classes (stocks, indices, currencies, commodities). The assets are: (i) the stocks of IBM, Boeing (BA), General Electric (GE), Coca-Cola (KO); (ii) the indices Dow Jones (DJI), Financial Times Stock Exchange 100 (FTSE), NASDAQ, S&P 500; (iii) the commodities Brent Crude Oil, West Texas Intermediate (WTI), Amsterdam-Rotterdam-Antwerp gasoline (ARA), Singapore gasoline (SING); and (iv) the exchange rates of the following currencies versus the U.S. Dollar: Danish Crone (DKK), British Pound (GBP), Yen, Swiss Francs (SWF). The full lines show the fitted q-exponentials, which are the same as in Figure 9a. From [106].

**Figure 12.**Binned inverse cumulative distribution of the county, $PI/P{I}_{0}$ (U.S.) and $GDP/GD{P}_{0}$ (Brazil, Germany and U.K.), where $PI$ denotes the Personal Income and $GDP$ denotes the Gross Domestic Product of countries. Three distributions are displayed for comparison: (i) q-Gaussian (with ${\beta}_{{q}^{\prime}}=0$) (dot-dashed); (ii) ($q;{q}^{\prime}$)-Gaussian (solid) and (iii) log-normal (dashed lines). (

**a**,

**b**) present insets with a linear-linear scale, to make more evident the quality of the fitting at the low region (in (

**c**,

**d**), the ($q;{q}^{\prime}$)-Gaussian and the log-normal curves are superposed and, so, are visually indistinguishable). The positions of the knees are indicated. The ankle is particularly pronounced in (c), though it is also present in the other cases. From [105], where further details are available.

**Figure 13.**Evolution of parameter q for the U.S. (squares), Brazil (circles), the U.K. (up triangles) and Germany (down triangles). The parameters ${q}^{\prime}$ (for each country) are constant for all years: ${q}_{Brazil}^{\prime}$ = 2.1, ${q}_{USA}^{\prime}$ = 1.7, ${q}_{Germany}^{\prime}$ = 1.5, ${q}_{UK}^{\prime}$ = 1.4. Lines are only guides to the eyes. As we verify, in some cases, the index q remains invariant along time, whereas in others, it evolves; the functional forms remain however the same as indicated in Figure 12. From [105].

**Figure 14.**Inverse cumulative probability distribution of Japanese land prices for the year 1998. The solid curve is a q-Gaussian with $q=2.136$, which corresponds to the slope $-1.76$, and $1/\sqrt{{\beta}_{q}}=$ 188,982 Yen. From [105], where further details are available.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tsallis, C.
Economics and Finance: *q*-Statistical Stylized Features Galore. *Entropy* **2017**, *19*, 457.
https://doi.org/10.3390/e19090457

**AMA Style**

Tsallis C.
Economics and Finance: *q*-Statistical Stylized Features Galore. *Entropy*. 2017; 19(9):457.
https://doi.org/10.3390/e19090457

**Chicago/Turabian Style**

Tsallis, Constantino.
2017. "Economics and Finance: *q*-Statistical Stylized Features Galore" *Entropy* 19, no. 9: 457.
https://doi.org/10.3390/e19090457