# How to statistically describe the Financial Markets

Any financial instrument that has a price today, tomorrow will have another price, higher or lower, the day after tomorrow will have another price and so on. The sequence of prices over time generates the so-called "historical series of prices" which describes its trend over time.

HISTORY In order to elaborate statistics, metrics, forecasts, and analyzes, Eugene Fama in 1965 elaborated the Random Walk Hypothesis, or rather the randomness of the price trend in the financial markets.

Although these theories had been advanced by Lois Bacheller in the early 1900s with his work "Theory of Speculation", the Nobel Prize winner Eugene Fama carried empirical demonstrations as well as theoretical to support this theory that the stock markets were random while also elaborating the Efficient Market Hypothesis, arguing that it was impossible to speculate on the financial markets.

PRACTICE History has taught us that these theories, even if from a mathematical point of view, are elegant and fascinating, in reality, they aren't able to correctly describe the financial markets. In my post of two weeks ago heuristics in finance based on the work of Prof. Gerd Gigerenzer, I explained the difference between the two environments, the one called "Risk" and the one called "uncertainty" (uncertainty).

Mathematical models of mean and variance, with all the subsequent implications, are good for describing an environment where all the variables are known, and it is also known the probability that a certain event may occur; the perfect example is the game of roulette, where it is known the possibility that comes out black instead of red (18/37), it is known the possibility that a number (1/37) comes out and the fact that there is 0, in the long run, the Casinò always wins.

Roulette is the perfect example of Efficient Market Hypothesis and also of "Risk" environment as defined by Prof. Gerd.

UNCERTAINTY But financial markets are not random, growth or loss of value, often, even if not always, are due to external factors, not always predictable, that influence the trend; we experienced it for the umpteenth time a week ago, where the Italian political uncertainty caused the stock market to lose more than 2.5% in just one session, but it is only one of the thousands of events that are conditioning the trend in the historical price series.

Therefore, considering the financial time series as random paths of which all the variables are known is a reductive approach that has a big advantage, I can use (with extreme elegance) a whole series of mathematical and statistical models to describe the past trend.

unfortunately, it is a false illusion because the markets are much more complex.

proponents

Nassim Nicholas Taleb, whom many of you know for the bestseller "the black swan", has often loudly criticized the limits of the mathematics used to describe the historical series, but has never proposed a real alternative.

In fact, it is very difficult to create an alternative that is robust and above all so elegant (as the math university professors claim) as the logic means and variance.

OUR COMMITMENT

Personally, with the help of my friend Prof. Ruggero Bertelli, I've been trying to build an alternative for some time, and the process started with the paper "the Diaman Ratio" which was published in 2014 on Willmott Magazine, a very technique newspaper that however is a must for the quantitative analyst around the world.

The idea is to create a set of statistical indicators, so-called deterministic, or that consider the historical series as a concatenation of events that has a precise order that would make no sense with a different sequence.

EXAMPLE I'll give an example: in 2007 the economy was fine and the banks exchanged billions of dollars derivatives on the loans of American citizens as if they were peanuts and without considering the risk that someone would not pay; then suddenly someone realized that it was risky this practice and no longer accepted these derivatives and broke the mechanism, August 2007, the market suddenly loses 15% and then recover it in part by the end of the month, random trend? Not exactly. The volatility of the VIX rises structurally meaning that the situation on the markets is no longer as rosy as in previous years, in January 2008 an employee of Societè Generale makes a hole of about 2 billion (I remember a similar amount), another thump of the stock exchange of 15%, the problems of subprime mortgages become known in America and house prices are starting to fall, the stock exchange goes down again and the big American banks are starting to have liquidity problems, the stock exchange goes down again and then we get to Lehman Brothers, defaulted in a weekend with the result that the market goes into fibrillation, VIX to 80 (with peaks of 100) and the market still collapses.

CONCATENATION

All these things that have happened have influenced the market, of course, and the concatenation of one with the other has led to consequences (deterministic) that could not be placed in time with a different order.

Is there anyone who can argue that the S & P index moved randomly in that period? Then the financial time series must be analyzed with deterministic and non-random indicators.

I give an example of teaching that I hope makes the idea:

Suppose a historical series of prices that fluctuate realizing a -10% and a 10% each month, the graphical trend would be as follows:

The average of this historical series is equal to zero and the variance is equal to 0.011111.

Already on the fact that the average is zero, but that the yields are negative at the end says a lot about the simplification of the Media and Variance approach ...

Now let's change the order of the succession of returns and first put all the negative returns and then all the positives and analyze the new historical series

The average of this new historical series is always 0, the variance is always 0.011111, but there is something in this historical series that does not come back, what is it?

DIFFERENCE

The loss that has had to endure the historical series number 2 is very different from the number 1, it does not matter if in the end, both have the same final result, it imports from the deterministic point of view, because I dare to say that for an investor the first or the second are indifferent.

THE SEQUENCE IS IMPORTANT

The sequence of returns is really important as well, so how the historical series is formed, with related ascending or descending trends, has an enormous importance that the traditional financial statistics completely omits and is not able to analyze.

This effect is even more evident in the world of Crypto, given that the volatility is very high, the rises and falls much more pronounced and therefore not have the right indicators (and volatility as a standard deviation I think is not appropriate for this asset class) it risks leading to incorrect analyzes and evaluations.

In the coming weeks, we will analyze a series of deterministic indicators, free from the mean and variance approach, which can be much more useful for the investor to make investment decisions.

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DB

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