What Is Stochastic Volatility?

Last reviewed: May 17, 2026 by Options Analysis Suite Research.

What Is Stochastic Volatility?

Black-Scholes assumes the underlying follows geometric Brownian motion with a constant volatility parameter sigma. If you fit Black-Scholes to a real option chain, the calibrated sigma changes by strike (skew/smile) and changes by tenor (term structure). The "constant" assumption is empirically false. Stochastic-volatility models replace the constant sigma with a random process: variance becomes its own state variable that evolves according to a stochastic differential equation, often correlated with the underlying.

The canonical specification is the Heston (1993) model:

dS = mu*S*dt + sqrt(v)*S*dW1
dv = kappa*(theta - v)*dt + nu*sqrt(v)*dW2
corr(dW1, dW2) = rho

Five parameters describe the joint dynamics: kappa (mean-reversion speed of variance), theta (long-run variance level), nu (vol-of-vol), rho (spot-vol correlation), and v0 (current variance). The variance process is the Cox-Ingersoll-Ross (1985) square-root diffusion, which keeps variance non-negative under the Feller condition (2*kappa*theta > nu^2). Heston pricing is closed-form via Fourier inversion of the characteristic function, making calibration tractable in milliseconds.

Why Does It Exist Empirically?

Three empirical facts that constant-vol models cannot explain but stochastic-vol models do:

• Implied volatility moves. If sigma were truly constant, calibrated IV would be flat across days. Instead, ATM IV varies daily by 1-3 vol points in calm regimes and 5-15+ during stress. The variance state has its own dynamics.
• Skew exists and persists. Equity-index options consistently price OTM puts at higher IV than equivalent calls. Constant-vol models cannot produce this; stochastic-vol models with negative rho can. The structural reason (Hull-White 1987, Heston 1993) is that negative spot-variance correlation fattens the priced left tail.
• Smile curvature persists. Even at-the-money options exhibit U-shaped IV across strikes: deep OTM options on either side trade at higher IV than ATM. Constant-vol models give a flat IV-by-strike line; stochastic-vol models with positive nu produce the U-shape via the conditional variance distribution.

Mean Reversion

The kappa*(theta - v)*dt drift is the mathematical expression of mean reversion. When current variance v exceeds long-run theta, drift is negative and v decays toward theta; when v is below theta, drift is positive and v rises. The half-life of a variance shock is ln(2)/kappa days. Calibrated SPX kappa values are typically 2-4 (half-life ~50-130 trading days), matching the empirical observation that VIX shocks decay over weeks-to-months rather than instantly.

Mean reversion is the structural reason IV crush happens: pre-event IV is elevated relative to long-run theta, and after the event resolves, variance mean-reverts back toward theta. It is also why vol-of-vol matters: nu controls how far variance can wander from theta in the meantime.

The Three Major Stochastic-Vol Models

• Heston (1993): CIR variance process with constant kappa, theta, nu, rho. Closed-form Fourier pricing. Dominant for full-surface fitting on equity indices. Five parameters.
• SABR (Hagan et al. 2002): log-normal forward process with stochastic alpha. Used per-expiration with the closed-form Hagan implied-vol approximation. Industry standard for interest-rate option smiles. Four parameters per expiration: alpha, beta, rho, nu.
• Hull-White (1987): the original stochastic-vol model with uncorrelated spot and vol processes. Captures smile but not skew. Largely superseded by Heston for empirical work.

Variants and extensions: Bates (1996) adds Poisson jumps to the spot process; SVCJ (Eraker 2004) adds jumps to variance; the Double-Heston model uses two correlated CIR processes for richer smile dynamics; rough volatility (Bayer-Friz-Gatheral 2016) replaces the standard Brownian variance driver with fractional Brownian motion to match empirical "volatility roughness."

How Does Stochastic Vol Connect to Surface Features?

• Skew comes from rho. Negative spot-vol correlation produces left-skewed return distributions: when spot falls, vol rises, fattening the left tail. SPX equity rho values typically calibrate to -0.5 to -0.8.
• Smile curvature comes from nu. Higher vol-of-vol fattens both tails of the conditional variance distribution, producing U-shaped IV. SPX nu calibrates to roughly 0.4-0.7 in normal regimes and 0.8-1.2 during stress.
• Term-structure shape comes from kappa and theta. Mean reversion drives long-tenor IV toward sqrt(theta) regardless of current variance. Short-tenor IV is dominated by current v plus jump-event premium.
• IV mean reversion comes from the kappa drift. Forward-starting IVs revert toward the calibrated long-run vol level under Heston dynamics.

Calibration in Practice

Heston calibration jointly estimates the five parameters from a full IV surface. The standard approach (Mikhailov-Nogel 2003, Andersen-Andreasen 2000): minimize the squared distance between observed and model-implied IVs across all (strike, expiration) pairs, optionally weighted by vega for liquidity. Optimization typically uses Levenberg-Marquardt or differential evolution; tractability hinges on a fast characteristic-function pricer. See calibration for methodology details.

Limitations and Caveats

• Cannot capture all jumps. Diffusion-only stochastic vol underestimates short-tenor skew because near-expiration smile shape is dominated by jump risk that diffusion alone cannot generate. Bates addresses this; pure Heston does not.
• Forward-smile dynamics may flatten too quickly. Calibrated Heston with constant parameters tends to produce forward smiles that decay faster than observed. This motivates rough volatility extensions.
• Five parameters can be unstable. Calibration on sparse single-name surfaces sometimes yields parameter sets that fit equally well, reflecting calibration ambiguity rather than economic reality.

Related Concepts

Heston Model · SABR Model · Local Volatility · Vol of Vol · Leverage Effect · Volatility Smile · Calibration · Pricing Model Landscape

References & Further Reading

• Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options." Review of Financial Studies, 6(2), 327-343. The canonical stochastic-volatility model.
• Hull, J. and White, A. (1987). "The Pricing of Options on Assets with Stochastic Volatilities." Journal of Finance, 42(2), 281-300. The original stochastic-volatility paper.
• Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E. (2002). "Managing Smile Risk." Wilmott Magazine, September, 84-108. The SABR model and the closed-form Hagan smile formula.
• Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). "A Theory of the Term Structure of Interest Rates." Econometrica, 53(2), 385-407. The CIR square-root process used as the Heston variance dynamics.
• Fouque, J.-P., Papanicolaou, G. and Sircar, K. R. (2000). Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press. Practitioner reference for stochastic-vol pricing and asymptotic expansions.

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