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Graduate-Level Modeling · Derivatives Pricing & Volatility Modeling

Heston Stochastic Volatility Option Pricing & Smile Engine

Black-Scholes assumes volatility is a constant. Markets disagree, loudly, in the shape of the implied volatility smile. Heston (1993) lets variance itself follow a mean-reverting random process, and this engine prices it with the model's own semi-closed-form solution.

How To Use This Model

Reading This Tool

Set the five Heston parameters, initial variance, mean-reversion speed, long-run variance, volatility of variance, and the correlation between price and variance shocks, and this tool prices the full strike range using the characteristic-function integral, then inverts Black-Scholes at every strike to reveal the implied volatility smile the model generates.

Watch what each parameter actually does: correlation ρ tilts the smile into a skew, volatility-of-variance σ bends its curvature, and mean reversion κ controls how quickly the term structure flattens toward the long-run level. The Feller condition badge tells you when the variance process can touch zero.

Model Parameters

Option Maturity T 1.00 yrs

Risk-Free Rate 4.0%

Dividend Yield 1.5%

Initial Volatility √v₀ 20%

Mean-Reversion Speed κ 2.0

Long-Run Volatility √θ 22%

Volatility Of Variance σ 0.50

Price-Variance Correlation ρ −0.70

Spot is normalized to $100 so strikes read directly as moneyness. Prices come from the “little Heston trap” characteristic-function formulation integrated with 96-point Gauss-Legendre quadrature, the same semi-closed-form the original 1993 paper made famous.

Model Prices & Smile Diagnostics

Feller: –

ATM Call (K = $100)

ATM Implied Volatility

90–110 Skew

Smile Curvature (Butterfly)

Implied Volatility Across Strikes At Maturity T

Derivatives Pricing & Volatility Modeling

The Model

dS_t = (r−q)S_tdt + √v_t·S_tdW_t^S
dv_t = κ(θ−v_t)dt + σ√v_tdW_t^v, d⟨W^S,W^v⟩ = ρdt
C = S e^−qTP₁ − K e^−rTP₂, P_j = ½ + (1/π)∫₀^∞ Re[ e^{−iφln K}f_j(φ)/(iφ) ]dφ

Variance follows a Cox-Ingersoll-Ross square-root diffusion, so it mean-reverts and stays non-negative. The two risk-neutral probabilities P₁ and P₂ come from inverting the model’s characteristic function, no simulation and no lattice, which is why Heston became the workhorse stochastic-volatility model of every derivatives desk. This implementation uses the branch-cut-safe “little Heston trap” algebra of Albrecher et al. (2007) with 96-point Gauss-Legendre quadrature, verified against an independent adaptive-quadrature implementation to eight decimal places, and implied volatilities are recovered by bisection against Black-Scholes using a double-precision normal CDF.

When To Actually Use This Model

Teaching why implied volatility smiles and skews exist, and which economic force each Heston parameter maps to.
Understanding volatility term structures: how today’s variance level converges toward its long-run mean, and at what speed, in traded option prices.
As the reference point for calibration exercises, Heston is the standard first stochastic-volatility model fitted to real smile data before jumps or local-volatility layers are added.

Key Assumptions & Limitations

Pure diffusion: no jumps in price or variance, so very short-dated smiles are flatter than markets show, practitioners add jumps (Bates, 1996) for that regime.
Parameters are constant through time; real desks recalibrate daily and the parameters wander, which is itself evidence of model incompleteness.
A single variance factor: the model cannot decouple the short and long ends of the volatility surface independently, multi-factor extensions exist for that.

Foundational References

Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327-343.

Albrecher, H., Mayer, P., Schoutens, W., & Tistaert, J. (2007). The Little Heston Trap. Wilmott Magazine, January 2007, 83-92.

Gatheral, J. (2006). The Volatility Surface: A Practitioner’s Guide. Wiley.

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