Heston Model Overview The Heston model is a stochastic volatility model used to price European options. Unlike Black–Scholes, which assumes constant volatility, the Heston model treats instantaneous variance as a random process that mean-reverts over time. This allows the model to capture common market features such as the volatility smile and skew. Key features
Stochastic variance (volatility is random and time-varying).
Mean reversion of variance toward a long-term level.
Possible correlation between asset returns and variance (leverage effect).
Semi-analytical (closed-form) pricing via characteristic functions and Fourier inversion, enabling efficient computation for European options. Explore More Resources
Model formulation The Heston model specifies a system of stochastic differential equations (SDEs) for the asset price S_t and its instantaneous variance V_t: dS_t = r S_t dt + sqrt(V_t) S_t dW1_t
dV_t = k (θ − V_t) dt + σ sqrt(V_t) dW2_t
corr(dW1_t, dW2_t) = ρ dt Explore More Resources
Parameters and variables
S_t: asset price at time t
V_t: instantaneous variance at time t (V_t ≥ 0)
r: risk-free interest rate
k (kappa): speed of mean reversion of variance toward θ
θ (theta): long-term variance (mean reversion level)
σ (sigma): volatility of variance (volatility-of-volatility)
ρ (rho): correlation between the Brownian motions driving S_t and V_t
dW1_t, dW2_t: Brownian motions (Wiener processes) Important condition (positivity)
* The Feller condition, 2kθ > σ^2, is sufficient (but not necessary) to keep V_t strictly positive. If violated, numerical schemes must handle the possibility that V_t approaches zero. Explore More Resources
How pricing works Heston derived a closed-form expression for European option prices in terms of the characteristic function of log(S_T). That characteristic function can be inverted numerically (via Fourier inversion or FFT methods) to compute option prices efficiently. Alternative numerical approaches include Monte Carlo simulation and finite-difference PDE solvers. Advantages of Heston pricing
Captures implied volatility smile/skew observed in markets.
Incorporates correlation between returns and volatility (important for equity options).
* Faster and more accurate than brute-force Monte Carlo when using the characteristic-function approach. Explore More Resources
Comparison with Black–Scholes Black–Scholes assumes constant volatility and lognormal asset returns, which simplifies pricing but cannot reproduce the volatility smile or skew. Heston relaxes the constant-volatility assumption and models variance dynamics explicitly, allowing it to fit a broader range of option market prices. However, Heston is more complex to calibrate and simulate. Calibration and extensions
* Calibration: Model parameters (k, θ, σ, ρ, initial V_0) are typically estimated by minimizing the difference between model-implied and market-implied volatilities across strikes and maturities. Calibration can be numerically intensive and sensitive to input data.
* Extensions: To capture sudden large moves (jumps) or more complex dynamics, practitioners often extend Heston with jumps or combine it with other stochastic-volatility specifications.
Limitations and practical considerations
* Primarily designed for European-style options; pricing American options requires additional numerical techniques or approximations.
* Calibration stability: parameter estimates may change over time and across maturities.
* If the Feller condition fails, specialized numerical methods are needed to preserve variance positivity.
* No built-in jump component: large discontinuous moves in prices require adding jumps to the model.
When to use the Heston model Use Heston when you need a tractable stochastic-volatility framework that:
Reproduces the volatility smile/skew,
Accounts for correlation between returns and volatility,
* Allows relatively fast pricing for European options via semi-analytical methods. Explore More Resources
For markets where jumps or more complex volatility dynamics dominate, consider Heston variants (e.g., Heston plus jumps) or alternative models better suited to those features.