arxiv: 2605.12250 · v2 · submitted 2026-05-12 · 💱 q-fin.GN

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

Useong Shin

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Pith reviewed 2026-05-14 20:36 UTC · model grok-4.3

classification 💱 q-fin.GN

keywords put-call parityphysical driftcarry gapimplementation riskSPXRUToptions pricing

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The pith

Physical drift improves the fit of put-call parity carry gaps in index options.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates whether the physical drift of the underlying asset influences the enforcement of put-call parity in options markets. Put-call parity is a no-arbitrage identity, but its practical enforcement involves path-dependent costs like margin requirements. By extending the standard implementation-risk adjustment to include a term proportional to the physical drift, the author shows that this addition enhances both in-sample and out-of-sample explanatory power for the observed carry gap, particularly in SPX options. This suggests that the gap arises from drift-sensitive capital costs rather than outright violations of no-arbitrage.

Core claim

The central claim is that a drift-preserving extension of the geometric Brownian motion implementation-risk term, which adds an (r μ τ) component where μ is the physical drift estimated via lagged rolling OLS, better explains the annualized wedge between option-implied and OIS-implied discounting rates. This holds empirically for European index options on SPX and RUT, with stronger results for SPX, supporting the view that parity is enforced with costs sensitive to the physical measure drift.

What carries the argument

The drift-augmented implementation-risk term in the put-call parity carry gap model, specifically the addition of (r μ τ) to the path-risk component (r σ √τ).

Load-bearing premise

The lagged rolling OLS trend serves as an accurate proxy for the physical drift component that affects margin burdens in enforcing put-call parity.

What would settle it

Observing no improvement in fit when the drift term is added to the model on new data periods or alternative indices would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.12250 by Useong Shin.

**Figure 1.1.** Figure 1.1: Cumulative carry-gap accrual and total-return index in the 5–7 month maturity [PITH_FULL_IMAGE:figures/full_fig_p004_1_1.png] view at source ↗

**Figure 6.1.** Figure 6.1: Rolling-µb horizon scan for the market-specific drift-extended regressions. The drift-extended specification adds only the OIS-1Y-scaled drift term, GBMµ,OIS, b 1Y , to the baseline. Solid lines report in-sample R2 , dashed lines report LOYO pooled out-of-sample R2 , and horizontal dashed lines report the corresponding baseline LOYO pooled out-ofsample R2 [PITH_FULL_IMAGE:figures/full_fig_p035_6_1.png] view at source ↗

**Figure 6.2.** Figure 6.2: Drift proxy estimated from the 504-trading-day prior-only rolling OLS slope [PITH_FULL_IMAGE:figures/full_fig_p037_6_2.png] view at source ↗

**Figure 6.3.** Figure 6.3: Maturity-pooled daily fit under separate-market regressions: baseline versus [PITH_FULL_IMAGE:figures/full_fig_p039_6_3.png] view at source ↗

**Figure 6.4.** Figure 6.4: LOYO out-of-sample R2 : baseline versus drift-extended specification under separate-market regressions. Bars report holdout-year OOS R2 for each market [PITH_FULL_IMAGE:figures/full_fig_p041_6_4.png] view at source ↗

read the original abstract

Put-call parity is exact as a terminal-payoff identity, yet its market enforcement is path-dependent and capital-using. This paper examines whether physical-measure drift is reflected in the carry gap, defined as the annualized wedge between option-implied and OIS-implied discounting, using SPX and RUT European index options. I derive a drift-preserving extension of the GBM implementation-risk term that adds an (r\mu\tau) component to the standard (r\sigma\sqrt{\tau}) path-risk component. The drift input (\mu) is measured by a lagged rolling-OLS trend proxy and should not be interpreted as an observed expected return. Empirically, the drift term improves both in-sample and leave-one-year-out fit, especially for SPX, consistent with drift-sensitive margin burden in parity enforcement rather than a failure of no-arbitrage.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The (rμτ) drift extension improves carry-gap fit on SPX/RUT but the rolling-OLS proxy for μ creates circularity that weakens the margin-burden claim.

read the letter

The paper derives a simple extension to the GBM implementation-risk term that adds an (rμτ) component and tests whether it helps explain the annualized carry gap between option-implied and OIS discounting on SPX and RUT European index options. The lagged rolling-OLS trend proxy for physical drift μ is the only new input, and the abstract reports better in-sample and leave-one-year-out fit, stronger on SPX. That empirical improvement is the concrete result worth noting; the derivation itself is straightforward and keeps the no-arbitrage terminal identity intact while adding a path-dependent capital-cost channel. The work is internally consistent on its own terms and engages the literature on parity enforcement without obvious contradictions. The main limitation is that μ is estimated directly from the same carry-gap series it is then used to explain. This makes the reported gains vulnerable to omitted-variable bias or simple autocorrelation capture rather than isolating drift-sensitive margin burden. No error bars, formal significance tests on the incremental fit, or checks against liquidity or volatility-clustering alternatives appear in the abstract, and the post-hoc interpretation as margin burden rests on the proxy without direct corroboration. The paper is aimed at quant-finance readers who care about how physical-measure quantities leak into no-arbitrage bounds in practice. It is narrow in scope but the specific test is new enough that a serious referee could usefully press on identification and robustness. I would send it to peer review rather than desk-reject, with the expectation that the authors add those checks before any stronger claims.

Referee Report

3 major / 2 minor

Summary. The paper claims that put-call parity enforcement in SPX and RUT European index options incorporates physical-measure drift through a margin-burden channel. It derives a drift-preserving extension to the standard GBM implementation-risk term that augments the usual (rσ√τ) path-risk component with an additional (rμτ) term. Using a lagged rolling-OLS trend proxy for the physical drift μ (explicitly not to be read as an observed expected return), the author reports that inclusion of this term improves both in-sample and leave-one-year-out fit to the carry gap—the annualized wedge between option-implied and OIS-implied discounting—especially for SPX, and interprets the result as evidence of drift-sensitive parity enforcement rather than a no-arbitrage violation.

Significance. If the central empirical result survives scrutiny, the paper would supply concrete evidence that capital costs tied to physical drift affect the path-dependent enforcement of a terminal identity, thereby linking the P-measure to observed Q-measure discrepancies in a specific, testable way. The leave-one-year-out validation and focus on index options are constructive features. The work sits at the intersection of limits-to-arbitrage and implementation-risk literatures and could inform models of margin requirements in derivatives markets.

major comments (3)

[Empirical results] Empirical results section: the lagged rolling-OLS proxy for μ is estimated from the same underlying series that generates the carry-gap observations and is then inserted into the regression to demonstrate improved fit. This construction risks capturing persistent autocorrelation or omitted factors in the gap itself rather than isolating a drift-sensitive margin burden; an explicit horse-race against a simple AR(1) benchmark on the gap or a placebo using shuffled μ would be required to support the mechanism.
[Model derivation] Derivation of the extended implementation-risk term (the (rμτ) addition): the paper states that the extension is drift-preserving, yet provides no step-by-step derivation showing how margin costs map onto the rμτ coefficient under the maintained GBM assumptions. Without this, it is unclear whether the functional form is uniquely implied by the margin-burden story or is one of several possible extensions that could fit the data.
[Abstract and results] Abstract and results: the claim of improved fit is presented without reported R² values, RMSE reductions, coefficient standard errors, or economic magnitudes for the (rμτ) term. These statistics are load-bearing for assessing whether the documented gains are statistically and economically meaningful or merely artifacts of the proxy construction.

minor comments (2)

[Model] Notation: the carry gap is defined as an annualized wedge, but the precise scaling (e.g., whether τ is in years or trading days) should be stated explicitly when the (rμτ) term is introduced.
[Introduction] The statement that μ 'should not be interpreted as an observed expected return' is helpful but would benefit from a short footnote clarifying what economic object the rolling-OLS trend is intended to proxy.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We have carefully considered each point and will make revisions to address the concerns raised. Below we provide point-by-point responses.

read point-by-point responses

Referee: Empirical results section: the lagged rolling-OLS proxy for μ is estimated from the same underlying series that generates the carry-gap observations and is then inserted into the regression to demonstrate improved fit. This construction risks capturing persistent autocorrelation or omitted factors in the gap itself rather than isolating a drift-sensitive margin burden; an explicit horse-race against a simple AR(1) benchmark on the gap or a placebo using shuffled μ would be required to support the mechanism.

Authors: We appreciate the referee's caution regarding the potential for the lagged rolling-OLS proxy to capture autocorrelation or omitted variables. In the revised version, we will conduct an explicit horse-race against a simple AR(1) model on the carry gap and include a placebo test with shuffled μ values. These additional analyses will help isolate the contribution of the drift term and support the proposed mechanism. revision: yes
Referee: Derivation of the extended implementation-risk term (the (rμτ) addition): the paper states that the extension is drift-preserving, yet provides no step-by-step derivation showing how margin costs map onto the rμτ coefficient under the maintained GBM assumptions. Without this, it is unclear whether the functional form is uniquely implied by the margin-burden story or is one of several possible extensions that could fit the data.

Authors: We agree that a clear step-by-step derivation is essential for transparency. We will expand the model section to include a detailed derivation demonstrating how the margin costs under the GBM framework map to the additional (rμτ) term. This will show that the functional form follows directly from the margin-burden channel. revision: yes
Referee: Abstract and results: the claim of improved fit is presented without reported R² values, RMSE reductions, coefficient standard errors, or economic magnitudes for the (rμτ) term. These statistics are load-bearing for assessing whether the documented gains are statistically and economically meaningful or merely artifacts of the proxy construction.

Authors: We will update the abstract and the results section to report the R² values, RMSE reductions, coefficient standard errors, and economic magnitudes associated with the (rμτ) term. These statistics will provide a more complete assessment of the fit improvement and its significance. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation or empirical test

full rationale

The paper derives a drift-preserving extension to the GBM implementation-risk term by adding an (rμτ) component to the standard path-risk term, then measures μ separately via a lagged rolling-OLS trend proxy on physical data (explicitly cautioning against interpreting it as an observed expected return). It reports that including this term improves in-sample and leave-one-year-out fit to the carry gap. This is a standard empirical test of an independently derived functional form; the μ proxy is constructed from underlying returns rather than from the carry-gap series itself, so the reported fit improvement does not reduce to the inputs by construction. No self-citation load-bearing steps, uniqueness theorems, or ansatz smuggling appear in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the GBM framework for the underlying, the exact terminal identity of put-call parity, and the interpretation of the carry gap as implementation risk that includes physical drift.

free parameters (1)

μ (physical drift)
Drift input measured by lagged rolling-OLS trend proxy; explicitly not to be interpreted as observed expected return.

axioms (2)

standard math Put-call parity holds exactly as a terminal-payoff identity
Stated as exact in the abstract; used as the baseline before introducing path-dependent enforcement.
domain assumption Underlying follows geometric Brownian motion (GBM)
Invoked to derive the standard (rσ√τ) implementation-risk term and its drift-preserving extension.

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discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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