# Liquidity Management for Token Issuers

## Introduction
Decentralized Finance (DeFi) has revolutionized financial transactions through peer-to-peer interactions, relying heavily on tokens as digital assets. **Liquidity**, the ease with which tokens can be bought or sold without significantly affecting price, is crucial for efficient price discovery, reduced costs, and a healthy ecosystem. Insufficient liquidity leads to price volatility, hindering adoption. Token issuers face the challenge of effectively managing token liquidity.

### Problem Definition
The "Token Issuer's Liquidity Management Problem," a central DeFi challenge, concerns how token issuers manage liquidity on Decentralized Exchanges (DEXs), often using Automated Market Makers (AMMs). The core issue is balancing desired outcomes like price stability with token holder trading. Effective navigation of this problem is crucial for token issuers to create a stable token environment.

### Key Agents
This problem involves two agents with potentially conflicting objectives:

* **The Token Issuer:** The project or entity creating and distributing the token. Issuers aim to maintain stable prices within a budget, using reserves and bootstrapping liquidity. Objectives include:
* Maintaining price stability.
* Managing budget efficiently.
* **The Token Holder:** Individuals or entities acquiring tokens through airdrops, incentives, etc. Holders aim to maximize returns through trading. Objectives include:
* Liquidating holdings at favorable prices.
* Maximizing profit.

Tension arises from these competing objectives. Issuers seek to minimize volatility, while holder trading can cause fluctuations. This dynamic is the crux of the Token Issuer's Liquidity Management Problem.

### Existing Solutions
Examples of solutions and frameworks include:

* **Arrakis Pro:** Automates liquidity provisioning in concentrated liquidity pools, demonstrating dynamic liquidity management.
* **OlympusDAO:** A DeFi protocol aiming to establish a low-volatility reserve asset (OHM) through mechanisms like staking and bonding. It can be analyzed as a stochastic control system focused on price stabilization, illustrating a theoretical approach to liquidity management.

## Static Liquidity Management
*Static liquidity management* involves establishing a pre-determined liquidity profile on an Automated Market Maker (AMM). This means deciding in advance how much liquidity to provide at various price levels and maintaining that configuration over time. The token issuer sets the parameters of the liquidity pool and does not actively adjust them in response to short-term market fluctuations. For simplicity, we assume that the token holder sells their incentive rewards immediately.

### Setup
Here, we assume the token issuer manages their liquidity via Uniswap v3. In Uniswap v3, liquidity providers can specify price ranges within which they want to provide liquidity, effectively creating a static liquidity profile.

The concept of a liquidity profile can be represented mathematically as $p \mapsto \ell(p)$, where $p$ denotes the price and $\ell(p)$ represents the amount of liquidity provided at that price. This function defines the relationship between price and the quantity of liquidity available.

**Note:** A Constant Function Market Maker (CFMM) setup (\( f(x, y) = L \)) can also be considered.

### Token Issuer's Objective
The token issuer aims to optimize their position by maximizing their cash reserves over time while minimizing deviations of the token's market price from a desired target price. This objective reflects the issuer's need to maintain financial stability and price predictability for the token.

The token issuer's objective can be mathematically represented as:

$$
\max_{u \in \mathcal{A}_{[0,T]}} \mathbb{E} \left[ y_T - \lambda \int^T_0 (p_t - \bar{p})^2 dt \right]
$$

where:
* $u_t$ represents the trading strategy employed by the token issuer to influence the token price.
* $y_t$ denotes the quantity of the token issuer's cash position at time $t$, with $y_0$ representing their initial budget for liquidity management.
* $p_t$ is the AMM price of the token at time $t$.
* $\bar{p}$ represents the target token price that the issuer aims to maintain.
* $\lambda$ is a parameter that quantifies the penalty for price fluctuations, reflecting the issuer's sensitivity to deviations from the target price.

### Price and Reserve Dynamics
To analyze the dynamics of reserves within a static liquidity management framework, it's necessary to normalize the trading strategies of the token issuer. Let $u_t$ represent the trading strategy of the token issuer. These strategies can be normalized as:

\begin{split}
\frac{dp_t}{p_t} &= (u_t - \frac{2 p_t^{\frac12}}{\ell(p_t)}r) dt + \sigma dW_t \\
d y_t &= -\frac12 \ell(p_t) p_t^{\frac12} \left(v_t dt + \sigma dW_t \right)
\end{split}

where
* $p_t$ is the price of the token at time $t$.
* $r$ is the rewards emission rate.
* $\sigma$ represents the volatility of the token price.
* $dW_t$ is a Wiener process representing random price fluctuations.
* $y_t$ represents the token issuer's cash position at time $t$.

#### Key Questions
A key question in static liquidity management is determining the optimal shape of the liquidity profile $\ell(p)$ and the corresponding trading strategy to achieve the desired objectives, such as price stability and capital efficiency.

## Dynamic Liquidity Management
Dynamic liquidity management involves actively adjusting the amount of liquidity provided on a DEX in response to changing market conditions. This approach requires continuous monitoring of the market and active intervention to maintain liquidity, denoted as $\ell_t$, around the current price, denoted as $p_t$. Token issuers using dynamic liquidity management strategies aim to optimize their liquidity provision in real-time.

### Token Holder's Objective
Unlike the setup in static liquidity management, which assumes the token holder sells their rewards immediately, here we assume the token holder seeks to maximize their cash received from selling their tokens, taking into account the potential impact of their sales on the token's price. This objective reflects the token holders' goal of maximizing their returns from trading activities.

The token holder's objective can be mathematically represented as:

$$
\max_{v \in \mathcal{A}_{[0,T]}} \mathbb{E} \left[ \tilde{y}_T - \psi \tilde{x}_T - \kappa \int^T_0 \tilde{x}^2_t dt \right]
$$

where:
* $v_t$ represents the trading strategy employed by the token holder.
* $\tilde{x}_t$ denotes the quantity of the token holder's token at time $t$.
* $\tilde{y}_t$ represents the quantity of the token holder's cash position at time $t$.
* $\kappa$ and $\psi$ are parameters that quantifies the penalty for holding the token inventory, reflecting the holder's preference for liquidating their position.

### Price and Reserve Dynamics
Similar to static liquidity management, the analysis of reserve dynamics in dynamic liquidity management requires normalizing the trading strategies of the token issuer and token holders. The normalization can be represented as:

$$
\frac{dp_t}{p_t} = (u_t - v_t) dt + \sigma dW_t
$$

It follows that the reserve dynamics are:

\begin{split}
d y_t &= -\frac12 \ell_t p_t^{\frac12} \left(v_t dt + \sigma dW_t \right) \\
d \tilde{y}_t &= \frac12 \ell_t p_t^{\frac12} v_t dt \\
d \tilde{x}_t &= \left(r - \frac12 \ell_t p_t^{-\frac12} v_t \right)dt
\end{split}

where:
* $\ell_t$ represents the dynamically adjusted liquidity profile at time $t$.
* $y_t$ represents the token issuer's cash position at time $t$.
* $\tilde{y}_t$ represents the token holder's cash position at time $t$.
* $\tilde{x}_t$ represents the token holder's token holdings at time $t$.
* $r$ is the rewards emission rate.

### Key Questions
A key question in dynamic liquidity management is how to formulate the problem as a Stackelberg or mean-field game, where the token issuer and token holders interact strategically over time. This involves determining the equilibrium strategies among agents.

## References
* Bergault, P., & Sánchez-Betancourt, L. (2024). *A mean field game between informed traders and a broker*. arXiv preprint arXiv:2401.05257.
* Cartea, Á., & Sánchez-Betancourt, L. (2022, November 2). *Brokers and informed traders: Dealing with toxic flow and extracting trading signals*. SSRN. [https://doi.org/10.2139/ssrn.4265814](https://doi.org/10.2139/ssrn.4265814)
* Cartea, Álvaro, Sebastian Jaimungal, and Leandro Sánchez-Betancourt. 2024. *Nash Equilibrium between Brokers and Traders*. arXiv preprint arXiv:2407.10561.
* Chitra, T., Kulkarni, K., Angeris, G., Evans, A., & Xu, V. (2022). *DeFi liquidity management via optimal control: Ohm as a case study*. Retrieved from https://angeris.github.io/papers/ohm-staking.pdfhttps://angeris.github.io/papers/ohm-staking.pdf
* OlympusDAO. *OlympusDAO doc*. Retrieved from https://docs.olympusdao.finance/main/overview/intro