Arbitrage with Bounded Liquidity

Research
1

In this short note, I have calculated the arbitrage gains, or equivalently, LVR when arbitraging between two markets with bounded liquidity: https://arxiv.org/pdf/2507.02027
This adjustment is, for example, useful for studying arbitrage for smaller market cap coins where slippage cost due to limited liquidity becomes important.

The main adjustment in the LVR formula comes through a term that depends on the relative liquidity in the two markets:

\text{ARB}_T=\text{LVR}_T=\int_0^T\ell(\sigma,Q_t)dt \quad\text{with}\quad \ell(\sigma,Q):=\tfrac{\sigma^2Q^2}{2}(1-\tfrac{|x^{*\prime}_1(Q)|}{|x_2^{*\prime}(Q)|})|x_1^{*\prime}(Q)|

where Q is the equilibrium ex-change rate post trade, \sigma its volatility and x^{*\prime}_1(Q) and x^{*\prime}_2(Q) denote the marginal liquidity in the two exchanges.

Thus, everything else being equal, arbitrage gains are smaller with bounded liquidity and depend on the imbalance in liquidity between the two exchanges. From the point of view of passive LPs, LVR per unit of capital deployed is worse in the less liquid exchange, so that, in the absence of other effects, we would expect liquidity to concentrate in one dominant DEX.

I also briefly discuss in this talk around min 8:40 in the context of cross chain arbitrage.

Comments and suggestions are appreciated.