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,x_{m}) = k x_{m}^{k}x^{-k-1}

for x greater or equal to x_{m}. x_{m} 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,x_{m}) times x from x_{m} 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 x_{m})

Now to the less definitive stuff. To the best of my knowledge as of now the total daily forex turnaround is about $3 trillions. A reasonable definition of a trader is someone who trades at least one standard size lot a day, that is, $100,000. Thus T=3× 10^{12} and x_{m}=10^{5}. There is a fair amount of uncertainty as to what the Pareto index k is for this market. There is a famous 80-20 rule applicable in many contexts. In this case it would mean that 20% of traders contribute 80% of the turnaround. The 80-20 rule is just a particular instance of a Pareto distribution corresponding to Pareto index of 1.161. With this input, we estimate to have about **four million forex transactions a day.** That’s what we’ve called L. The number of traders creating L transactions a day depends on the structure of the relationships between them, that is, who trades with whom. Imagine a trader as a point on a plane, the L transactions being the links connecting the points, and you get the picture — there are various ways of connecting the points. Our problem is that we’ve estimated (with Pareto’s help) the number of links but we do not know the number of points they connect. One situation, extreme in a sense and only applicable for the sake of academic argument, would be to assume that each trader has a unique partner (kind of a monogamous marriage). Then the number of traders n is simply L times 2. Of course such a system is not capable of moving money. But as you will see, it requires the highest number of traders (**eight million traders?!**) to create a given turnaround, and is interesting as a limiting case. Another situation is the egalitarian one where every trader is equally likely to trade with every other trader. This is also not realizable in practice, but it corresponds to the equation:

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.