Optimal entry and exit for variance swaps: closed-form rules for the perpetual contract

Battle Count: 63

Highly relevant for desks trading variance swaps, VIX futures, and listed variance futures (Cboe S&P 500 Variance Futures launched September 2024). Provides actionable closed-form entry and exit thresholds, identifies the critical role of carrying costs and idle capital charges, and reveals the fundamental asymmetry between entry and exit timing. The finding that entry requires a cost of idle capital while exit does not is directly applicable to trading desk policy design.

The closed-form solutions require computing confluent hypergeometric functions M(a,ν;ςv) and U(a,ν;ςv) and solving transcendental smooth-pasting equations numerically (bisection). The dated contract requires a fully implicit finite-difference scheme with Howard iteration. Parameter calibration (Feller condition, stationary law percentiles) adds complexity. However, the core logic reduces to two threshold comparisons against the current variance state.

The paper provides complete mathematical derivations, explicit parameter calibrations (Section 8.1), and detailed numerical results. All closed-form solutions are expressed in terms of standard confluent hypergeometric functions. However, no code repository is provided. The numerical solver for the dated contract (finite-difference scheme) is described but not released. Parameters are fully specified for replication.