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

Technical Summary: The P behind Q: Empirical Evidence from Physical Drift in Put–Call Parity

Problem Statement
Put–call parity is a terminal-payoff identity that holds under risk-neutral (Q-measure) pricing, independent of the physical (P-measure) drift of the underlying asset. However, enforcing this parity before maturity requires capital commitment, variation margin, and funding, making the process path-dependent. While quoted put–call parity residuals in price space are often tightly compressed in liquid markets like SPX and RUT, a systematic wedge remains in the "synthetic–traded forward channel" when option-implied discount factors are compared against OIS benchmarks.

The central problem addressed is whether this residual "carry gap" (the annualized wedge between option-implied and OIS discount factors) reflects only diffusion-driven path risk (volatility) or if it also incorporates the cost of maintaining parity under a physical price path with non-zero drift. Specifically, the paper investigates if the direction of the physical price path creates asymmetric margin and capital burdens for arbitrageurs, thereby exposing the enforcement mechanism to physical drift even when the terminal payoff remains risk-neutral.

Methodology
The paper employs a reduced-form empirical approach using minute-level NBBO quotes for SPX and RUT European-style index options, matched with OIS discount factors.

1. Carry Gap Construction: The dependent variable is the carry gap ($CG$), defined as the annualized log-difference between the OIS discount factor and the option-implied discount factor derived from put–call parity.
2. Baseline Specification: The authors first establish a benchmark regression based on a Geometric Brownian Motion (GBM) path-risk term. This term, $r\sigma\sqrt{\tau}$, represents the support capital required to survive random price diffusion, scaled by opportunity cost rates (OIS rates). The baseline model controls for diffusion path risk (using 1-year and 10-year OIS rates), trading frictions (bid-ask spreads), and broad financial conditions (NFCI).
3. Drift-Preserving Extension: The core theoretical contribution is the derivation of a drift-preserving extension to the support-capital argument. While the terminal payoff is deterministic, the pre-maturity margin path depends on the direction of the futures leg exposure. The authors derive that preserving physical drift $\mu$ adds a first-order directional burden proportional to $r\mu\tau$.
   • Since true physical drift $\mu$ is unobserved, the paper uses a reduced-form proxy $\hat{\mu}$, defined as the prior rolling-OLS slope of the log total-return index (using a 504-trading-day lookback to avoid look-ahead bias).
   • The empirical regressor is constructed as $GBM_{\hat{\mu}, OIS, 1Y} \propto r \hat{\mu} \tau$.
4. Empirical Design: The study estimates separate regressions for SPX and RUT. It evaluates performance using in-sample fit ($R^2$, RMSE, MAE) and leave-one-year-out (LOYO) out-of-sample validation. Inference is conducted using date-based HAC (Newey–West) standard errors with a 21-day lag.

Key Contributions

1. Reframing the Carry Gap: The paper reframes the carry gap not as a failure of put–call parity, but as an "implementation wedge." It demonstrates that while price-space residuals are compressed, the cost of enforcing parity in carry space remains visible and state-dependent.
2. Drift-Preserving GBM Term: The paper derives and tests a new empirical term, $r\hat{\mu}\tau$, which extends the standard diffusion-based path-risk term ($r\sigma\sqrt{\tau}$). This term captures the directional margin burden associated with physical price trends, distinguishing between the costs of enforcing parity from the long-futures side versus the short-futures side.
3. Physical Drift in Enforcement: The study provides evidence that physical drift enters the process of enforcing risk-neutral parity (via margin and capital costs) rather than the option payoffs themselves. This supports the "P behind Q" thesis: the capital-support technology required to implement a risk-neutral relation is realized along the physical price path.

Results

• In-Sample Fit: Adding the drift term ($GBM_{\hat{\mu}, OIS, 1Y}$) improves in-sample fit for both markets, with a significantly larger gain in SPX. For SPX, $R^2$ increases from 0.312 to 0.348; for RUT, it rises from 0.281 to 0.287. The improvement is monotonic across maturity buckets in SPX, becoming more pronounced at longer tenors (e.g., a 0.106 $R^2$ gain in the 21m+ bucket), consistent with the $\tau$-scaling of the drift term.
• Out-of-Sample Performance: The drift-extended specification improves LOYO out-of-sample performance. In SPX, pooled OOS $R^2$ rises from 0.221 to 0.310, with notable improvements during volatile regimes (e.g., 2020, 2021). In RUT, the improvement is smaller (pooled OOS $R^2$ from 0.171 to 0.188) but consistent.
• Coefficient Structure: The coefficient on the drift term ($\psi$) is positive and statistically significant at the 1% level in both markets under HAC(21) inference.
  • SPX: $\psi \approx 0.119$
  • RUT: $\psi \approx 0.054$
  • The positive sign indicates that the observable drift proxy aligns with the carry gap after controlling for diffusion risk and frictions.
  • The inclusion of the drift term reduces the magnitude and significance of the long-horizon OIS path-risk term in SPX, suggesting the drift proxy absorbs low-frequency variation previously attributed to long-rate path risk.
• Stability: Coefficient signs remain stable across all LOYO folds, and the drift term's significance holds across different lookback horizons, particularly around the 504-day (two-year) medium-horizon window.

Significance and Claims
The paper claims modestly that it does not challenge the validity of risk-neutral pricing or the put–call parity terminal identity. Instead, it identifies a distinct "implementation layer" behind the Q-measure relation.

• The "P behind Q": The significance lies in demonstrating that the capital burden of enforcing a risk-neutral relation is exposed to physical-measure price dynamics. The "P" (physical drift) influences the "Q" (risk-neutral enforcement) not by altering the payoff, but by altering the cost of the capital required to survive the path to maturity.
• Implementation Premium: The carry gap is interpreted as an implementation premium. Even when quoted parity is tight, the synthetic-forward channel may remain misaligned with traded futures due to the finite nature of arbitrage capital, funding constraints, and the asymmetric margin burdens created by physical drift.
• Practical Interpretation: The fitted carry gap serves as a state-dependent "price-space hurdle." It suggests that arbitrageurs should compare observed parity residuals not just against zero, but against a hurdle that accounts for volatility, funding rates, trading frictions, and the directional margin burden implied by recent physical drift.

The paper concludes that while a full structural model of parity-enforcement pricing remains for future research, the reduced-form evidence confirms that the cost of enforcing risk-neutral parity is partially aligned with physical-path information.