Interaction Between P and Q Measures: A Put-Call Parity Perspective

Option pricing is typically conducted under the risk-neutral (Q) measure, whereas the drift of the underlying asset in the real world is specified by the physical (P) measure. The interaction between these two measures is subtle and has been the subject of several studies, including research on delta hedging and the volatility risk premium.

Reference [1] contributes to this literature by examining put-call parity. Specifically, the author argues that while put-call parity is a terminal payoff condition operating under the Q measure, enforcing it in practice is not costless, and these costs are influenced by the real-world drift of the underlying asset.

The author pointed out,

This paper studies the carry-space wedge in put–call parity from the perspective of drift-sensitive implementation risk. Put–call parity is a static no-arbitrage relation over terminal payoffs and plays a central role in Black–Scholes–Merton-type risk-neutral pricing. In actual markets, however, the position that enforces parity is not held as a frictionless static identity. It must be maintained until maturity while being exposed to interim price paths, variation margin, trading frictions, funding conditions, and finite-capital constraints. The observed carry-space wedge may therefore reflect the path-dependent capital burden of parity enforcement rather than a terminal pricing error.

The main contribution of the paper is to add a drift-preserving component to the standard diffusion-based GBM path-risk term. The baseline support-capital argument implies an rσ√τ burden under zero-drift Brownian motion. When drift is preserved, a first-order approximation produces an additional directional margin-burden component of the form rµτ. I derive this term and test its empirical counterpart in SPX and RUT index options. The empirical µ used in the regression is not an observed future expected return. Because future physical drift is unobservable, I proxy for it using the rolling OLS slope of past log total-return paths.

In short, using SPX and RUT index options, the paper finds that the drift term adds explanatory power for the carry gap beyond volatility-based path-risk measures, bid-ask spreads, and financial-condition controls. The author interprets this as evidence that the capital burden of parity enforcement may be influenced by physical-market trends, even though option pricing itself remains risk-neutral.

The results bring me to the thought that hedgers and arbitrageurs have to assume and manage margin requirements, funding costs, and capital constraints. This is not always convenient. Hence, their returns reflect, at least in part, these costs.

Let us know what you think in the comments below.

References

[1] Shin, U. (2026). The P behind Q: Empirical Evidence from Physical Drift in Put–Call Parity, arXiv:2605.12250v2.