A primer on perpetuals

TL;DR

This paper introduces and analyzes two types of perpetual contracts in continuous-time markets, providing model-free and semi-robust formulas for funding and discount rates, along with replication strategies, connecting them to traditional financial instruments.

Contribution

It presents novel definitions, formulas, and replication strategies for perpetual contracts in both continuous and jump market models, extending existing financial derivatives theory.

Findings

Model-free expressions for funding and discount rates in continuous markets.

Semi-robust formulas accounting for jumps in asset prices.

Explicit replication strategies when volatility is independent of asset jumps.

Abstract

We consider a continuous-time financial market with no arbitrage and no transactions costs. In this setting, we introduce two types of perpetual contracts, one in which the payoff to the long side is a fixed function of the underlyers and the long side pays a funding rate to the short side, the other in which the payoff to the long side is a fixed function of the underlyers times a discount factor that changes over time but no funding payments are required. Assuming asset prices are continuous and strictly positive, we derive model-free expressions for the funding rate and discount rate of these perpetual contracts as well as replication strategies for the short side. When asset prices can jump, we derive expressions for the funding and discount rates, which are semi-robust in the sense that they do not depend on the dynamics of the volatility process of the underlying risky assets, but do depend on the intensity of jumps under the market's pricing measure. When asset prices can jump and the volatility process is independent of the underlying risky assets, we derive an explicit replication strategy for the short side of a perpetual contract. Throughout the paper, we illustrate through examples how specific perpetual contracts relate to traditional financial instruments such as variance swaps and leveraged exchange traded funds.