Constant-volatility pricing under empirical market behavior

Research significance

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 context

Criteria for replacing a flat-volatility baseline

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.

The study measured a frozen SPY adjusted-close sample from January 2015 through July 2026. Constant-volatility geometric Brownian motion leaves three fingerprints. Rolling volatility should fluctuate around a stable level. Standardized returns should be approximately Gaussian. Return magnitudes should not stay autocorrelated. The sample contradicted each one.

Direction stayed hard to predict, but magnitude persisted. Lag-1 autocorrelation of absolute returns was about 0.35, and it survived at longer lags. Fat tails ran alongside a nearly thirty-fold realized-volatility range. A single constant parameter describes that observed return process poorly.

The Heston result demonstrates a mechanism. Negative price–volatility correlation and stochastic variance can generate an equity-like strike skew. A single Black–Scholes volatility cannot. The parameters were illustrative, not calibrated to a dated live option chain.

• Read implied volatility across strike and maturity instead of applying one number everywhere.
• 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.
• Attach model-risk language to Greeks and prices. Parameter precision does not remove specification error.

Inspect the technical study → Open the option-valuation model → Inspect the research record → Reproduce from the executed notebook ↗

Tail calibrationHistorical VaR under coverage and independence testsVolatility regimes and clustered losses challenge a slowly adapting empirical quantile. Numerical uncertaintyMonte Carlo convergence, discretization bias, and variance reductionSeparate estimator noise from the discretization bias of richer dynamics.

The 1973 model makes one structural bet: that a single number, \(\sigma\), describes the randomness of an asset for the life of an option. Assume geometric Brownian motion with constant volatility, continuous frictionless hedging, and a constant rate. Then every European option on the same underlying and expiry must price off the same \(\sigma\). This is not a minor implication — it is the testable implication. Invert the formula on market prices and the implied volatility should come back flat across strikes:

$$\sigma_{\text{imp}}(K, T) = \text{const} \quad \text{for all } K.$$

The relevant historical claim is narrower than “the smile began in 1987.” Rubinstein’s S&P 500 option study documents a pronounced, persistent post-crash skew in the 1987–1992 sample. It emphasizes that the observed surface is inconsistent with the constant- volatility model.²

Take the model's assumptions in order and hold them against an ordinary day of market data. Returns are not Gaussian: daily index returns show excess kurtosis and negative skew. Moves the model rates as once-in-millennia arrive within careers (the arithmetic is in Note VI). Volatility is not constant: it clusters, with quiet weeks following quiet weeks and violent days following violent days. It also moves inversely with price, the leverage effect. Prices jump: earnings, policy surprises, and liquidity spirals produce discontinuities no continuous diffusion generates. Each violation leaves a fingerprint in option prices. The market prices repricing risk whether or not the model does, and it records every fingerprint in one curve:

Out-of-the-money puts trade at implied volatilities several points above at-the-money. Out-of-the-money calls trade below. Traders are paying up for crash protection. The market quotes that premium through the Black-Scholes formula, using implied volatility as the unit. The smile is the market's standing annotation on the model: the Gaussian tail is too thin on the left, and the constant σ is not constant.

Heston (1993) promotes variance from parameter to process:¹

$$dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S, \qquad dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v, \qquad d\langle W^S, W^v \rangle_t = \rho\,dt.$$

Three mechanisms do the work. Mean reversion (\(\kappa\), \(\theta\)): variance is pulled toward a long-run level, so volatility shocks decay. That produces clustering, and it makes long-dated smiles flatter than short-dated ones, as observed. Vol-of-vol (\(\xi\)): variance itself is risky. That fattens both tails of the return distribution and gives the smile its curvature. Correlation (\(\rho < 0\)): the leverage effect, wired in explicitly. Negative return shocks arrive with rising variance, which skews the distribution left and tilts the smile downward toward high strikes. With \(\rho\) near \(-0.7\), the model generates the equity index skew almost as a matter of course. The model also has a closed-form characteristic function, so whole volatility surfaces calibrate quickly. That speed contributed to its broad use in derivatives practice.

Below, the flat-σ model meets Heston-generated target prices. The Heston call values are precomputed offline from the Heston characteristic function (Heston 1993, "little trap" formulation, Simpson-rule integration of the P₁/P₂ integrals; parameters in the table note). The Black-Scholes prices are computed live at the selected flat volatility. No single σ reconciles the two columns — that is the entire point.

Scope boundary No dated option chain, bid–ask spread, or observed implied-volatility surface enters this example. The fixed Heston grid is the target and flat Black–Scholes is the candidate.

Objective and residual definition

For the displayed one-parameter comparison, the diagnostic objective is the vega-weighted implied-volatility root-mean-square error

$$J(\sigma)=\sqrt{\frac{\sum_i \nu_i\,[\sigma-\sigma_i^{H}]^2}{\sum_i \nu_i}}, \qquad \nu_i=\operatorname{Vega}_{BS}(K_i,\sigma_i^{H}).$$

A genuine market calibration would replace the generated targets with timestamped option quotes. It should also weight residuals by measurement quality such as bid–ask spread. Exposing the objective is essential. Without it the parameter vector cannot be interpreted or reproduced.

Heston parameters: S₀ = 100, r = 2%, q = 0, T = 0.5y, v₀ = 0.04, κ = 2.0, θ = 0.05, ξ = 0.6, ρ = −0.7. Values computed offline in Python (Simpson rule, 4,000 nodes on [0, 200]) and hardcoded; implied vols recovered by bisection. Inspect the Heston implementation at commit ad24c499958 and the derivatives notebook at the same commit.

Heston is a better description, not a true one. Its diffusive variance cannot move fast enough to explain the steep smiles of short-dated options. A market pricing overnight jump risk needs jumps. Practice therefore moved to Bates (Heston plus Merton jumps) and beyond. Its single variance factor forces the whole term structure of skew to move in lockstep, while real surfaces twist. Its calibrated parameters, re-fit each morning, drift in ways the model says they should not. That drift indicates residual misspecification. The modern frontier (rough volatility, with Hurst exponents near 0.1) suggests volatility's memory is structurally unlike anything a Markovian diffusion can produce. These limitations motivate jump-diffusion, multifactor, and rough-volatility alternatives.

Black–Scholes remains useful as a quoting convention even though its literal market dynamics are rejected. Implied volatility expresses observed prices relative to the constant-volatility baseline and makes deviations across strike and maturity comparable.

The distinction is operational. The formula can organize prices and sensitivities without establishing that returns are Gaussian, volatility is constant, or hedging is continuous. Those assumptions require separate empirical tests, monitoring, and model-risk controls. Model output stays conditional on the assumptions, calibration data, numerical method, and market regime behind it.