Optimal Liquidation of Perpetual Contracts

TL;DR

The paper addresses optimal liquidation of a position in a perpetual contract under permanent and temporary price impact and funding costs. For the identity payoff $ψ(s)=s$, it derives a closed-form optimal trading speed $ν^{*}$ that depends on inventory and a funding-rate state, with the functions $ξ$ and $π$ solving Riccati-type ODEs. For general payoffs, it develops asymptotic approximations in the small funding-rate parameter $β$ and in short horizon $T$, including a second-order expansion and a short-horizon closed-form surrogate that remains near-optimal. The results bridge optimal execution with perpetual-contract dynamics and are supported by simulations showing how funding-rate effects alter liquidation patterns and the relative performance of the approximations.

Abstract

An agent holds a position in a perpetual contract with payoff function $ψ$ and attempts to liquidate the position while managing transaction costs, inventory risk, and funding rate payments. By solving the agent's stochastic control problem we obtain a closed-form expression for the optimal trading strategy when the payoff function is given by $ψ(s) = s$. When the payoff function is non-linear we provide approximations to the optimal strategy which apply when the funding rate parameter is small or when the length of the trading interval is small. We further prove that when $ψ$ is non-linear, the short time approximation can be written in terms of the closed-form trading strategy corresponding to the case of the identity payoff function.

Figures (5)

• Figure 1: Cross sectional density plots of inventory when trading according to the optimal strategy given in \ref{['eqn:closed-form nu']}. The thick dotted curve shows the Almgren-Chriss liquidation strategy. Thin curves represent the $5^{th}$ and $95^{th}$ percentile and the mean. In each panel, the initial spot price is $S_0 = 100$, but the initial perpetual price is $P_0 = 101$ (left), $P_0 = 100$ (middle), and $P_0 = 99$ (right). Parameter values are $T=1$, $k = 0.1$, $b = 0.1$, $\alpha = 100$, $\phi = 0.5$, $\beta = 5$, $\sigma = 1$, $\eta = 1$, $\rho = 0.3$.
• Figure 2: Sample paths of the process $A$ defined in Proposition \ref{['prop:behavior']} (left panel) and inventory (right panel) for various values of temporary price impact parameter $k$. Other parameter values are $T=5$, $b = 0.1$, $\alpha = 100$, $\phi = 0.5$, $\beta = 5$, $\sigma = 1$, $\eta = 1$, $\rho = 0.3$.
• Figure 3: Cross sectional density plots of the process $A$ defined in Proposition \ref{['prop:behavior']}. The temporary price impact parameter in each panel is $k=2\cdot 10^{-1}$ (left), $k=2\cdot 10^{-3}$ (middle), $k=2\cdot 10^{-5}$ (right). Other parameter values are $T=5$, $b = 0.1$, $\alpha = 100$, $\phi = 0.5$, $\beta = 5$, $\sigma = 1$, $\eta = 1$, $\rho = 0.3$.
• Figure 4: The payoff functions use to demonstrate asymptotic accuracy of trading strategies. The left and right panels add a logistic and quadratic function, respectively, to the identity.
• Figure 5: Strategy performance for various values of $T$. The left and right panels use the logistic and quadratic payoff functions, respectively, from Figure \ref{['fig:funding_functions']}. Other parameter values are $k = 0.1$, $b = 0.1$, $\alpha = 0.1$, $\phi = 0.5$, $\beta = 5$, $\sigma = 1$, $\eta = 1$, $\rho = 0.3$, $Q_0 = 10$, $P_0 = 100$, $S_0 = 100$.

Theorems & Definitions (10)

• Proposition 1: Value Function for Identity Payoff Function
• Theorem 2: Optimal Trading Strategy for Identity Payoff Function
• Proposition 3
• Theorem 5: Asymptotic Approximation of Value Function
• Theorem 6: Asymptotic Approximation of Optimal Trading Speed
• Theorem 7: Asymptotic Approximation of Value Function
• Theorem 8: Asymptotic Approximation of Optimal Trading Speed
• Proposition 9: Closed-form Approximation of Optimal Trading Speed
• Lemma 10
• Lemma 11