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I wrote a program which finds negative weight cycles in a graph to find triangular arbitrage opportunities, using the bellman ford algorithm.

The basic principle is this, given three currencies ($x$, $y$, $z$), and the exchange rates between the three ( $x_r$, $y_r$, $z_r$), if $ x_r * y_r * z_r > 1 $ then an arbitrage opportunity exists.

This can then be phrased in terms of a graph by first inverting

$$ 1 / (x_r * y_r * z_r) < 1 $$

and then taking the natural log, which gives

$$ log(1 / x_r ) + log(1 / y_r) + log( 1/ z_r) < 0 $$

I can then weight the edges of the graph with these values and expect it to give me the correct result.

The trouble I'm having is factoring trade fees (which is a constant percentage e.g. 0.02%) into this. What I've figured is if

$$ (x_r - x_r*fee)* (y_r - y_r*fee) * (z_r - z_r*fee) > 1 $$

then there is an opportunity. So it follows that

$$ log(1 / (x_r *(1 - fee)) ) + log(1 /( y_r*(1 - fee))) + log( 1/ (z_r*(1 - fee)) ) < 0 $$ However using these values as my edge weights gives the wrong results.

How can I properly take fees into account?

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    $\begingroup$ I asked a somewhat similar question, but never received any replies. Not sure if what I have so far helps or not, I can try posting a more thorough answer tomorrow. Essentially you need to treat the edge weights as functions and a path as a composition of those functions. See: math.stackexchange.com/questions/370310/… $\endgroup$ Jun 1, 2013 at 1:03

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Interestingly, you can ignore the fees. All the fees do is increase the minimum starting amount necessary to have an arbitrage for a given negative-weight cycle.

Determine if any negative-weight cycles exist without fees. If a negative-weight cycle exists without fees, you can determine the minimum starting amount needed to have a negative weight cycle with the fees included by solving:

$f_{x_r}\circ f_{y_r}\circ f_{z_r}(n_0) = 0$ where $f_{x_r}$,$f_{y_r}$,and $f_{z_r}$ represent the edge-weight functions: $f_a(n)=a*(n-fee)$

Also, keep in mind that arbitrage opportunities may exist in cycles of lengths 2 to $\infty$, not just 3.

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  • $\begingroup$ This does not seem true to me. Say I have euro, usd and gbp and the exchange rates between them are 1, 1 and 1.1, if there is a cumulative fee greater then 10% then no matter my starting value I would come out with less money, yet this would be returned as a negative weight cycle. $\endgroup$
    – Loourr
    Jun 5, 2013 at 17:01
  • $\begingroup$ Sorry, I misunderstood the fee - I assumed it was a fixed amount. If the fee is a percentage, not fixed, then you subtract that percentage from each rate. So, if the cumulative fee is 12%, your rates are 0.96, 0.96, 1.06, and there is no negative weight cycle. $\endgroup$ Jun 6, 2013 at 11:02
  • $\begingroup$ That's what I figured, but thats what I think I'm doing, and it is still giving me false positives $\endgroup$
    – Loourr
    Jun 6, 2013 at 22:22

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