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If I have multiple markets (let's say 5, but the solution should be generic) trading the same stock/commodity/whatever, and the markets differ in both variable fees (which are in % of the trade order) and fix fees (which are in absolute number of $ per trade order), and suppose there exists an arbitrage opportunity on more than 2 markets at the same time, how do you calculate the absolutely most profitable sequence of market orders? (order of orders matters)

The variable fees are not a problem, but the fix fees complicate the whole algorithm tremendously. Is this a traveling salesman type of problem? Or, is there any paper which deals with this problem.

The fees might be something like this (shown as an example):

  • 1st market: $5 + 1 %

  • 2nd market: $4 + 2 %

  • 3rd market: $0 + 5 %

  • 4th market: $10 + 0 %

  • 5th market: $3 + 3 %

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I find your question slightly confusing. Why you need more than 2 markets to present an arbitrage opportunity, 2 or more are already sufficient. And why is it complicated to add in your execution related fees, whether stated as a percentage of notional traded or as fixed fee per trade? – Matt Wolf Jul 10 '13 at 1:20
Example: Market1(fixed fee: 1, variable fee: 2%, BBO: 105/107), Market2(fixed fee: 0.5, variable fee 1%, BBO: 98/100). Variable Fees -> Buy asset at M2 for (100 + 1) and sell to M1 for (105-2.1). PnL with fixed fees applied: (105-0.5) - (100+1). -> Apply same logic to all markets and chose the most profitable one. – Matt Wolf Jul 10 '13 at 1:27
Well, the idea is that the arbitrage opportunity can be at more than 2 markets at the same time. And the market depth at each exchange is finite IN VOLUME. What I mean is that at price 107, there might be only 3 pieces of the stock. – Paya Jul 10 '13 at 2:00
that does not change the kind of algorithm you need to run the markets over. You need to include your all-in execution related costs and may find out that low volume will not push you over the "hurdle-rate", which may rank this particular arbitrage lower or even net-unprofitable. – Matt Wolf Jul 10 '13 at 3:37
Have you made any progress on this question? Is there anything in my answer which you have trouble following? – Cristian Dima Jul 15 '13 at 6:18

Why can't you just adjust your book prices by the variable price and then subtract the fixed price off the PNL when calculating that up?

For exchange, with a 2% variable fee, a book 98 bid 100, resting offer at $100 would go up too $102 and bid go down to $96.04.

Note evaluate your arbs like you normally would and subtract off the fixed fees from the corresponding venues.

This assumes the fixed fee is per transaction. If the fixed fee is per unit volume, then you can augment the book like the variable fees instead.

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