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.
4 Likes
As some extensions have been added to the paper cited above, I wanted to provide additional context and description of results.
Model
The model assumes that there are two exchanges with at least one of the exchanges having a quadratic cost function. The cost that is considered is specifically slippage cost - i.e. the cost that arises from moving the price with your trade. The choice of specifically 2 exchanges is motivated by the idea that we can extend the results to more exchanges by treating one exchange as an aggregate of multiple other exchanges.
Quadratic Costs
The choice of quadratic costs is motivated partly by the observation that most AMMs either explicitly bake in a quadratic cost or have been shown to have approximately quadratic costs for small trades, and partly by the observation that most limit order books can also be approximated by a quadratic cost function (i.e. marginal cost increases linearly in trade size). Quadratic costs is also an assumption found in other finance papers. The main exception we see empirically is major pairs (e.g. WETH-USDC) being traded on limit order books. Since the tick sizes are quite large liquidity tends to concentrate at the “top of the book” (highest bid & lowest ask price), the slippage cost may be constant for small trade sizes. Hence, our theory applies generally except when we consider major pairs where price movements happen on the CLOB.
Price Process
Previous work assumed that the price movement of the asset followed a Geometric Brownian Motion (GBM) - a standard assumption - and that the price was completely determined on one, completely liquid market. The justification for the choice of GBM was that the arrival of “noise trades” according to a random process gives a GBM price movement in the limit.
In our case, we also assume a GBM, but need a slightly different justification. Instead of noise only arriving on one market, we generalise to the case where noise trades arrive on both markets and that the implied price (i.e. the price that one gets after both markets are arbed to equality) follows a GBM. In order to justify this, we show that it doesn’t matter on which market noise trades arrive as long as the size and direction follow a certain random process and arbs happen whenever there is a mismatch in the markets.
Results
The relationship between LVR and block times and volatility don’t change in our new setting. However, our setting allows us to make statements about the distribution of value across different actors. The summarised key predictions of our model are that:
- Losses per unit of liquidity are worse on lower liquidity markets.
- Overall LVR grows the more noise trades happen on lower liquidity market. I.e. liquidity providers and noise traders benefit at the expense of arbitrageurs when more noise trades are routed to the more liquid exchange