Stock market volatility: An approach based on Tsallis entropy

arXiv:0809.4570v1 [q-fin.ST] 26 Sep 2008 Stock market volatility: An approach based on Tsallis entropy S´onia R. Bentes1∗, Rui Menezes2, Diana A. Mendes2 1ISCAL, Av. Miguel Bombarda, 20, 1069-035 Lisboa, Portugal 2ISCTE, Av. Forcas Armadas, 1649-025 Lisboa, Portugal. Abstract One of the major issues studied in finance that has always intrigued, both scholars and practitioners, and to which no unified theory has yet been dis- covered, is the reason why prices move over time. Since there are several well-known traditional techniques in the literature to measure stock market volatility, a central point in this debate that constitutes the actual scope of this paper is to compare this common approach in which we discuss such popular techniques as the standard deviation and an innovative methodology based on Econophysics. In our study, we use the concept of Tsallis entropy to capture the nature of volatility. More precisely, what we want to find out is if Tsallis entropy is able to detect volatility in stock market indexes and to compare its values with the ones obtained from the standard deviation. Also, we shall mention that one of the advantages of this new methodology is its ability to capture nonlinear dynamics. For our purpose, we shall basically fo- cus on the behaviour of stock market indexes and consider the CAC 40, MIB 30, NIKKEI 225, PSI 20, IBEX 35, FTSE 100 and SP 500 for a comparative analysis between the approaches mentioned above. PACS (2008): 87.23.Ge, 89.65.Gh, 89.70.Cf, 89.90.+n Keywords: Stock market volatility; standard deviation; nonlinear dynamics; Tsallis entropy; econophysics ∗E-mail: soniabentes@clix.pt 1 Introduction In the last few years there has been an increasing debate on the subject of stock market volatility. In spite of its present relevance, this is not an entirely new issue and has emerged in a systematic way when Shiller [1] first argued that the observed stock market volatility was inconsistent with the predictions of the present value models, quite popular in the past. Moreover, Grossman and Shiller [2] found out that the intemporal variation appeared to be inex- plicably high and could not be rationalized even in models with a stochastic discount factor. Even though some authors questioned the conclusion of exces- sive volatility, like Flavin [3] or Kleidon [4], latter tests accounting for dividend nonstationarity and small sample bias continued to lend support to Shiller’s initial claim (see Refs. [5], [6], [7], [8], [9]). A new insight into this was brought by Schwert [10], who asked the seminal question ”Why does stock market volatility change over time?”, having reached the conclusion that only a small amount of fluctuations could be explained by models of stock valuation. In this light, many other studies have appeared with the aim of studying every single aspect of stock market volatility, giving rise to an intense debate on the theme. Recognizing its relevance, Daly [11] summarizes some of the major reasons pointed out for its study: (i) Firstly, when market exhibits an excess volatility, investors may find it difficult to explain it based only upon the in- formation about the fundamental economic factors. As a result an erosion of confidence and a reduced flow of capital into equity markets may occur. (ii) Secondly, for firms individually considered, volatility is an important factor in determining the probability of bankruptcy. The higher the volatility for a given capital structure, the higher the probability of default. (iii) Thirdly, volatility is also an important factor in determining the bid-ask spread. So, the higher the volatility of the stock the wider will be the spread between bid and ask prices, thus affecting the market liquidity. (iv) Fourthly, hedging techniques such as portfolio insurance are affected by the volatility level, with the prices of insurance increasing with volatility. (v) Fifthly, if consumers are risk averse, as the financial theory suggests, an increase in volatility will therefore imply a reduction in economic activity with adverse consequences for investment. (vi) Finally, increased volatility over time may induce regulatory agencies and providers of capital to force firms to allocate a larger percentage of available capital to cash equivalent investments, to the potential detriment of allocation efficiency. In this brief overview we have tried to shed some light on the theme and to unfold some of its major implications. Nevertheless, given the impracticability of analyzing volatility as a whole we focus on its particular aspect of measure- ment. Here, however we face an obstacle: since volatility is not observed, there has been no agreement on how to measure it, thus emerging a plethora of techniques. Another conclusion that appeared to have arisen is that volatility 2 is volatile. The main contribution of this paper is to compare two different approaches: one based on the statistical measure of the standard deviation or variance and another cen

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