Stochastic Volatility Model (Heston)
Also known as: Heston model, SV model, continuous-time stochastic volatility, Stokastik Volatilite Modeli (Heston, SV)
The stochastic volatility model is a continuous-time option-pricing and risk framework in which volatility follows its own random process rather than staying constant. The Heston model, introduced by Steven Heston in 1993, gives the variance a mean-reverting square-root (CIR) dynamic and yields a closed-form option price; it is the continuous-time counterpart of GARCH.
Key highlights
- Provides a closed-form option-pricing formula, making calibration and pricing fast.
- Captures the volatility smile, volatility clustering, and the leverage effect through the correlation ρ.
- Mean-reverting CIR variance is the natural continuous-time analogue of GARCH, with economically interpretable parameters.
Intuition
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How it works
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When to use it
Use the Heston stochastic volatility model when pricing options or measuring risk on a continuous, time-series asset whose volatility clearly changes over time, with at least about 50 observations or a cross-section of option quotes for calibration. It is appropriate when you need to reproduce the volatility smile and the leverage effect, and when the mean-reversion assumption for variance is reasonable. The Feller condition (2κθ > σ²) should hold so that variance stays positive. It is less suitable when only constant-volatility behaviour is present or when calibration data are too sparse to identify the parameters.
Strengths & limitations
- Provides a closed-form option-pricing formula, making calibration and pricing fast.
- Captures the volatility smile, volatility clustering, and the leverage effect through the correlation ρ.
- Mean-reverting CIR variance is the natural continuous-time analogue of GARCH, with economically interpretable parameters.
- Calibration requires either option prices or a sufficiently long return history; the five parameters can be hard to identify from sparse data.
- If the Feller condition (2κθ > σ²) is violated, the variance process can approach zero and numerical pricing becomes unstable.
- Assumes a single square-root variance factor, so it can struggle to fit very short-maturity smiles without added jumps.
Common pitfalls
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Applications
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Frequently asked
How is the Heston model different from Black-Scholes?
Black-Scholes assumes a single constant volatility, while the Heston model lets variance follow its own mean-reverting random process. This lets Heston reproduce the volatility smile and the leverage effect that Black-Scholes cannot.
What is the Feller condition?
The Feller condition 2κθ > σ² guarantees that the square-root variance process stays strictly positive and does not reach zero. When it is violated, variance can hit zero and numerical pricing becomes unstable.
Why is the correlation ρ usually negative?
A negative correlation between the price and variance shocks produces the leverage effect: when prices fall, volatility tends to rise. This asymmetry is needed to match the shape of equity volatility smiles.
How does Heston relate to GARCH?
Heston is essentially the continuous-time counterpart of GARCH. Both describe time-varying, mean-reverting volatility, but Heston works in continuous time and yields a closed-form option price, whereas GARCH is a discrete-time model fitted to return series.
Sources
- 1.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.
- 2.Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.ISBN 978-0471792512
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ScholarGate. (2026, June 1). Stochastic Volatility Model. ScholarGate. https://scholargate.app/finance/stochastic-volatility-model