Automated Market Making and Arbitrage Profits in the Presence of Fees (2305.14604v1)

Abstract: We consider the impact of trading fees on the profits of arbitrageurs trading against an automated marker marker (AMM) or, equivalently, on the adverse selection incurred by liquidity providers due to arbitrage. We extend the model of Milionis et al. [2022] for a general class of two asset AMMs to both introduce fees and discrete Poisson block generation times. In our setting, we are able to compute the expected instantaneous rate of arbitrage profit in closed form. When the fees are low, in the fast block asymptotic regime, the impact of fees takes a particularly simple form: fees simply scale down arbitrage profits by the fraction of time that an arriving arbitrageur finds a profitable trade.

• The paper extends the LVR framework by incorporating trading fees and discrete arbitrageur arrivals to quantify how fees reduce arbitrage profits in AMMs.
• It derives the stationary distribution of the log mispricing process and provides closed-form expressions for the probability of trade and arbitrage profit rates.
• The framework offers actionable insights for setting AMM fees, managing liquidity provider risks, and understanding blockchain block time effects on DeFi protocols.

This paper investigates the impact of trading fees on the profits earned by arbitrageurs interacting with Automated Market Makers (AMMs), which directly corresponds to the adverse selection costs borne by Liquidity Providers (LPs). It extends the "loss-versus-rebalancing" (LVR) framework developed in prior work (Saeed et al., 2020), which quantified arbitrage costs in a frictionless, continuous-time setting, by incorporating both trading fees and discrete arbitrageur arrival times.

Model Setup

The model considers an AMM trading a risky asset against a numeraire.

1. Market Price: The external market price $P_t$ of the risky asset follows a geometric Brownian motion (GBM) with volatility $\sigma$.
2. AMM: The AMM is a general Constant Function Market Maker (CFMM) defined by a bonding function $f(x,y)=L$, where $x$ and $y$ are the reserves of the risky asset and numeraire. The pool's state can be represented by the reserves $(x,y)$ or by the implied price $\tilde{P}_t$ such that the reserves are $(x^*(\tilde{P}_t), y^*(\tilde{P}_t))$, derived from the pool value function $V(P) = \max_{f(x,y)=L} (Px + y)$.
3. Trading Fees: Proportional trading fees are charged on trades, paid in the numeraire. The fees are parameterized by $\gamma_+$ and $\gamma_-$ (in log-price units) for buying and selling the risky asset from the pool, respectively. A symmetric fee structure ($\gamma_+ = \gamma_- = \gamma$) is assumed for much of the analysis for simplicity.
4. Arbitrageur Arrivals: Arbitrageurs can only trade against the AMM at discrete times $\tau_i$, modeled as arrivals of a Poisson process with rate $\lambda$. This rate $\lambda$ is linked to the blockchain's mean interblock time ($\Delta t = \lambda^{-1}$).
5. Arbitrageur Behavior: Arriving arbitrageurs are myopic profit maximizers. If the market price $P_t$ is outside the range $$[\tilde{P}_{t^-} e^{-\gamma_-}, \tilde{P}_{t^-} e^{+\gamma_+}]$$, where $\tilde{P}_{t^-}$ is the AMM price just before arrival, the arbitrageur trades with the AMM (and hedges on the external market) until the AMM's marginal price, adjusted for fees, equals the market price. This brings the AMM's implied price $\tilde{P}_t$ to either $P_t e^{-\gamma_+}$ or $P_t e^{+\gamma_-}$. If $P_t$ is within the no-trade range, no trade occurs.

Mispricing Process Dynamics

The core of the analysis focuses on the log mispricing process $z_t = \log(P_t / \tilde{P}_t)$.

• Between Arrivals: $z_t$ follows a diffusion process driven by the GBM of the market price: $dz_t = (\mu - \sigma^2/2) dt + \sigma dB_t$.
• At Arrivals ($\tau_i$): $z_t$ jumps. If $z_{\tau_i^-} > +\gamma_+$, it jumps to $+\gamma_+$. If $z_{\tau_i^-} < -\gamma_-$, it jumps to $-\gamma_-$. Otherwise, it stays the same.

This makes $z_t$ a Markovian jump-diffusion process, reflected within the "no-trade region" $[-\gamma_-, +\gamma_+]$.

Key Results

1. Stationary Distribution: Under the symmetry assumption ($\mu = \sigma^2/2, \gamma_+ = \gamma_- = \gamma$), the mispricing process $z_t$ is ergodic and converges to a unique stationary distribution $\pi(\cdot)$. This distribution has a density $p_\pi(z)$ that is:
   • Uniform over the no-trade region $[-\gamma, +\gamma]$.
   • Exponentially decaying outside this region (tails are exponential with parameter $\eta/\gamma = \sqrt{2\lambda}/\sigma$). The total probability mass within the no-trade region is $\pi_0 = \frac{\eta}{1+\eta}$, and outside is $\pi_+ + \pi_- = \frac{1}{1+\eta}$, where $\eta = \sqrt{2\lambda}\gamma/\sigma$.
2. Probability of Trade ($\rho$): The long-run fraction of blocks containing an arbitrage trade (i.e., the probability that an arriving arbitrageur finds $z_t$ outside the no-trade region) is $$\rho = \pi_+ + \pi_- = \frac{1}{1 + \eta} = \frac{1}{1 + \sqrt{2\lambda}\gamma/\sigma}$$. This probability decreases with higher fees ($\gamma$), more frequent blocks ($\lambda$), and lower volatility ($\sigma$).
3. Rate of Arbitrage Profit ($\mathcal{A}$): The expected instantaneous rate of arbitrage profit, $\mathcal{A} = \lim_{T\to\infty} E[\mathcal{A}_T]/T$, is derived in a semi-closed form involving an integral (Laplace transform) of the profit function over the tails of the stationary distribution.
4. CPMM Example: For a Constant Product Market Maker ($xy=k$), the normalized rate of arbitrage profit is given in explicit closed form: $$\frac{\mathcal{A}}{V(P)} = \frac{\sigma^2}{8} \times \rho \times \frac{\cosh(\gamma/2)}{1 - \sigma^2/(8\lambda)}$$. This shows that for small fees and fast blocks ($\lambda \to \infty$), $\mathcal{A} \approx \rho \times \text{LVR}$, where LVR = $\frac{\sigma^2}{8} V(P)$ is the frictionless arbitrage profit rate.
5. Asymptotic Analysis ($\lambda \to \infty$): In the fast block limit, for general AMMs (under mild technical conditions):
   • $$\mathcal{A} \approx \frac{\sigma^2 P}{2} \times \bar{y}^{*\prime}(P, \gamma) \times \rho$$, where $\bar{y}^{*\prime}(P, \gamma)$ is the average marginal liquidity at the edges of the no-trade region $$\frac{y^{*\prime}(Pe^{-\gamma}) + y^{*\prime}(Pe^{+\gamma})}{2}$$. For small $\gamma$, this simplifies to $\mathcal{A} \approx \text{LVR} \times \rho$.
   • This implies arbitrage profits scale as $$\mathcal{A} \propto \sqrt{\lambda^{-1}} \sigma^3 / \gamma$$. Faster blocks reduce arbitrage profit.
   • The instantaneous rate of fee generation $\mathcal{F}$ is also derived, showing that for small $\gamma$, $\mathcal{F} \approx \text{LVR} \times (1 - \rho)$.
   • Combining these, $\mathcal{A} + \mathcal{F} \approx \text{LVR}$ in this regime, meaning the total potential arbitrage profit (LVR) is split between actual arbitrageur profit ($\mathcal{A}$) and fees paid ($\mathcal{F}$), with the split determined by $\rho$.

Implications for Implementation

• Quantifying LP Costs: The model provides a more realistic way to estimate LP losses due to adverse selection (arbitrage) by accounting for fees and discrete trading opportunities. This can be used for ex post performance analysis or ex ante risk assessment using realized volatility.
• AMM Fee Setting: The formulas, especially the asymptotic results and the probability of trade $\rho$, give quantitative guidance on how fees impact arbitrage. Higher fees reduce $\mathcal{A}$ but also increase the time the AMM price deviates from the market price (increasing $\sigma_z$, the standard deviation of mispricing). The paper quantifies this trade-off (shown in Figure 1).
• Blockchain Design: The result that $\mathcal{A} \propto \sqrt{\lambda^{-1}}$ (for $\gamma > 0$) suggests that blockchains with faster block times (higher $\lambda$) inherently reduce the potential for arbitrage exploitation of AMMs, without necessarily impacting fee revenue from noise traders.
• Relationship to LVR: The paper shows that LVR represents an upper bound on arbitrage profits. Fees and discrete block times effectively act as frictions that reduce realized arbitrage profits below the LVR level. The factor $\rho$ quantifies this reduction in the fast-block, low-fee regime.

In summary, the paper provides a tractable mathematical framework for analyzing arbitrage against AMMs with fees and discrete trading times. It yields closed-form or semi-closed-form results for the stationary distribution of mispricing and the expected rate of arbitrage profits and fees, offering practical insights for LP risk management, AMM fee design, and understanding the impact of blockchain speed on DeFi protocols.