I've tried to get an answer for this question elsewhere but with no luck, so I would appreciate a mathematical analysis of it.
Assume $0\%$ commission. Consider three currencies, let's say the Zong, the Yabba and the Xoxo. The Zong is trading against the Yabba at say z:y, the Yabba against the Xoxo at y:x and Xoxo against the Zong at x:z.
Then surely:
$$\dfrac{z}{x}\times\dfrac{y}{z}\times\dfrac{x}{y}=1$$
, but this doesn't seem to be the case. Is the commission keeping this stable?
If you're assuming a completely fair currency market, then it makes perfect sense: essentially what your equation is telling you is the rate at which the Xoxo is trading against itself. That is, if you take $x$ Xoxo's, you can trade it in for $z$ Zongs which you can then turn in for $y$ Yabbas which you can then turn in for $x$ Xoxo's. So your overall rate becomes, $x:x$ Xoxo's, which simplifies to $1.$
In the real world, however, these rates aren't going to be totally fair, so you would have to use 3 totally independent ratios, e.g., $z_1:y_1$ Zongs to Yabbas, $y_2:x_1$ Yabbas to Xoxo's, and $x_2:z_2$ Xoxo's to Zongs. And so then your rate of Xoxo's to Xoxo's by trading Xoxo's to Zongs to Yabbas to Xoxo's would become: $$\frac{z_2}{x_2}\cdot\frac{y_1}{z_1}\cdot\frac{x_1}{y_2}$$ Which may be greater than, less than, or equal to $1$.