Paper: arXiv 2310.11771
Authors: Damien Ackerer, Julien Hugonnier, Urban Jermann
Abstract
Perpetual futures are contracts without expiration date in which the anchoring of the futures price to the spot price is ensured by periodic funding payments from long to short. We derive explicit expressions for the no-arbitrage price of various perpetual contracts, including linear, inverse, and quantos futures in both discrete and continuous-time. In particular, we show that the futures price is given by the risk-neutral expectation of the spot sampled at a random time that reflects the intensity of the price anchoring. Furthermore, we identify funding specifications that guarantee the coincidence of futures and spot prices, and show that for such specifications perpetual futures contracts can be replicated by dynamic trading in primitive securities.
Complexity vs Empirical Score
- Math Complexity: 8.0/10
- Empirical Rigor: 2.5/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper provides a rigorous mathematical foundation for perpetual futures pricing across various settlement structures (linear, inverse, quantos) using continuous and discrete-time no-arbitrage models. While highly theoretical with limited empirical backtesting on live market data, it fills a major theoretical gap in crypto-derivatives literature.
Research Flowchart
flowchart TD
A["Research Goal:<br>Derive No-Arbitrage Pricing & Replication<br>for Perpetual Futures"] --> B["Inputs:<br>Spot Price Dynamics & Funding Rate Specifications<br>(Linear, Inverse, Quantos)"]
B --> C["Methodology:<br>Model Discrete & Continuous-Time<br>Funding Mechanisms"]
C --> D["Computational Process:<br>Evaluate Risk-Neutral Expectation<br>at Random Anchoring Times"]
D --> E["Key Outcome 1:<br>Futures Price = Expected Spot<br>at Random Time Intensity"]
D --> F["Key Outcome 2:<br>Identified Funding Rules for Exact Spot-Futures<br>Coincidence & Dynamic Replication"]