A mathematical framework for modelling CLMM dynamics in continuous time

Introduction to the show: ident: Quantitative Finance. The day in quantitative finance: models of markets, risk and prices.

Ruth: We're looking at the paper: "A mathematical framework for modelling CLMM dynamics in continuous time".

Warren: Here's the gist of the paper. A MATHEMATICAL FRAMEWORK FOR MODELLING CLMM DYNAMICS IN CONTINUOUS TIME develops a rigorous mathematical framework for analyzing Concentrated Liquidity Market Makers (CLMMs) in a continuous-time setting, demonstrating that the presence of trading fees fundamentally constrains the admissible price processes to those with finite variation to avoid infinite fee generation Modeling Liquidity Profiles The paper introduces a framework for modeling liquidity profiles as measure-valued processes, which precisely characterizes how concentrated liquidity affects market behavior and trading outcomes It establishes that the pool reserves are expressed in terms of the liquidity profile and its integral, providing a direct relationship between the reserves and the distribution of liquidity The reserve curve is both decreasing and convex, ensuring that the price impact of trades increases with trade size, which is a crucial property for market stability Furthermore, as in standard CFMMs, the instantaneous price P corresponds to the negative slope of the tangent to the reserve curve Analysis of Arbitrage Dynamics The analysis covers three distinct arbitrage models: myopic arbitrage, finite-horizon optimization, and infinite-horizon optimization with discounted or ergodic controls The study derives closed-form solutions for optimal arbitrage strategies under each scenario, providing insights into how rational actors interact with CLMMs and influence price discovery In the myopic arbitrage model, the mispricing process Zt can be decomposed as Zt = ln St − ln P0 + Lt − Ut and remains within the range

ln γ, − ln γ: for all t ≥ 0 Constraints Imposed by Fees A critical finding is that the inclusion of trading fees precludes the existence of diffusion terms in the price process to avoid infinite fee generation For myopic arbitrage, with fees, the arbitrageur’s profit is given by dARBt = (St − γ−1Pt)dx+t + (γPt − St)dx−t This suggests that the quadratic variation of Pt vanishes due to the finite variation assumption, meaning the arbitrageur’s profit is driven by terms involving dLt and dUt rather than the quadratic variation of the price process Optimal Control Strategies The paper derives optimal controls for different arbitrage scenarios, including a finite-horizon optimization problem formulated as sup u∈A

0,T: E

Z T 0 (St − Pt)dxt: For the infinite-horizon ergodic control model, the optimal Markov control is found to be u∗(z) = rτλz + 1 − 1/√λτ µ Under this optimal strategy, the mispricing process Zt follows an Ornstein-Uhlenbeck process dZt = rτλµ/τ− Zt dt + σdWt Implications for Market Design

Title and authors: Ruth: Alright, let's talk about the title and authors of this paper again: "A mathematical framework for modelling CLMM dynamics in continuous time." We need to make sure we nail down what that means in practical terms for us as market analysts.

Warren: I think the title is very descriptive because it immediately tells you the scope—it’s a mathematical framework, not just an application. It’s trying to build the underlying rules for how these concentrated liquidity market makers actually move when trading is happening continuously, which is a big step up from just looking at snapshots.

Iris: I agree. When we look at the authors, Shen-Ning Tung and Tai-Ho Wang, they are bringing in this specific mathematical rigor to analyze CLMMs within a continuous time setting. This allows them to use calculus on their models instead of relying solely on discrete step analyses which often introduce approximations.

Ruth: So when I think about the implication for us, it’s that we can move away from purely heuristic models and start building something that has a firm mathematical foundation for how liquidity profiles evolve.

Warren: Right. It moves the discussion from "what happens in this scenario" to "what are the fundamental constraints on what *can* happen in this scenario." That constraint, as they show, is heavily tied to trading fees.

Iris: The authors explicitly state that they model the evolution of liquidity profiles as measure-valued processes. This is a sophisticated way to capture the continuous nature of how liquidity itself changes across different price levels.

Ruth: So when we translate that for our listeners, it means we are looking at how the distribution of liquidity tokens evolves smoothly over time, not just jumping from one state to the next.

Warren: And they link those reserves directly to this profile and its integral, which is a powerful way to connect the actual assets in the pool to that abstract liquidity distribution.

Iris: That integral relationship is what allows them to define that reserve curve as both decreasing and convex, which mathematically enforces that price impact increases with trade size.

Ruth: So we get this elegant structure where bigger trades cause bigger price movements, which is a key stability insight for any market maker.

Warren: And they tie the instantaneous price P to the negative slope of that reserve curve—that’s our operational definition of price in this context.

The paper's summary: Ruth: Now, let's unpack what the paper actually summarizes as its core findings. Basically, it summarizes how concentrated liquidity affects market behavior and trading outcomes by using this continuous-time framework we just discussed.

Warren: The summary highlights that they examine three distinct arbitrage models: myopic, finite-horizon, and infinite-horizon with discounted or ergodic controls. It’s a comprehensive look at how different rational actors interact with these CLMMs.

Iris: They derive closed-form solutions for the optimal arbitrage strategies under all three scenarios, which gives us concrete answers about what a rational actor should be doing depending on their time horizon and control structure.

Ruth: In the myopic model, they show that the mispricing process Zt is decomposed into ln St minus ln P0 plus Lt minus Ut and it stays confined within a specific range of

ln gamma, − ln gamma: .

Warren: That confinement is significant because it shows that even in a simple myopic view, there are hard boundaries for how far the price can drift from its initial state. It limits the potential for runaway mispricing.

Iris: And then they show how fees force the price process to have finite variation to avoid infinite fee generation, which is a major constraint on what kind of dynamics we can observe.

Ruth: So the summary boils down to this: concentrated liquidity creates a specific reserve structure, and fees impose mathematical boundaries on the price dynamics that we need to respect.

Warren: And they show that under optimal controls—whether finite horizon or ergodic—there are specific, derived control strategies u* that lead to predictable outcomes for mispricing Zt.

Iris: For the infinite-horizon case, this results in an Ornstein-Uhlenbeck process for Zt, which points toward a natural mean-reversion tendency in CLMM markets over the long run.

The paper's improvements: Ruth: So we’ve covered what they found, and now let's discuss the suggested improvements or extensions to this framework. They aren't just static results; they propose ways to make this model even more useful.

Warren: One major improvement is in how they suggest we can optimize fees earned. They rewrite the liquidity provider’s value function using a complex expression involving k, r, and p terms that depends on which range k falls into—whether it’s less than r minus one, between r minus one and r, or greater than r.

Iris: That rewritten value function is what allows AI systems to develop optimal liquidity provision strategies by optimizing those fees earned under this framework. It’s a direct path for an AI to figure out the best way to set their liquidity levels given the fee structure.

Ruth: And another powerful improvement is in predicting inventory dynamics, where they give us formulas based on myopic arbitrage: dxt equals l(Pt)P minus one over two t squared (dLt minus dUt), and similarly for dy.

Warren: That formula is really useful because it shows that the inventory x only increases when Zt is at ln gamma, and it decreases only when Zt is at negative ln gamma. It gives us a very specific rule for predicting how the inventory changes based on where mispricing sits.

Iris: That relationship between the mispricing process and inventory change is what makes predicting CLMM inventory dynamics much more tractable for an econometrician because it provides a clear, deterministic link.

Ruth: Furthermore, they show we can design robust arbitrage strategies by solving the Hamilton-Jacobi-Bellman equation using that specific optimal control u* we mentioned earlier.

Warren: That optimal control u* is z minus partial z V lambda, and it’s found to be r tau lambda z plus one minus one over sqrt(lambda tau) mu <ref:2412.18580#pg1>. It gives us a concrete, actionable strategy for designing robust arbitrage plans under the finite horizon problem.

Conclusion: Ruth: We've covered a lot about "A mathematical framework for modelling CLMM dynamics in continuous time," summarizing how fees constrain price processes and how different control strategies yield predictable mispricing dynamics.

Warren: The main implication for me is that if you want to model CLMMs with real trading costs, you absolutely have to account for those fees because they fundamentally change the admissible price paths.

Iris: And from an econometric standpoint, the work provides a solid way to identify the mechanisms at play by showing how variation in reserves does the work and what falsifies this identification.

Ruth: So, if we take everything we've discussed—the liquidity profile modeling, the fee constraints on diffusion terms, and those optimal controls—it suggests a much more realistic picture of CLMM behavior.

Warren: It gives us a way to design systems that are better equipped to handle the frictions inherent in concentrated liquidity provision.

Iris: Ultimately, this framework helps us understand the structural tendencies in these markets by providing a rigorous mathematical language for those dynamics, even if it doesn't capture every single real-world noise.

Ruth: That’s it for today on this paper. We’ve talked about how modeling CLMM dynamics in continuous time requires respecting the constraints imposed by fees to keep the price processes finite and predictable.