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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

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