Estimating Arbitrage-Free Prices in a Fragmented Market
Summary
The document proposes estimating equilibrium exchange rates across three traded asset pairs without simulating arbitrage trades through every order book. It formulates a constrained optimization that minimizes the volume-weighted squared deviations between observed rates and fitted equilibrium rates. The constraints require the three rates to be mutually consistent, eliminating triangular arbitrage.
The formulation uses trading volume over a chosen period as a proxy for each pair’s influence. The text presents this as a question rather than a validated method: it supplies no solution, empirical test, or comparison with alternatives. It also does not address bid-ask spreads, order-book depth, timing, or how volumes should be chosen, all of which may affect equilibrium estimates in a high-frequency setting. The idea is a starting point for projecting inconsistent pair prices onto an arbitrage-free set.
Key ideas
- Triangular arbitrage-free rates must satisfy consistency relationships across all three asset pairs.
- The proposed estimate minimizes squared deviations from observed rates subject to those relationships.
- Trading activity is used to weight each pair’s deviation in the objective.
- The document does not demonstrate performance or resolve practical market microstructure choices.
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Full text
# Price at equilibrium in a market with arbitrage opportunities
# Price at equilibrium in a market with arbitrage opportunities
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 ?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.