Paper: arXiv 2609.19102

Authors: Jun Maeda

Abstract

Variance swaps are a convenient instrument for trading vega and convexity, and a listed contract now trades on Cboe. We ask when a trader should put such a position on and when she should take it off, and for a perpetual, continuously settled contract we answer both in closed form: each threshold is the unique root of a smooth-pasting equation in confluent hypergeometric functions. Under the pricing measure the question has no content, the mark-to-market being a martingale. Under the physical measure with a variance risk premium it becomes meaningful, and then reduces: the accrued variance separates exactly, the maturity, strike and costs are absorbed into a single forcing term whose sign fixes the geometry of the exercise region, and what is left on the perpetual is an affine reward on a CIR process, which the optimal-stopping literature already solves. Entering the position and exiting it are not mirror images. An exit rule follows from the premium and the trading spread, both observable. For the short — the only side worth opening under the empirical sign of the premium — an entry rule exists only for an interval of carrying charges, and even there triggers only deep in the upper tail of the physical law: a trader who is out of the market pays nothing to stay out, so an operational entry rule needs a cost of idle capital that the exit rule does not.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents highly advanced mathematical derivations for optimal entry/exit rules, leveraging complex stochastic calculus and confluent hypergeometric functions. While the theoretical framework is robust and novel, the empirical validation is limited to numerical examples rather than extensive backtesting on real market data. The core contribution lies in the mathematical reduction of the problem to a known optimal stopping framework.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal entry/exit for perpetual variance swaps?] --> B{Methodology: Optimal stopping theory under physical measure};
    B --> C[Data/Inputs: Variance risk premium, trading spread, carrying charges];
    C --> D{Computational Process: Solving smooth-pasting equations with Confluent Hypergeometric Functions for CIR process};
    D --> E[Key Finding 1: Closed-form entry/exit rules derived];
    D --> F[Key Finding 2: Exit rule depends on premium & spread; Entry rule exists for interval of carrying charges];
    E & F --> G[Outcome: Practical guidance for Vega trading strategies];