## Central European Time

Central European time is chosen for the following reason. Forex week begins, roughly speaking (since the volume increase is gradual) on Sunday 5pm and ends Friday 6pm Eastern time. It is convenient to define this week to consist of 5 full days, from 6pm Sunday to 6pm Friday New York time. When it’s 6pm in New York, it’s midnight in Berlin, Paris, Madrid, Rome, Geneva and Frankfurt. These cities use Central European Time or CET. Therefore, the convenience of using CET is that one gets 5 non-interrupted, full 24-hour long trading days per week. Table 1 compares four time zones including major trading centers of the world.

 Tokyo Central Europe Greenwich Eastern US 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

## Profile Histogram

A profile histogram provides an economic representation of two-dimensional (X vs Y) data. Unlike a two-dimensional histogram, the profile histogram is unbounded in the Y (vertical) dimension. Like any histogram, it has discrete bins along the X axis to aggregate the data. The data are represented graphically as a series of (X,Y) points on a plot, with the point position in Y corresponding to the mean and the vertical bars typically indicating the measure of the spread such as the RMS, or the measure of the precision of the mean, whereby larger bars correspond to less precision.

None of the projection techniques is perfect, since the reduction of information involved in all projections is not guaranteed to be “intelligent”. But they do solve the problem, even though it is necessary to look from more than one point of view and try more than one path to optimization.

## High Order Cumulants

Cumulants are statistical measures of correlation designed to go to zero whenever any one or more quantities under study become statistically independent of the rest. Cumulants generalize the concept of a correlation measure; in particular, a correlation of two bodies, quantities and so on, the most intuitive one, can be represented and measured by the second-order cumulant. Higher orders can be conceived.

Financial time series come as sequences of bars or “candles”, one bar per time step of the series. The bar has an open, close, low and high levels of price. In the liquid market like forex, close, low and high should be sufficient while open is typically not too different from the previous close and is believed to be redundant.

To judge the quality of market predictions, we are interested in multivariate cumulants, since for each of the three essential components of a candle there is a prediction. Because we make predictions for each of these components, the number of variables we would like to correlate is even, and therefore we are interested in even-order cumulants.

The simplest of these is second order cumulant also known as covariance:

 c2 = E[x1x2] – E[x1]E[x2] (1)

Here E[] stands for the averaging (a.k.a. expectation) operator.

In general, n-th order cumulant is constructed by making a correlation term E[x1x2…xn] and subtracting all terms which are not “genuinely” n-th order correlations, but are composed of lower order ingredients.

Take for example the daily candle. We can generate predictions for daily changes in low and high, such that the correlation between the real and predicted change for high will be positive. Same for low. If we take a trading position having yesterday’s low as a stop-loss and yesterday’s high as a profit target, we want to make sure that not only there is a tendency for low not to be hit when the prediction says so (attested to by the positive correlation between the daily change in low and its forecast), and not only there is a tendency for the high to be hit when the prediction says so (attested to by the positive correlation between the daily change in high and its forecast), but that these two things tend to happen “simultaneously” within the same trade. This is the essence of the difference between the “genuine” fourth order correlation and a mere superposition of two second order ones.

Forth order cumulant is defined as:

 c4 = E[1,2,3,4] – E[1,2,3]E[4] – E[1]E[2,3,4] – E[1,3,4]E[2] – E[1,2,4]E[3] – E[1,2]E[3,4] – E[1,3]E[2,4] – E[1,4]E[2,3] + 2(E[1,2]E[3]E[4] + E[1,3]E[2]E[4] + E[1,4]E[2]E[3] + E[2,3]E[1]E[4] + E[2,4]E[1]E[3] + E[3,4]E[1]E[2]) – 6E[1]E[2]E[3]E[4]. (2)

Here we use the notation: E[x1x2…] gets replaced by E[1,2…] for the sake of brevity.

What is being subtracted is in fact products of lower order cumulants, which in turn subtract their lower order cumulants, which is why there are terms with both plus and minus sign alternating in a certain order. A recurrence relation exists allowing one to express higher order cumulants in terms of lower order ones.

A cumulant of order higher than 2 will go to zero if any two quantities are proportional to each other:

 x1=ax2. (3)

The fact that it will also go to zero whenever any one quantity is statistically independent of the rest, combined with the additivity of cumulants, implies that a higher order cumulant will go to zero whenever any pair of quantities has even a less deterministic, randomized form of that equation:

 x1=ax2 + r (4)

where r is a random number independent of x2.

A non-zero higher order cumulant indicates that a relationship between the data is not merely Eq. (4), with its familiar visualization as a diagonally elongated cloud in the x1, x2 space — even though one may see such a cloud and other signatures of two-point correlations when subjecting higher-order correlated data to a lower order analysis.

To keep the cumulant independent of the units in which the underlying quantities are expressed, we sometimes normalize it:

 C4 = c4/(Var[1]Var[2]Var[3]Var[4])1/2, (5)

where Var is variance.

## Investment

“An investment operation is one which, upon thorough analysis promises safety of principal and an adequate return. Operations not meeting these requirements are speculative.” Apparently for Graham and Dodd, the term “speculation” has no positive connotations.

## What Do You Mean By “Predictable” Or “Predictability” When You Talk About Forex?

Imagine tossing a coin which is slightly bent in a way which is known to you. The bend is almost unnoticeable but it does exist and this fact will become obvious after a long enough series of coin tosses, if you do the book-keeping accurately. Trading the markets with a good trading system is similar. The amount of predictability is small, making it justifiable for people who do not have access to an analysis technique of sufficient sensitivity to talk about “efficient markets” — the markets indeed look “efficient” to their methods of observation. The fact of predictability only becomes undeniable after hundreds or thousands of “coin tosses” (trades).

The world economy is never in equilibrium. “I assume that markets are always wrong” said George Soros. A market actor (investor, trader, central banker) requires finite time to analyze the changing environment and to make a decision. Yet more time and resources are needed to turn it into action visible by the market. No analysis is perfect, and some are less perfect than others. In addition, people who share common educational and cultural background tend to make similar, rather than the most efficient, decisions. Groupthink exists inside communities and large organizations. For these and other reasons, the so-called market inefficiency exists. Once the market inefficiency is quantified, predictive power (and a winning strategy) is only a step away, since such inefficiency — always specific and quantifiable — is in some sense but a deviation from the perfect symmetry which random noise (aka “efficient market”) would possess.

## Anticorrelation

Anticorrelation, or anti-correlation, means the same as negative correlation. To say that A and B are anticorrelated means that A tends to move up when B moves down and vice versa. This can be said when discussing both autocorrelation or cross-correlation. The term is not as widely used in financial contexts as it is in physics.

## How many people trade forex every day? A Pareto estimate.

To estimate how many people trade forex every day, we use Pareto distribution. This is a power-law, highly asymmetric distribution, encountered in many contexts, including distribution of wealth. Pareto’s probability density distribution is

f(x|k,xm) = k xmkx-k-1

for x greater or equal to xm. xm is the minimum value of x, in our case — the minimum value contributed by a trader into the total turnaround. This number defines who we call a trader for the purpose of this estimate, a threshold so to speak.

According to the definition of a mathematical expectation E[x], it is an integral of f(x|k,xm) times x from xm to infinity and

L E[x] = T

where T is the total turnaround and L is the number of trades that create it. The integral is easy to deal with for k > 1 and we obtain

L = T(k-1)/(k xm)

L = n(n-1)/2

Because n is much larger than one, n is simply

n = (2L)1/2

If such topology of trading links were the case, only about 3 thousand traders (with Pareto wealth distribution) could create our present trading volume. Such topology may be considered an idealized limit, the Holy Grail of the online trading business. The reality is probably in between these two extremes, with traders being not completely isolated, but forming relatively isolated clusters around brokers, banks, hedge funds and similar institutions. The relationships between those higher level entities are then much closer to the egalitarian model — a tightly knit community where every member knows every other member.

If the estimates of the number of traders, given the trading volume, differ so much depending on the ogranization of the market, then the really interesting conclusion is the converse: the real reason for the spectacular growth in the forex trading volume seen in the past few years probably has at least as much to do with changes in the organization of the market as it does with purely economic reasons.