**On the Efficiency of Markets (part I)**

When one inspects financial time series, it is obvious to recognize that the time evolution is unpredictable. Price of any financial product is essentially indistinguishable from a stochastic process. However, a quantitative understanding of financial market might be possible.

The first attempt to understand has been done a long time ago, in 1900 by Bachelier. In his Ph.D., entitled "Théorie de la spéculation", he treated with the pricing of options in speculative markets, an activity that today is extremely important in financial markets where derivative securities – those what value depends on the values of other more Basic underlying variables – are regularly traded on many different exchanges.

To put Bachelier's work into perspective, the Black & Scholes option pricing model – considered the stone in option-pricing theory – was published in 1973, almost three-quarters of a century after the publication of his thesis. Moreover, theorists and practitioners are aware that the Black & Scholes model needs correction in its application, meaning that the problem of which stochastic process describes the changes in the logarithm of prices in a financial market is still an open one.

After Bachelier's seminal work, the problem of the distribution of price changes has been considered by several authors since the 1950s. The original proposal of random walk prices with a Gaussian distribution of price changes was soon replaced by a model in which stock prices are log-normal distributed, ie, stock prices are performing what is called a geometric Brownian motion.

In a geometric Brownian motion, the differences of the logarithms of prices are Gaussian distributed. This model is known to provide only a first approximation of what is observed in real data. For this reason, a number of alternative models have been proposed with the aim of explaining the real conditions observed on financial markets.

We have just done our first steps in quantitative finance. It follows that, even if no forecast is acceptable for financial assets, it does not mean that no quantitative knowledge is possible. This activity is usually called economophysics: as we have started to explain, it's a subject applying and proposing ideas and methods of statistical physics and complexity into data coming from economic phenomena. Economics deals with human behavior related with the management of the resources, finances, imports etc. And can be regarded as a part of social or natural science. Thus, the aim of econophysics is to apply the ideas of natural science as far as well into economics.

Let's come back to the line of the discussion and our attempt to model on a quantitative basis financial markets. Time series of asset prices are unpredictable and we have given few hints because it is surely a correct issue. This is the cornerstone of the description of price dynamics as stochastic processes. Since the 1980s it has been recognized in the physical sciences that unpredictable time series and stochastic processes are not synonymous.

Specifically, chaos theory has shown that unpredictable time series can arisen from deterministic nonlinear systems. Here, from an empirical point of view, it is quite unquestionably that it will be possible to discriminate between the random and the chaotic hypotheses. However, we have made a decisive progress as we know that we can not forecast financial stochastic time series but we can build a well define knowledge of financial markets, under simple and powerful ideas.

**On the Efficiency of Markets (part II)**

"I'd be bum in the street with a tin cup if the markets were efficient" (Warren Buffet).

From the previous part, we can conclude that all traders and external news result collectively into large fluctuations of any price of a financial product. It follows that any predictability pattern is either detectable nor exploitable. This is basically what is called the "Efficient Market Hypothesis" [EMH] that Buffet contests: for him markets are not efficient, arbitrage opportunities exist and it's possible to make money out of them!

Of course, we have to consider any proposal by Buffet with the highest level of attention. Markets are complex systems that incorporated information about a given asset in the time series of its price. As mentioned above, a standard paradigm in finance is that the market is efficient in the determination of the most rational price of the traded asset.

The EMH was originally formulated in 1965 by Samuelson. A market is said to be efficient if all the available information is instantly processed when it reaches the market and it is immediately reflected in a new value of prices of the assets traded.

The theoretical motivation for the efficient market hypothesis has its roots in the pioneering work of Bachelier (see above), who suggested that the price of assets in a speculative market could be described as a stochastic process.

Samuelson showed that prices of speculative financial assets fluctuate randomly and that the best prediction for price at time t according to the all history is the price at time t-1. Stochastic processes obeying this "law" are called martingales. The notion of a martingale is, intuitively, a probabilistic model of a 'fair' game. In gambler's terms, the game is fair when gains and losses cancel, and the gambler's expected future wealth coincides with the gambler's present assets.

The fair game conclusion about the price changes observed in a financial market is equivalent to the statement that there is no way of making a profit on an asset by simply using the recorded history of its price fluctuations.

The conclusion of this 'weak form' of the efficient market hypothesis is then that price changes are unpredictable from the historical time series of those changes. Then, Fama (1970) developed the concept of EMH and made a distinction between three forms of EMH: (a) the weak form (of Samuelson), (b) the semi-strong form and (c) the strong form.

The strong form suggests that securities prices reflect all available information, even private information. Seyhun (1986, 1998) provides sufficient evidence that insiders profit from trading on information not already incorporated into prices. Here the strong form does not hold in a world with an uneven playing field.

The semi-strong form of EMH asserts that security prices reflect all publicly available information. There are no undervalued or overvalued securities and thus, trading rules are incapable of producing superior returns. When new information is released, it is fully incorporated into the price rather speedily. Again, no arbitrage opportunity exists.

What next? Certainly, the weak and semi-strong forms of the EMH are not fully correct and Buffet is right. Then, one can start from EMH and incorporated deviations from rational expectations in the behavior of agents in an attempt to explain anomalies of financial markets.

It raises the question of finding arbitrage opportunities! In addition, it is not so obvious that even if an arbitrage is present, we could exploit it. Since the 1960s, a great number of empirical investigations have been devoted to testing the limits of the EMH, which has been put on trial and subjected to a constant critical re-examination.

Below, we give a few known arbitrage opportunities or anomalies, well discussed financial financial publications, for example by Philip, Russel and Torbey (2002).

The January Effect: Rozeff and Kinney (1976) were the first to document evidence of higher mean returns in January as compared to other months. Using NYSE stocks for the period 1904-1974, they find that the average return for the month of January was 3.48% as compared to only 0.42% for the other months. Later studies document the effect persists in more recent years: Bhardwaj and Brooks (1992) for 1977-1986 and Eleswarapu and Reinganum (1993) for 1961-1990.

The effect has been found to be present in other countries as well (Gultekin and Gultekin, 1983). More recently, Bhabra, Dhillon and Ramirez (1999) document a November effect, which is observed only after the Tax Reform Act of 1986. They also find that the January effect is stronger since 1986. Taken together, their results support a tax-loss Selling explanation of the effect.

The Weekend Effect (or Monday Effect): French (1980) analyzes daily returns of stocks for the period 1953-1977 and finds that there is a tendency for returns to be negative on Mondays whereas they are positive on the other days of the week. He notes that those negative returns are "caused only by the weekend effect and not by a general closed-market effect".

A trading strategy, which would be profitable in this case, would be to buy stocks on Monday and sell them on Friday. Kamara (1997) shows that the S & P 500 has no significant Monday effect after April 1982, yet he finds the Monday effect undiminished from 1962-1993 for a portfolio of smaller US stocks.

Internationally, Agrawal and Tandon (1994) find significant negative returns on Monday in nine countries and on Tuesday in eight countries, yet large and positive returns on Friday in 17 of the 18 countries studied.

However their data does not extend beyond 1987. Steeley (2001) finds that the weekend effect in the UK has disappeared in the 1990s.

Over / Under Reaction of Stock Prices to Earnings Announcements: There is substantial documented evidence on both over and under-reaction to earnings announcements. DeBondt and Thaler (1985, 1987) present evidence that is consistent with stock prices overreacting to current changes in earnings. They report positive (negative) estimated abnormal stock returns for ports that previously generated inferior (superior) stock price and perception performance.

This could be construed as the prior period stock price behavior overreacting to earnings developments (Bernard, 1993). Thus, the evidence suggests that information is not impounded in prices instantaneously as the EMH would predict.

Standard & Poor's (S & P) Index effect: Harris and Gurel (1986) and Shleifer (1986) find a surprising increase in share prices (up to 3 percent) on the announcement of a stock's inclusion into the S & P 500 index. Since in an efficient market only information should change prices, the positive stock price reaction appears to be contrary to the EMH because there is no new information about the firm other than its inclusion in the index.

The Weather: Few would appeal that sunshine puts people in a good mood. People in good moods make more optimistic choices and judgments. Saunders in 1993 shows that NYSE indices tend to be negative when it is cloudy. More recently, Hirshleifer and Shumway (2001) analyze data for 26 countries from 1982-1997 and find that stock market returns are positively correlated with sunshine in almost all the countries studied. Interestingly, they find that snow and rain have no predictive power! That's enough for the examples of anomalies. More details are given by Philip, Russel and Torbey (2002).

**Discussion**

We are convinced that opportunities can be present on the market, sometimes for a short period of time, sometimes with intermittency. Our main challenge is to find such opportunities that are statistically relevant and that can be exploited on a regular basis of course. None on the examples above fulfil these conditions! In addition, we have to be sure that, in presence of such an opportunity, we have the ability to exploit it.

First, let's come back on the modeling side. It is interesting to follow our discussion of the EMH at a fundamental level in order to derive some ideas that could have been useful in the daily practice. Summers in 1986 suggests that pricing should combine a random walk plus a fad variable. The fad variable is modeled as a slowly mean-reverting stationary process.

That is, stock prices will exercise some temporary aberrations, but will always return to their equilibrium price levels. One may argument that market mechanisms may be able to correct the individual decision biases, and that individual differences may not matter in the aggregate. In such cases, it would be quite hard to exploit any such biases.

However, the transition from micro to macro behavior is still not well established. For example, in their study of price differences among similar consumer products, Pratt, Wise and Zeckhauser (1979) demonstrate the failure of the market to correct individual biases. Then, maybe, we keep a chance.

Let's conclude this post on the EMH and its limits by the most convincing argument.

The stock market crash of 1987 continues to be problematic for the supporters of EMH. Any attempt to accommodate a 22.7% depreciation of the stocks (in one day) within the theoretical framework of EMH would be a formidable challenge. It seems reasonable to assume that the decline did not occur due to a major shift in the perceived risk or expected future dividend.

The crash of 1987 provides further credence to the argument that the market includes a significant number of speculative investors who are guided by "non-fundamental" factors. Thus, the assumption of rationality in conventional models needs to be rethought and reformulated (to conform to reality).