The Cost of a Free Lunch: Evidence from U.S. Derivatives Markets

This paper demonstrates that while put-call parity holds in terminal payoff, daily settlement and capital constraints create a systematic, path-dependent implementation wedge between option-implied discount factors and the OIS curve, which is quantitatively explained by volatility-driven carry costs and trading frictions.

The Gist

The Big Idea: The "Free Lunch" That Isn't Free

Imagine you are at a restaurant. The menu says a specific meal costs $0. You see other people eating it for free, and the price tag clearly reads zero. You decide to order it, thinking you've found a "free lunch."

However, the paper argues that while the price tag is zero, the experience of eating it is not free. There are hidden costs: you have to stand in a long line, you might have to pay a deposit that gets locked up for a while, and if the restaurant gets crowded, you might have to pay a fee just to keep your spot in line.

In the world of finance, this "free lunch" is a rule called Put-Call Parity. It's a mathematical guarantee that says if you combine a few specific financial contracts (options and futures), you should end up with a guaranteed profit with zero risk.

The Paper's Discovery:
The author, Useong Shin, looked at the U.S. stock market (specifically the S&P 500 and Russell 2000) and found that while the "price tag" of this free lunch is indeed zero (the math works out perfectly), there is a hidden "carry cost" or "survival fee" to actually hold the trade until it finishes.

The Analogy: The Tightrope Walker

To understand the hidden cost, imagine a tightrope walker (the trader) trying to cross a canyon (the time until the contract expires).

1. The Destination (Maturity): The walker knows exactly where they will land on the other side. The math says they will make it. This is the "terminal payoff."
2. The Journey (The Path): To get there, the walker must balance on a rope that sways in the wind.
   • If the wind blows one way (the market goes up), the walker has to pay a "margin fee" immediately to stay on the rope.
   • If the wind blows the other way (the market goes down), the walker gets a little cash back, but they can't spend it immediately; it's locked up.

The Problem: Even though the walker knows they will make it to the other side, they need a "survival fund" to pay the fees during the walk. If they run out of cash in the middle of the canyon, they fall, even if they were destined to succeed.

The paper calls this the "Path Risk." The cost isn't about where you end up; it's about how much cash you need to keep your balance while you get there.

What the Paper Actually Measured

The author looked at millions of data points from stock options and compared them to a standard "risk-free" interest rate (called the OIS curve).

• The "Carry Gap": This is the difference between the cost of holding the trade in the real world and the theoretical "risk-free" cost.
• The Finding: The author found a consistent "gap" of about 37 basis points (roughly 0.37%) per year.
  • Think of this as a "toll fee" that the market charges you for the privilege of walking the tightrope.
  • This fee is always positive. It's never negative. It's like a tax on the journey.

Why Does This Gap Exist?

The paper suggests this gap exists because of three main things:

1. The Volatility of the Wind (Path Risk): The more the market swings (volatility), the more cash you need to keep your balance. The paper found that the "toll fee" gets higher when the market is wilder and when the trip takes longer.
2. The Cost of the Rope (Funding): If you need to borrow money to pay the fees while walking, and interest rates are high, the trip becomes more expensive.
3. Traffic Jams (Trading Frictions): Sometimes, it's hard to buy or sell the contracts quickly without paying a higher price (the bid-ask spread). This adds to the cost.

The "Free Lunch" Myth

The paper concludes that the "free lunch" isn't actually free.

• The Old View: "Look, the math says the profit is guaranteed! It's free!"
• The New View: "The math says the profit is guaranteed, but you have to pay a survival fee to keep the trade alive while you wait for that profit."

The "profit" traders see isn't a magic trick; it's actually compensation for the risk of having to manage their cash flow, pay margin fees, and survive the daily swings of the market.

Summary of Key Points

• The Puzzle: The price of the trade is zero, but the cost to hold it is not.
• The Measurement: The author found a systematic "carry gap" of about 37 basis points in the US market.
• The Cause: It's caused by the need for cash to survive daily market swings (path risk), not by a mistake in the math.
• The Proof: The author tested this over many years and different market conditions (including the pandemic and rate hikes). The "toll fee" remained consistent, proving it's a real feature of the market, not just a glitch.

In short: You can't get a free lunch. You can get a lunch that looks free on the menu, but you have to pay a "survival fee" to keep your seat at the table until the meal is served.

Technical Summary

Technical Summary: The Cost of a Free Lunch: Evidence from U.S. Derivatives Markets

Problem Statement
Put–call parity is a fundamental no-arbitrage identity stating that a portfolio of a European call, a put (same strike/maturity), the underlying asset, and a risk-free bond yields a deterministic terminal payoff. In U.S. index options markets (SPX and RUT), quoted residuals against traded futures are tightly compressed around zero, suggesting efficient pricing. However, this paper argues that the absence of a visible price-space residual does not imply that enforcing parity is economically costless.

The core problem is the distinction between the terminal-payoff identity (which is exact by construction) and the implementation strategy (which is path-dependent). Enforcing parity requires an arbitrageur to survive daily settlement, variation margin calls, interim funding needs, and finite capital constraints. While the terminal payoff is deterministic, the path to maturity involves interim cash-flow risks. If the index moves adversely, the short futures leg generates immediate margin outflows that must be funded, while offsetting option gains are not immediately liquid. This creates a "survival-capital" burden. The paper posits that while price-space residuals are arbitraged away, a systematic "carry gap"—a wedge in the cost of capital required to survive the path—remains visible.

Methodology
The study utilizes minute-level NBBO data for S&P 500 (SPX) and Russell 2000 (RUT) options from January 4, 2016, to October 31, 2025.

1. Identification of Option-Implied Discount Factors:
   Following Azzone and Baviera (2021), the paper extracts market-implied discount factors ($\hat{B}_t(T)$) directly from the option cross-section. By exploiting the linear relation between the synthetic forward ($C_t - P_t$) and the strike price $K$, the method jointly estimates the forward value and the discount factor. This approach avoids external inputs (spot, dividends, futures) that introduce nonsynchronicity and estimation noise, ensuring the discount factor is identified purely within the option market.

2. Construction of the Carry Gap:
   The paper compares the option-implied discount factor against a bootstrapped Overnight Indexed Swap (OIS) discount factor ($D^{OIS}_t(T)$), the standard benchmark for derivatives discounting. The Carry Gap (CG) is defined as the annualized deviation:
   $$CG_t(T) = \frac{1}{\tau_t(T)} \log \left( \frac{D^{OIS}_t(T)}{\hat{B}_t(T)} \right)$$
   A positive CG indicates the options market embeds a higher implied carry cost than the risk-free OIS curve.

3. Regression Specification:
   To explain the carry gap, the paper introduces a reduced-form GBM path-risk term. Motivated by geometric Brownian motion, the expected support capital required to keep a parity-enforcement position solvent is proportional to volatility ($\sigma$) and the square root of time ($\sqrt{\tau}$). The specification includes:

   • GBM Terms: Two rate-scaled terms ($OIS_{1Y}$ and $OIS_{10Y}$) multiplied by $\sigma \sqrt{\tau}$, representing short-to-medium and long-term opportunity costs of capital.
   • Trading Frictions: Bid-ask spread scaled by maturity ($BA_{med}/\tau$).
   • Financial Conditions: The NFCI (National Financial Conditions Index) to proxy for funding stress.

   The baseline regression is:
   $$CG^{bp}_{i,t} = \alpha + \delta D^{SPX}_i + \phi_1 GBM_{1Y} + \phi_{10} GBM_{10Y} + \beta \frac{BA_{med}}{\tau} + \gamma NFCI_t + \varepsilon_{i,t}$$

Key Results

• Distributional Properties: The carry gap is systematically positive. The sample median is approximately 37 basis points (bp), with 98.4% of observations being positive. This holds for both SPX (median ~36 bp) and RUT (median ~39 bp). The distribution is not centered at zero, ruling out simple measurement noise.
• Maturity Structure: The carry gap remains positive across all maturities (30–40 bp range) but exhibits higher dispersion at the short end due to microstructure frictions and annualization effects.
• Time-Series Variation: The gap exhibits low-frequency persistence and regime-dependent shifts (e.g., elevated levels in 2018 and 2020–2021), suggesting a link to broader financial conditions rather than transient noise.
• Regression Findings:
  • The GBM path-risk terms are statistically significant with opposite signs: the short-horizon term ($\phi_1$) is negative, and the long-horizon term ($\phi_{10}$) is positive.
  • The bid-ask term is positive (frictions increase the gap), and NFCI is negative (tighter financial conditions compress the gap).
  • The model explains approximately 30% of the variance ($R^2 \approx 0.31$) in the pooled sample.
• Robustness and Validation:
  • Leave-One-Year-Out (LOYO): While the model is not a strong high-frequency forecasting tool (mean out-of-sample $R^2$ is modest), the signs of all key coefficients remain stable across all LOYO folds. The model fails primarily in specific regimes (e.g., 2020 for SPX), but the structural relationship persists.
  • Residual Stationarity: Engle–Granger tests confirm that the residuals of the level relation are stationary, mitigating concerns that the results are spurious alignments of persistent trends.
  • Alternative Benchmark: Replacing OIS with Treasury constant-maturity yields (DGS) weakens the fit but preserves the core sign structure, confirming the result is not an artifact of the OIS curve choice.

Significance and Contributions
The paper makes three primary contributions, modestly framed as identifying a systematic empirical object rather than discovering a new arbitrage opportunity:

1. Decoupling Identity from Implementation: It separates the theoretical terminal-payoff identity of put–call parity from its practical enforcement. It demonstrates that a small quoted price-space residual does not imply risk-free enforcement; the "cost of a free lunch" is hidden in the path-dependent capital requirements.
2. Documentation of the Carry Gap: It identifies the carry gap as a systematic, state-dependent wedge in carry space. This gap has a positive center, low-frequency persistence, and is linked to implementation risk, trading frictions, and financial conditions.
3. Reduced-Form Path-Risk Explanation: It provides a reduced-form explanation for this wedge using a GBM-based path-risk term. The evidence suggests that the apparent return to parity enforcement is not a literal free lunch but rather compensation consistent with implementation risk, cash-flow timing, and finite capital constraints.

The paper concludes that while the post-GFC environment (stable OIS conventions, granular data) makes this wedge measurable, the phenomenon itself may not be new. The contribution lies in making the "shadow cost" of successful parity enforcement visible and linking it to a specific path-risk structure.