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I have a fragmented market with multiple assets which are traded with each other and some times triangular arbitrage can occur. The question is how to predict the price of those assets once the triangular arbitrage opportunity is gone ? One obvious way would be to stop time and perform all the arbitrage trades on the order books of the market and then compute the midpoint or weighted midpoint price for each order book. But this is hard and inefficient in a HFT environment.

I have re written the problem as the following constraint optimization problem.

Suppose you have an economy with 3 assets $A, B, C$, with $r_{AB}$ the rate of $A$ against $B$. The rates $r_{AB}^*, r_{AC}^*, r_{BC}^*$ at equilibrium can be found by minimizing

$\frac{N_{AB} (r_{AB} - r_{AB}^*)^2 + N_{AC}(r_{AC} - r_{AC}^*)^2 + N_{BC}(r_{BC} - r_{BC}^*)^2}{N_{AB} + N_{AC} + N_{BC}}$

Where $N_{AB}, N_{AC}, N_{BC}$ are the volumes of the traded pairs over an arbitrary period, and the constraints are

$r_{AB}^* = \frac{r_{AC}^*}{r_{BC}^*}$

$r_{AC}^* = r_{AB}^*r_{BC}^*$

$r_{BC}^* = \frac{r_{AC}^*}{r_{AB}^*}$

which are the necessary (redundant) constraints for a market without arbitrage opportunities.

Basically, this amount to minimizing the difference between the rates at non-equilibrium and the rates at equilibrium weighted by their trading activities. Would this method work ? Is there any literature where this problem is explored ?

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