Tuning in to Frequencies: How Global Assets Align with U.S. Put-Call Parity Residuals

This paper demonstrates that the pricing gap between the SPX and RUT, traditionally viewed through risk-neutral lenses, is significantly explained by residual physical-measure investment opportunities captured by global assets (IEFA, IGOV, IAU), suggesting that finite-capital put-call parity enforcement reflects real-world investment dynamics rather than simple arbitrage failures.

The Gist

Imagine the financial world as a giant, high-stakes game of "Price Tag Matching."

In this game, there's a golden rule called Put-Call Parity. It's like a mathematical law that says: "If you buy a specific combination of options (contracts to buy or sell later) and a futures contract, the total price must equal the price of the stock itself." If the prices don't match, a smart trader (an "arbitrageur") should be able to buy the cheap side, sell the expensive side, and lock in a risk-free profit.

In a perfect world, this rule would be followed perfectly, and the prices would always match. But in the real world, they don't match exactly. There's a tiny gap between the two prices. The paper calls this the "Carry Gap."

The Mystery: Why is there a gap?

For a long time, economists thought this gap was just about the cost of borrowing money (interest rates) and the cost of trading (fees). They built a model using only U.S. interest rates (called OIS rates) to explain the gap.

The Analogy: Imagine you are a delivery driver (the arbitrageur) trying to move a package from Point A to Point B. You know the distance (the math) is fixed. But you also have to pay for gas (interest rates) and tolls (trading fees). The old model said, "The cost of the trip is just gas and tolls."

The Problem: The author, Useong Shin, noticed that even after accounting for gas and tolls, there was still a "mystery cost" left over. The delivery driver was still paying more than the math said they should.

The New Idea: What else is the driver thinking about?

Shin asks a simple question: What if the driver isn't just thinking about the cost of gas for this specific trip, but also about all the other jobs they could have taken?

If the driver is tied up moving this package, they can't take a different, potentially more profitable job. That "lost opportunity" is a real cost, even if no money changes hands right now.

To test this, Shin looked at what other "jobs" (investments) were available to these drivers. He didn't just look at U.S. interest rates. He looked at three other big buckets of money:

1. IEFA: Stocks from developed countries outside the U.S. (like Europe and Japan).
2. IGOV: Government bonds from those same foreign countries.
3. IAU: Gold.

The Discovery: The "Outside Option" Effect

Shin found that when he added these three global assets to his model, the mystery gap suddenly made much more sense.

• The Analogy: It turns out the "mystery cost" wasn't just about gas. It was about the driver worrying that while they are stuck delivering this package, the price of gold might skyrocket, or foreign stocks might surge, and they are missing out on those gains.
• The Result: When foreign stocks (IEFA) were doing well, the gap got smaller. When gold (IAU) was acting as a "safe haven" (people were scared), the gap got bigger.

This suggests that the "Carry Gap" isn't a failure of the math. It's a reflection of the opportunity cost for the people enforcing the rule. Because these traders have limited money (finite capital), they have to choose between enforcing this rule and investing elsewhere. The "gap" is the price of that choice.

Why Foreign Assets?

You might wonder, "Why look at foreign stocks and gold instead of just U.S. stocks?"

Shin explains that the basic model already includes a lot of U.S.-specific information (like U.S. interest rates and U.S. market volatility). It's like the driver already knows the local gas prices. The foreign assets provide new information that the U.S. model was missing. They represent the "global" opportunities the driver is giving up.

The "Drift" vs. The "Global View"

The paper also revisits an idea from a previous study: that the "gap" might be explained by the U.S. stock market's own momentum (its "drift").

Shin's new test shows that while the U.S. market's momentum does matter, it's actually just a small part of the bigger picture. Once you look at the global picture (Foreign Stocks + Bonds + Gold), the U.S. momentum doesn't add much new information. It's like realizing that worrying about the local weather is less important than worrying about the global climate.

The Bottom Line

The paper concludes that Put-Call Parity is still a perfect mathematical rule at the very end of the day. However, getting there is a journey that costs money and time.

The "Carry Gap" is the price tag for the journey, not the destination. It tells us that the people keeping the market honest (the arbitrageurs) are human. They have limited capital, and they are constantly weighing the cost of keeping this specific trade open against all the other exciting investment opportunities available in the global market.

In short: The gap isn't a mistake in the math; it's a reflection of the real-world cost of having your money tied up while better opportunities might be popping up elsewhere in the world.

Technical Summary

Technical Summary: Tuning in to Frequencies: How Global Assets Align with U.S. Put–Call Parity Residuals

1. Problem Statement
Put–call parity in European index options (specifically SPX and RUT) represents one of the cleanest no-arbitrage relations in financial markets. While visible price-space deviations are typically compressed by arbitrage, a systematic "carry gap" often remains. This gap is defined as the annualized difference between the option-implied discount factor and the Overnight Indexed Swap (OIS) benchmark.

Previous research (Shin, 2026a; Shin, 2026b) established that this carry gap is not merely a violation of terminal-payoff identity but a path-dependent, capital-using phenomenon. It is partially explained by OIS-based funding costs, volatility (GBM path-risk terms), trading frictions, and financial conditions. Furthermore, a proxy for the own-index physical drift ($\hat{\mu}$) was found to add explanatory power.

The central problem addressed in this paper is whether the physical-measure component driving the carry gap is best viewed as a narrow, own-index drift effect or as part of a broader "outside-option" state. Specifically, the paper investigates whether the opportunity cost of the finite capital required to enforce parity extends beyond domestic OIS rates to include low-frequency investment opportunities available to arbitrageurs in global asset classes (equities, bonds, and gold).

2. Methodology
The study employs a reduced-form regression framework to explain the daily carry gap ($CG^{bp}_{i,t}$) for SPX and RUT markets.

• Data: The sample covers January 4, 2016, to October 31, 2025, utilizing minute-level NBBO quotes from ThetaData and OIS data.
• Baseline Specification: The baseline model regresses the carry gap against:
  • OIS-based GBM path-risk terms (1-year and 10-year OIS rates scaled by volatility and time).
  • Option-market trading frictions (ATM bid-ask spread scaled by time).
  • The Chicago Fed National Financial Conditions Index (NFCI).
• The 3ETF Extension: The core innovation is the introduction of a parsimonious block of three Exchange-Traded Funds (ETFs) to proxy for outside-option states:
  • IEFA: Developed ex-U.S. equity.
  • IGOV: Developed ex-U.S. sovereign bonds (FX-unhedged).
  • IAU: Gold.
  • Construction: Each ETF term is constructed as a GBM-style variable: $104 \cdot b^{(n)}_{a,t} \cdot \frac{2}{3} \cdot \frac{Vol_{i,t}}{100} \cdot \sqrt{\frac{2\tau_{i,t}}{\pi}}$, where $b^{(n)}_{a,t}$ is the prior $n$-day rolling OLS slope of the log price path (computed using information up to $t-1$).
• Model Comparison: The paper compares four specifications: (1) Baseline (OIS only), (2) Drift (Baseline + own-index $\hat{\mu}$), (3) 3ETF (Baseline + IEFA, IGOV, IAU; dropping OIS 10Y), and (4) 3ETF+Drift.
• Evaluation: Performance is assessed using in-sample $R^2$, RMSE, and MAE. Crucially, out-of-sample (OOS) stability is tested via a "Leave-One-Year-Out" (LOYO) validation procedure, ensuring the results are not driven by a single calendar year or overfitting.
• Robustness: The study includes broad-dollar neutralization (subtracting the broad-dollar index from ETF prices), Principal Component Analysis (PCA) to test for single-factor collapse, nested horizon selection to prevent look-ahead bias, and alternative asset blocks (U.S.-centered and Emerging-market).

3. Key Results

• Improved Fit: Adding the 3ETF block significantly improves model fit. In SPX, in-sample $R^2$ rises from 0.312 to 0.402; in RUT, it rises from 0.281 to 0.361.
• Out-of-Sample Stability: The improvement persists in LOYO validation. Pooled OOS $R^2$ for SPX increases from 0.221 (baseline) to 0.379 (3ETF); for RUT, it increases from 0.171 to 0.318. The 3ETF model notably repairs the baseline's severe failure during the 2020 stress period.
• Coefficient Structure:
  • IEFA: Enters with a negative and highly significant coefficient, suggesting that stronger developed ex-U.S. equity opportunities are associated with a narrower carry gap (lower opportunity cost of capital).
  • IAU: Enters with a positive and significant coefficient, indicating that safe-haven or risk-off states (gold) widen the carry gap.
  • IGOV: Enters positively, acting as a slow-moving duration-related component, though its significance is more market-dependent.
  • OIS 1Y: Remains negative and significant, confirming that short-horizon funding costs remain a distinct driver.
• Drift Absorption: When the 3ETF block is included, the own-index drift proxy ($\hat{\mu}$) adds negligible incremental explanatory power (e.g., SPX $R^2$ increases by only 0.002). This suggests the drift proxy is a scalar projection of the broader physical-measure state captured by the 3ETF block, rather than an independent primitive.
• Robustness:
  • Dollar Neutrality: Broad-dollar adjustment of the ETF prices does not diminish the explanatory power, ruling out the result as a disguised broad-dollar cycle.
  • Factor Structure: PCA and residualization tests confirm the 3ETF block is not a single latent factor but a multi-channel structure with distinct frequencies (IEFA is fast, IGOV is slow, IAU is intermediate).
  • Alternative Blocks: U.S.-centered and Emerging-market blocks also improve upon the baseline, but the developed ex-U.S. block (IEFA/IGOV/IAU) provides the most balanced and robust fit, likely because the baseline already controls for U.S.-specific factors.

4. Significance and Claims
The paper claims to provide evidence for reduced-form P–Q alignment. It argues that while put–call parity is a risk-neutral (Q-measure) terminal-payoff identity, its enforcement in real markets is path-dependent and capital-intensive. Finite-capital intermediaries must fund the trade and meet margin calls before maturity. Consequently, the "carry gap" reflects the opportunity cost of this capital.

The paper's primary contribution is demonstrating that this opportunity cost is not fully summarized by domestic OIS rates or a single index drift. Instead, it aligns with a broader set of physical-measure (P-measure) outside-option states, specifically global equity, sovereign bond, and gold opportunities.

The author explicitly states that this is not a structural model of P-to-Q transmission, nor does it imply a failure of risk-neutral pricing. Rather, it shows that a nearly closed no-arbitrage relation can still retain a "carry-space trace" of the physical investment opportunities faced by the arbitrageurs who enforce it. The 3ETF block serves as a parsimonious proxy for these residual outside-option costs, offering a more complete explanation of the carry gap than OIS-based funding variables alone.