Decision brief · Derivatives and model risk
Why one volatility number cannot price every strike
Observed volatility variation, tail behavior, and return dependence were inconsistent with the tested constant-volatility model.
Depth 1 · Answer
The tested Black–Scholes dynamics were inconsistent with the observed sample.
SPY 21-day realized volatility ranged from roughly 3% to 93%, daily excess kurtosis was about 14, the worst day was near a 10σ event under a Gaussian yardstick, and lag-1 absolute- return autocorrelation was about 0.35. An equity-like, Feller-satisfying Heston specification generated a short-dated implied-volatility skew from roughly 25% to 16% across strikes.
Published 8 August 2026 · Underlying investigation last updated 6 July 2026
- Realized vol range
- ≈3%–93%
- Excess kurtosis
- ≈14
- Worst-day yardstick
- ≈10σ
- Illustrative skew
- ≈25%–16%
Why it matters
Derivatives users, risk teams, and model validators
The result matters wherever one volatility estimate drives option prices, Greeks, hedges, scenario losses, or claims about the likelihood of extreme market moves.
Decision affected
When a flat-volatility baseline needs a richer model or surface
Use Black–Scholes for the job it can do, while making strike, maturity, tail, and volatility-dynamics risk visible. A precise price under a rejected dynamic assumption is not the same as a reliable decision.
Depth 2 · Evidence
The rejection came from several independent fingerprints
The study measured a frozen SPY adjusted-close sample from January 2015 through July 2026. Under constant-volatility geometric Brownian motion, rolling volatility should fluctuate around a stable level, standardized returns should be approximately Gaussian, and return magnitudes should not remain autocorrelated. The sample contradicted each fingerprint.
Direction remained difficult to predict, but magnitude persisted: lag-1 autocorrelation of absolute returns was about 0.35 and remained present at longer lags. Fat tails and a nearly thirty-fold realized-volatility range made a single constant parameter an inadequate description of the observed return process.
The Heston result is a mechanism demonstration. Negative price–volatility correlation and stochastic variance can generate an equity-like strike skew that a single Black–Scholes volatility cannot. The parameters were illustrative, not calibrated to a dated live option chain.
Depth 3 · Application
Separate a useful quoting convention from a claim about market dynamics
- Inspect implied volatility across both strike and maturity instead of applying one volatility to every contract.
- Monitor realized volatility and return-magnitude persistence as state variables rather than constants.
- Stress hedge and valuation outputs to skew, term structure, jumps, and volatility-of-volatility.
- Validate a richer model against a dated option surface before treating its calibration as empirical evidence.
- Keep model-risk language attached to Greeks and prices; parameter precision does not remove specification error.
Depth 4 · Limits
The evidence rejects fingerprints; it does not crown a true model
- The empirical study covers one underlying and one historical window.
- Close-to-close returns omit intraday price and volatility structure.
- The Heston parameters are illustrative rather than calibrated to contemporaneous market quotes.
- Heston still treats parameters as constants and omits jumps and rough-volatility effects.
- Simulated stochastic-volatility paths carry discretization bias that must be budgeted separately from sampling error.
Depth 5 · Method and code
Follow the model from closed form to empirical rejection
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