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Information Measure for Financial Time Series: Quantifying Short-Term

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Information Measure for Financial Time Series: Quantifying Short-Term Information measure for financial time series: quantifying short-term market heterogeneity Linda Ponta1 and Anna Carbone2 1Universit´aCattaneo LIUC, Castellanza, Italy 2Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy A well-interpretable measure of information has been recently proposed based on a partition obtained by intersecting a random sequence with its moving average. The partition yields disjoint sets of the sequence, which are then ranked according to their size to form a probability distribution function and finally fed in the expression of the Shannon entropy. In this work, such entropy measure is implemented on the time series of prices and volatilities of six financial markets. The analysis has been performed, on tick-by-tick data sampled every minute for six years of data from 1999 to 2004, for a broad range of moving average windows and volatility horizons. The study shows that the entropy of the volatility series depends on the individual market, while the entropy of the price series is practically a market-invariant for the six markets. Finally, a cumulative information measure - the `Market Heterogeneity Index'- is derived from the integral of the proposed entropy measure. The values of the Market Heterogeneity Index are discussed as possible tools for optimal portfolio construction and compared with those obtained by using the Sharpe ratio a traditional risk diversity measure. 1. INTRODUCTION In this scenario, new tools for portfolio optimization and asset pricing should be designed on the basis of the expected volatility to evaluate market performances at Several connections between economics and statisti- microscopic level, beyond the figures of risk provided by cal thermodynamics have been suggested over the years. more traditional techniques. Marginal utility and disutility have been related respec- In this work, the information measure approach pro- tively to force, energy and work[1]. Differential pressure posed in [21{23] is applied to prices and volatilities of and volume in an ideal gas have been linked to price and tick-by-tick data, recorded from 1999 to 2004, of six fi- volume in financial systems [2]. Analogies have been put nancial markets. The investigation is performed over a forward between money utility and entropy [3] and be- broad range of volatility and moving average windows. tween temperature and velocity of circulation of money Remarkably, it is found that the entropy of the volatility [4]. As a result of the growing diversification and glob- series takes different values for the different markets as alization of economy, applications of entropy concepts to opposed to the entropy of the prices, which is mostly con- finance and economics are receiving renewed attention stant and consistent with a homogeneous random walk as instruments to monitor and quantify market diver- structure of the time series. In order to provide a com- sity [5{15]. Heterogeneity of private and institutional pact visualization of the results and a clear procedure investments, might result at a microscopic level, among for practically uses of the proposed approach, a market other effects, in imperfect and asymmetric flow of infor- heterogeneity index (MIX) is introduced, defined on the mation, ultimately undermining the theory of efficient basis of the integral of the entropy functional. MIX es- market with its assumption of a homogeneous random timates are provided for prices and volatilities of the six process underlying the stock prices dynamics [16]. financial markets, over different volatility horizons and In the last decades, the interplay of noise and prof- moving average windows. The values of the index are itability with specific focus on volatility investment finally compared with the results obtained by using the strategies has therefore gained interest and is under in- Sharpe ratio, a traditional estimate of portfolio risk. arXiv:1710.07331v2 [q-fin.ST] 19 Feb 2018 tense scrutiny [17{20]. Volatility series exhibit remark- able features suggesting the existence of some amount of `order' out of the seemingly random structure. The degree of `order' is intrinsically linked to the informa- 2. ENTROPY tion, embedded in the volatility patterns, whose extrac- tion and quantification might shed light on microscopic The definition of entropy, adopted in the framework of phenomena in finance [8{15]. The Volatility Index (VIX), this work [21{23], stem from the idea of Claude Shan- defined as the near-term volatility conveyed by S&P 500 non to quantify the expected information contained in a stock index option prices, has been suggested to monitor message extracted from a sequence xt [24] by using the investor sentiment. VIX futures and options have been functional: introduced as trading instruments. The Chicago Board Options Exchange (http://www.cboe.com/) calculates M X and updates the values of more than 25 indexes designed H[P ] = −S[P ] = − pj log pj : (2.1) to measure the expected volatility of different securities. j=1 2 with P a probability distribution function associated with the factor F (τ; n) taking the form exp(−τ=n), to with the time sequence xt. Different approaches have account for the finite size effects when τ n, resulting been proposed for the evaluation of the entropy of a ran- in the drop-off of the power-law and the onset of the dom sequence (see Refs. [25{30] for a few examples of exponential decay. entropy measures). The preliminary but fundamental By using Eq. (2.3), Eq. (2.1) writes (the details of the step is the symbolic representation of the data, through derivation can be found in [21, 23]): a partition suitable to map the continuous phase-space τ S(τ; n) = S + log τ α + ; (2.4) into disjoint sets. The method commonly adopted for 0 n partitioning a sequence is based on a uniform division where S is a constant, log τ α and τ=n are related respec- in blocks having equal size. Then the entropy is esti- 0 tively to the terms τ −α and F(τ; n). mated over subsequent partitions corresponding to dif- The constant S in Eq. (2.4) can be evaluated as fol- ferent block sizes [25]. The choice of the optimal par- 0 lows: in the limit n ∼ τ ! 1, S ! −1 consistently tition is not a trivial task, as it is crucial to effectively 0 with the minimum value of the entropy S(τ; n) ! 0 that discriminate between randomness/determinism of the en- corresponds to a fully ordered (deterministic) set of clus- coded/decoded data (see Section III of Ref.[31] for a short review of the partition methods). ters with same duration τ = 1. The maximum value of the entropy S(τ; n) = log N α is then obtained when Here, the partition is obtained by taking the intersec- n ∼ τ ! N with N the maximum length of the sequence. tion of fx g with the moving average fx g for different t et;n This condition corresponds to the maximum randomness moving average window n [21{23]. For each window n, (minimum information) carried by the sequence, when a the subsets fx : t = s; ::::; s − ng between two consec- t single cluster is obtained coinciding with the whole series. utive intersections are ranked according to their size to Since the exponent α is equal to the fractal dimension obtain the probability distribution function P . The ele- D = 2 − H with H the Hurst exponent of the time se- ments of these subsets are the segments between consec- ries, the term log τ D in Eq. (2.4) can be interpreted as utive intersections and have been named clusters (see the a generalized form of the Boltzmann entropy S = log Ω, illustration shown in Fig. 1). The present approach di- where Ω = τ D can be thought of the volume occupied by rectly yields either power-law or exponential distributed the fractional random walker. blocks (clusters), thus enabling us to separate the sets of The term τ=n in Eq. (2.4) represents an excess entropy inherently correlated/uncorrelated blocks along the se- (excess noise) added to the intrinsic entropy term log τ D quence. Moreover, the clusters are exactly defined as the by the partition process. It depends on n and is related to portions of the series between death/golden crosses ac- the finite size effect discussed above. This issue has been cording to the technical trading rules. Therefore, the in- discussed in [21], while the general issue of the finite size formation content has a straightforward connection with effects on entropy measure has been discussed in Refs. the trader's view of the price and volatility series. [32{35] For the sake of clarity, the main relationships relevant The method and Eqs. (2.1 - 2.4) have been applied for to the investigation carried in this paper will be shortly estimating the probability distribution function P (`; n) recalled in the next paragraphs. and the entropy S(`; n) of the 24 nucleotide sequences Consider the time series fxtg of length N and the mov- of the human chromosomes in [21]. It is worth noting ing average fxet;ng of length N − n with n the moving that the characteristic sizes ` and τ (respectively clus- average window. The function fxet;ng generates, for each ter length and duration) are equivalent from a statistical n, a partition fCg of non-overlapping clusters between point of view. However, from the meaning viewpoint, the two consecutive intersections of fxtg and fxet;ng. Each cluster duration τ is more suitable than the length ` when cluster j has duration: dealing with time series fxtg. Furthermore, the duration τ is particularly relevant to financial market series, due to its straightforward connection to the duration of the τj ≡ ktj − tj−1k (2.2) investment horizon via the volatility window T and the moving average window n, entering as parameters in the where the instance tj−1 and tj refer to two subsequent above calculation.
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