Pricing and Replicating Perpetual Contracts with Funding or Discounts

Summary

The document studies two perpetual contract designs in a continuous-time, no-arbitrage market without transaction costs. In one design, the long side receives a fixed payoff linked to underlying assets and pays funding to the short side. In the other, the payoff is adjusted by a time-varying discount factor and there are no funding payments. Under continuous, strictly positive asset prices, the paper derives model-free funding and discount rate expressions and replication strategies for the short side.

It then allows asset prices to jump. The resulting rate formulas do not depend on the underlying assets’ volatility dynamics, but they do depend on jump intensity under the pricing measure. An explicit short-side replication strategy is given when volatility is independent of the risky assets. Examples connect these contracts to variance swaps and leveraged exchange-traded funds. The conclusions rely on the stated idealized market assumptions, and the jump-case results retain dependence on jump intensity.

Key ideas

• The paper compares perpetual contracts with funding payments and contracts with a changing discount factor.
• Under continuous positive prices, it derives rate expressions and short-side replication strategies.
• With jumps, the formulas remain independent of volatility dynamics but depend on pricing-measure jump intensity.
• An explicit replication strategy in the jump setting requires volatility independence from risky assets.
• Examples relate perpetual contracts to variance swaps and leveraged exchange-traded funds.

Tags

# A primer on perpetuals


# A primer on perpetuals









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.