The P behind Q: Empirical Evidence from Physical Drift in Put-Call Parity

Abstract

Put-call parity is a terminal-payoff identity, but its enforcement is capital-using. I study the carry gap, the annualized wedge between option-implied and OIS discount factors, in SPX and RUT index options. Quoted parity is tightly compressed, while the synthetic-traded forward channel leaves a systematic wedge. I interpret this wedge as an implementation premium under finite arbitrage capital. A drift-preserving GBM term, r μ-hat τ, improves in-sample and leave-one-year-out fit, especially in SPX. The evidence suggests that physical drift enters not option payoffs, but the process enforcing risk-neutral parity.