Stabilizing effect of volatility in financial markets

Davide Valenti, Giorgio Fazio, and Bernardo Spagnolo

Phys. Rev. E 97, 062307 – Published 8 June 2018

Abstract

In financial markets, greater volatility is usually considered to be synonymous with greater risk and instability. However, large market downturns and upturns are often preceded by long periods where price returns exhibit only small fluctuations. To investigate this surprising feature, here we propose using the mean first hitting time, i.e., the average time a stock return takes to undergo for the first time a large negative (crashes) or positive variation (rallies), as an indicator of price stability, and relate this to a standard measure of volatility. In an empirical analysis of daily returns for 1071 stocks traded in the New York Stock Exchange, we find that this measure of stability displays nonmonotonic behavior, with a maximum, as a function of volatility. Also, we show that the statistical properties of the empirical data can be reproduced by a nonlinear Heston model. This analysis implies that, contrary to conventional wisdom, not only high, but also low volatility values can be associated with higher instability in financial markets. This proposed measure of stability can be extremely useful in risk control.

Received 28 April 2017
Revised 11 May 2018

DOI:https://doi.org/10.1103/PhysRevE.97.062307

Physics Subject Headings (PhySH)

Brownian motion Econophysics Fluctuations & noise Stochastic processes

Interdisciplinary PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Davide Valenti^1,2,*, Giorgio Fazio^3,4,†, and Bernardo Spagnolo^1,5,6,‡

¹Dipartimento di Fisica e Chimica, Group of Interdisciplinary Theoretical Physics and CNISM, Università di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy
²IBIM-CNR Istituto di Biomedicina ed Immunologia Molecolare “Alberto Monroy,” Via Ugo La Malfa 153, I-90146 Palermo, Italy
³Business School, Newcastle University, 5 Barrack Road, NE1 4SE Newcastle upon Tyne, United Kingdom
⁴SEAS, Università di Palermo, I-90128 Palermo, Italy
⁵Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-90123 Catania, Italy
⁶Radiophysics Department, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia

^*davide.valenti@unipa.it
^†giorgio.fazio@ncl.ac.uk
^‡bernardo.spagnolo@unipa.it

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Issue

Vol. 97, Iss. 6 — June 2018

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Images

Figure 1
Time series of returns and the corresponding first hitting time, that is, the time it takes for a stock price return to cross for the first time a large negative threshold $Θ_{f}$ starting from an initial position $Θ_{i}$ . By ensemble averaging on all the time series of the market, we obtain the proposed indicator of price return stability.
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Figure 2
Crashes: (a) MFHT as a function of volatility, with the thresholds $Θ_{i} = - 0.1 {\bar{σ}}^{r}$ and $Θ_{f} = - 1.5 {\bar{σ}}^{r}$ , from empirical time series (blue circles) and theoretical results (red triangles), obtained from a nonlinear Heston model [Eqs. (5) and (6)]. (b) MFHT vs volatility (real data) for a fixed difference between thresholds, $Θ_{f} - Θ_{i} = - 1.4 {\bar{σ}}^{r}$ , with $[Θ_{i}, Θ_{f}]$ ranging from $[+ 0.9 {\bar{σ}}^{r}, - 0.5 {\bar{σ}}^{r}]$ to $[- 1.6 {\bar{σ}}^{r}, - 3.0 {\bar{σ}}^{r}]$ . (c) MFHT vs volatility (real data) for a fixed starting threshold $Θ_{i} = - 0.1 {\bar{σ}}^{r}$ and a different final threshold $Θ_{f}$ , ranging from $- 0.5 {\bar{σ}}^{r}$ to $- 3.0 {\bar{σ}}^{r}$ .
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Figure 3
Rallies: (a) MFHT as a function of volatility, with the thresholds $Θ_{i} = + 0.1 {\bar{σ}}^{r}$ and $Θ_{f} = + 1.5 {\bar{σ}}^{r}$ , from empirical time series (blue circles) and theoretical results (red triangles), obtained from a nonlinear Heston model. (b) MFHT vs volatility (real data) for a fixed difference between thresholds, $Θ_{f} - Θ_{i} = + 1.4 {\bar{σ}}^{r}$ , with $[Θ_{i}, Θ_{f}]$ ranging from $[- 0.9 {\bar{σ}}^{r}, + 0.5 {\bar{σ}}^{r}]$ to $[+ 1.6 {\bar{σ}}^{r}, + 3.0 {\bar{σ}}^{r}]$ . (c) MFHT vs volatility (real data) for a fixed starting threshold $Θ_{i} = + 0.1 {\bar{σ}}^{r}$ and different final thresholds $Θ_{f}$ , ranging from $+ 0.5 {\bar{σ}}^{r}$ to $+ 3.0 {\bar{σ}}^{r}$ .
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Figure 4
(a) Cubic potential used in the dynamic equation for the variable $x (t)$ . The black circle denotes the starting position ( $x_{0} = 0.0$ ) used to obtain the theoretical results. The potential parameters are $m = 2$ and $n = 3$ . (b) PDF of the first hitting times of the returns for real data (blue circles) and the model (red triangles).
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Figure 5
PDF of the stock price returns for real data (blue circles) and the model (red triangles).
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Figure 6
(a) Probability distribution of the volatility for real data (blue circles) and the model (red triangles). We consider the whole set of price returns consisting of $N_{T} = 3030 \times 1071 = 3 245 130$ values and calculate, both for real data and theoretical results, the related PDFs. (b) Correlation function of the returns for real data and the model. Inset: Correlation function of the absolute returns for real data and the model. The values of the parameters are the same as in Fig. 3.
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