Applications of Entropy in Finance: A Review

Journal of Physics: Conference Series, 2012

When uncertainty dominates understanding stock market volatility is vital. There are a number of reasons for that. On one hand, substantial changes in volatility of financial market returns are capable of having significant negative effects on risk averse investors. In addition, such changes can also impact on consumption patterns, corporate capital investment decisions and macroeconomic variables. Arguably, volatility is one of the most important concepts in the whole finance theory. In the traditional approach this phenomenon has been addressed based on the concept of standard-deviation (or variance) from which all the famous ARCH type models -Autoregressive Conditional Heteroskedasticity Models-depart. In this context, volatility is often used to describe dispersion from an expected value, price or model. The variability of traded prices from their sample mean is only an example. Although as a measure of uncertainty and risk standard-deviation is very popular since it is simple and easy to calculate it has long been recognized that it is not fully satisfactory. The main reason for that lies in the fact that it is severely affected by extreme values. This may suggest that this is not a closed issue. Bearing on the above we might conclude that many other questions might arise while addressing this subject. One of outstanding importance, from which more sophisticated analysis can be carried out, is how to evaluate volatility, after all? If the standard-deviation has some drawbacks shall we still rely on it? Shall we look for an alternative measure? In searching for this shall we consider the insight of other domains of knowledge? In this paper we specifically address if the concept of entropy, originally developed in physics by Clausius in the XIX century, which can constitute an effective alternative. Basically, what we try to understand is, which are the potentialities of entropy compared to the standard deviation. But why entropy? The answer lies on the fact that there is already some research on the domain of Econophysics, which points out that as a measure of disorder, distance from equilibrium or even ignorance, entropy might present some advantages. However another question arises: since there is several measures of entropy which one since there are several measures of entropy, which one shall be used? As a starting point we discuss the potentialities of Shannon entropy and Tsallis entropy. The main difference between them is that both Renyi and Tsallis are adequate for anomalous systems while Shannon has revealed optimal for equilibrium systems.

Diskussionsbeiträge der Fakultät für Wirtschaftswissenschaft der FernUniversität in Hagen Herausgegeben vom Dekan der Fakultät Alle Rechte liegen bei den Verfassern In modern portfolio theory like that of Markowitz or Sharpe the investor follows a mean/variance-rationality. Even the founders of this theory observed unsatisfactory results because of symmetrical risk measures like variance or standard deviation. Post-modern theory then considers downside risk measures and takes into consideration the investor's specific goals. In this contribution we follow these ideas, but use an information theoreti-cal inference mechanism under Maximum Entropy and Minimum Relative Entropy, re-spectively. The approach results in a high performance Expert System under the shell SPIRIT, combining an index model with the new method. For three DAX listed blue chips and for varying risk attitudes of the investor the system's portfolio selection capacity is compared to that of classical Markowitz ...

arXiv: Statistical Finance, 2015

This thesis applies entropy as a model independent measure to address three research questions concerning financial time series. In the first study we apply transfer entropy to drawdowns and drawups in foreign exchange rates, to study their correlation and cross correlation. When applied to daily and hourly EUR/USD and GBP/USD exchange rates, we find evidence of dependence among the largest draws (i.e. 5% and 95% quantiles), but not as strong as the correlation between the daily returns of the same pair of FX rates. In the second study we use state space models (Hidden Markov Models) of volatility to investigate volatility spill overs between exchange rates. Among the currency pairs, the co-movement of EUR/USD and CHF/USD volatility states show the strongest observed relationship. With the use of transfer entropy, we find evidence for information flows between the volatility state series of AUD, CAD and BRL. The third study uses the entropy of S&P realised volatility in detecting ch...

2019

This paper introduces an intrinsic entropy model which can be employed as an indicator for gauging investors’ interest in a given exchange-traded security, along with the state of the overall market corroborated by individual security trading data. Although the syntagma of intrinsic entropy might sound somehow pleonastic, since entropy itself characterizes the fundamentals of a system, we would like to make a clear distinction between entropy models based on the values that a random variable may take, and the model that we propose, which employs actual stock exchange trading data. The model that we propose for the intrinsic entropy does not include any exogenous factor that could influence the level of entropy. The intrinsic entropy signals if the market is either inclined to buy the security or rather to sell it. We further explore the usage of the intrinsic entropy model for algorithmic trading, in order to demonstrate the value of our model in assisting investors’ intraday stock ...

Entropy

In this paper we investigate the relationship between the information entropy of the distribution of intraday returns and intraday and daily measures of market risk. Using data on the EUR/JPY exchange rate, we find a negative relationship between entropy and intraday Value-at-Risk, and also between entropy and intraday Expected Shortfall. This relationship is then used to forecast daily Value-at-Risk, using the entropy of the distribution of intraday returns as a predictor.

Entropy

In this paper, we propose an adaptive entropy model (AEM), which incorporates the entropy measurement and the adaptability into the conventional Markowitz’s mean-variance model (MVM). We evaluate the performance of AEM, based on several portfolio performance indicators using the five-year Shanghai Stock Exchange 50 (SSE50) index constituent stocks data set. Our outcomes show, compared with the traditional portfolio selection model, that AEM tends to make our investments more decentralized and hence helps to neutralize unsystematic risks. Due to the existence of self-adaptation, AEM turns out to be more adaptable to market fluctuations and helps to maintain the balance between the decentralized and concentrated investments in order to meet investors’ expectations. Our model applies equally well to portfolio optimizations for other financial markets.

Journal of Asset Management, 2016

Entropy, a term used in Physics to quantify the degree of randomness in a complex system, is shown to be relevant for portfolio diversification. The link between entropy and diversification lies in the notion of uncertainty. We introduce the concept of available diversification in an investment universe and of diversification curves. We build a framework for assembling a fully diversified risk parity-like portfolio with a fundamentalbased high-conviction strategy, through a constrained entropy-maximisation process by which a portion of potential portfolio return is swapped for extra diversification. The main results of this study are: • mean-variance optimised portfolios are highly concentrated and scarcely related to the asset return assumptions; • few basis points of expected returns can be converted into a huge amount of extra diversification that making the portfolio allocation more robust to parameter uncertainty; • on a more conceptual ground, we investigate the relationship between portfolio risk and diversification concluding that they should be managed distinctly. The empirical analysis presented in this work shows that entropy is a useful means to alleviate the lack of diversification of portfolios on the efficient frontier.

Arxiv preprint arXiv:0809.4570, 2008

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 discovered, 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 focus 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.

Entropy

The most known and used abstract model of the financial market is based on the concept of the informational efficiency (EMH) of that market. The paper proposes an alternative which could be named the behavioural efficiency of the financial market, which is based on the behavioural entropy instead of the informational entropy. More specifically, the paper supports the idea that, in the financial market, the only measure (if any) of the entropy is the available behaviours indicated by the implicit information. Therefore, the behavioural entropy is linked to the concept of behavioural efficiency. The paper argues that, in fact, in the financial markets, there is not a (real) informational efficiency, but there exists a behavioural efficiency instead. The proposal is based both on a new typology of information in the financial market (which provides the concept of implicit information—that is, that information ”translated” by the economic agents from observing the actual behaviours) and...

Proceedings of 2nd International Electronic Conference on Entropy and Its Applications, 2015

The application of entropy in finance can be regarded as the extension of the information entropy and the probability entropy. It can be an important tool in various financial methods such as measure of risk, portfolio selection, option pricing and asset pricing. A typical example for the field of option pricing is the Entropy Pricing Theory (EPT) introduced by Les Gulko [1996]. The Black-Scholes model [1973] exhibits the idea of no arbitrage which implies the existence of universal risk-neutral probabilities but unfortunately it does not guarantees the uniqueness of the risk-neutral probabilities. In a second step the parameterization of these risk-neutral probabilities needs a frame of stochastic calculus and to be more specific for example the Black and Scholes frame is controlled by Geometric Brownian Motion (GBM). This implies the existence of riskneutral probabilities in the field of option pricing and their uniqueness is vital. The Shannon entropy can be used in particular manners to evaluate entropy of probability density distribution around some points but in the case of specific events for example deviation from mean and any sudden news for stock returns up (down), needs additional information and this concept of entropy can be generalized. If we want to compare entropy of two distributions by considering the two events i.e. deviation from mean and sudden news then Shannon entropy [1964] assumes implicit certain exchange that occurs as a compromise between contributions from the tail and main mass of the distribution. This is important now to control this trade-off explicitly. In order to solve this problem the use of entropy measures that depend on powers of probability for example Tsallis [1988], Kaniadakis [2001], Ubriaco [2009], Shafee [2007] and Rényi [1961] provide such control. In this article we use entropy measures depend on the powers of the probability. We OPEN ACCESS 2 propose some entropy maximization problems in order to obtain the risk neutral densities. We present also the European call and put in this frame work.

Procedia Economics and Finance, 2015

The application of entropy in finance can be regarded as the extension of information entropy and probability theory. In this article we apply the concept of entropy for underlying financial markets to make a comparison between volatile markets. We consider in the first step Shannon entropy with different estimators, Tsallis entropy for different values of its parameter, Rényi entropy and finally the approximate entropy. We provide computational results for these entropies for weekly and monthly data in the case of four different stock indices.

2016

Uncertainty is one of the most important concept in financial mathematics applications. In this paper we review some important aspects related to the application of entropy-related concepts to option pricing. The Kullback-Leibler information divergence and the informational energy introduced by Onicescu are the main tools investigated in this paper. We highlight a necessary condition that must be verified when obtaining the probability distribution minimising the Kullback-Leibler information divergence. Deriving a probability distribution by optimising the information energy has some pitfalls that are discussed in this paper.

Advances in Business and Management Vol. 20., 2023

This chapter seeks to demonstrate the self-care mechanisms for the individual and society that the present capitalist economic system offers as an affordable solution. Social care systems can draw inspiration from what we are discussing, just as private people can use our findings to their advantage. This chapter aims to bring order to a complex system of investment decision-making that can then be used as a decisionsupport tool to help all concerned thrive. The chapter proves that a portfolio based on nature’s entropy is self-sustaining, since the criteria taken into account in its design give it the same system properties as the other systems that constitute our world. The universal conception of systems theory allows for interdisciplinary transformation. Using a pattern similar to the ordering of nature, the chapter would like to demonstrate the inherent self-sustaining power of this pattern within asset management. The chapter reports on an empirical survey covering 984 investment portfolios and 1148 investment elements and concentrates on three main research issues: investor decision-making, diversification mismatches, and recommendations for sustainable asset management. Based on the methodology, 6-6 investment portfolios per individual were constructed. The portfolios sought diversification and return divergences in the short term of a semester (three months) and a long term of a 10-year interval. The key questions of the research focus on when, why, and what: When is the winning formula really the winning formula? Why is the winning formula – by how much and for how long? What is the explanation for the winning formula and what strategies make it sustainable? Interesting conclusions can be drawn from the research by contrasting the long- and short-term primary results of behavioral finance. The demonstrated sustainability-based portfolio management can serve as a guideline for the development of individual and institutional actors in financial culture. Raising financial awareness and self-management to a higher level needs to be coupled with a behavioral understanding of utility, which is also illuminated by the behavioral and outcome focus of the current chapter. Keywords: behavioral finance, financial culture, portfolio management, selfsufficiency, sustainable wealth management

Journal of Quantitative Economics

We measure stock market efficiency by drawing the comprehensive sample from Asia, Europe, Africa, North-South America, and Pacific Ocean regions and rank the cross-regional stock markets according to their level of informational efficiency. The study period spans from January 1, 1994, to August 3, 2017. We employ the approximate entropy approach and find that stock market efficiency evolves over the period. The degree and nature of evolution vary across regions and the development stage of the markets. The global, regional, domestic economic, and non-economic factors influence the adaptive nature of the stock markets. The emerging stock markets have improved efficiency by financial liberalization policy but are adversely affected by global shocks. The estimates validate the relevance of the adaptive market framework to describe the rejection of random walk without excess returns. The results suggest the growing presence of technical analysis and active portfolio managers. The emerging markets in Asia hold policy lessons for their peers. The findings suggest that global investors need to overcome the homogeneity bias as returns opportunities exist within the region and types of markets.

Entropy

To take into account the temporal dimension of uncertainty in stock markets, this paper introduces a cross-sectional estimation of stock market volatility based on the intrinsic entropy model. The proposed cross-sectional intrinsic entropy (CSIE) is defined and computed as a daily volatility estimate for the entire market, grounded on the daily traded prices—open, high, low, and close prices (OHLC)—along with the daily traded volume for all symbols listed on The New York Stock Exchange (NYSE) and The National Association of Securities Dealers Automated Quotations (NASDAQ). We perform a comparative analysis between the time series obtained from the CSIE and the historical volatility as provided by the estimators: close-to-close, Parkinson, Garman–Klass, Rogers–Satchell, Yang–Zhang, and intrinsic entropy (IE), defined and computed from historical OHLC daily prices of the Standard & Poor’s 500 index (S&P500), Dow Jones Industrial Average (DJIA), and the NASDAQ Composite index, respecti...