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Equilibrium Liquidity and Risk Offsetting in Decentralised Markets

Fayçal Drissi, Xuchen Wu, Sebastian Jaimungal

Abstract

We study the economic viability of liquidity provision in decentralised exchanges (DEXs) within a structural framework in which market outcomes are endogenous. We formulate strategic interactions as a sequential game: a risk-averse liquidity provider (LP) sets the supply of liquidity in the DEX and a costly dynamic replication strategy in a centralised exchange (CEX), price-sensitive traders determine trading volumes, and arbitrageurs align prices. We establish existence of equilibrium under general trading functions. We show that DEX liquidity depth is a central instrument for risk management, because the LP adjusts liquidity ex ante to manage exposure. In addition to the classical trade-off between liquidity demand and adverse selection, we identify two further determinants of the viability of liquidity provision: the ratio of risk aversion to replication costs and private information. The ratio governs the aggressiveness of replication: greater relative risk aversion reduces risk but also lowers equilibrium liquidity and its mean profitability. Private information has a non-monotonic effect. For moderate price movements, speculative benefits increase liquidity. For large price movements, anticipated adverse selection and replication costs lead to thinner markets.

Equilibrium Liquidity and Risk Offsetting in Decentralised Markets

Abstract

We study the economic viability of liquidity provision in decentralised exchanges (DEXs) within a structural framework in which market outcomes are endogenous. We formulate strategic interactions as a sequential game: a risk-averse liquidity provider (LP) sets the supply of liquidity in the DEX and a costly dynamic replication strategy in a centralised exchange (CEX), price-sensitive traders determine trading volumes, and arbitrageurs align prices. We establish existence of equilibrium under general trading functions. We show that DEX liquidity depth is a central instrument for risk management, because the LP adjusts liquidity ex ante to manage exposure. In addition to the classical trade-off between liquidity demand and adverse selection, we identify two further determinants of the viability of liquidity provision: the ratio of risk aversion to replication costs and private information. The ratio governs the aggressiveness of replication: greater relative risk aversion reduces risk but also lowers equilibrium liquidity and its mean profitability. Private information has a non-monotonic effect. For moderate price movements, speculative benefits increase liquidity. For large price movements, anticipated adverse selection and replication costs lead to thinner markets.
Paper Structure (43 sections, 13 theorems, 175 equations, 6 figures)

This paper contains 43 sections, 13 theorems, 175 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General features of the model
  3. DEX price and liquidity.
  4. The model
  5. Stage one
  6. Stage two
  7. Stage three
  8. Stage three: trading volumes and fee revenue
  9. Assumptions
  10. Liquidity needs and trading volumes
  11. Fee revenue
  12. Stage two: risk offsetting
  13. Assumptions
  14. DEX holdings.
  15. CEX holdings.
  16. ...and 28 more sections

Key Result

Lemma 1

The performance criterion eqn:criterion2 can be written as $J[\nu]+H\,$, where $H=Q_0\,F_0+\mathbb{E}\![\int_{0}^{T}\{Q_0\,A_t\,F_t-\tfrac{\phi}{2}\,(Y_t+Q_0)^2\}\, {\mathrm{d} t}]$ is a well-defined real number that does not depend on $\nu$, and $J$ is the bounded linear-quadratic functional where $\Lambda$ is a symmetric bounded linear operator on ${\mathcal{A}}_2$ and $b$ is an element of ${\m

Figures (6)

  • Figure 1: Illustration of how iso-liquidity curves map reserve levels into execution prices.
  • Figure 2: Sample path of the LP's reserves $Y_t$ held in the DEX and the inventory $Q_t$ held in the CEX. The left panel corresponds to a ratio of risk aversion to trading costs $\psi = 5$, while the right panel corresponds to $\psi = 100$. Other default parameter values are trading costs $\eta=0.01$, profitability $\gamma = 0.05$, fundamental volatility $\sigma = 0.2$, and investment horizon $T = 1$.
  • Figure 3: Equilibrium liquidity supply $\kappa^{\star}$ in \ref{['eq:kappahedgeCPM']} plotted as functions of the model primitives. Default parameter values are: fundamental volatility $\sigma = 0.2$, investment horizon $T = 1$, private signal $A = 0$, CEX trading cost $\eta = 10^{-2}$, ratio $\psi = \sqrt{\phi/2\eta} = 10$, and profitability $\gamma = 0.05$.
  • Figure 4: Violin plots of the empirical distributions of the equilibrium profits of liquidity provision expressed in units of the reference asset $X$ (top panels) and in percentage term (bottom panels). The distribution is obtained from $100$,$000$ market simulations, with the time interval discretised into $100$ steps. Default parameter values are $\sigma = 0.2$, $T = 1$, $A = 0$, $\eta = 10^{-2}$, $\psi = 10$, and $\gamma = 0.05$.
  • Figure 5: Sample paths of the fundamental price $F_t$ (top panels), and the LP's reserves $Y_t$ held in the DEX and the inventory $Q_t$ held in the CEX (bottom panels). The left panels correspond to $A=-0.2$ and the right panels to $A=0.2.$ Other default parameter values are ratio $\psi=0.1,$ profitability $\gamma = 0.05$, fundamental volatility $\sigma = 0.2$, and investment horizon $T = 0.3$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • ...and 8 more