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.

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

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.

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.

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.

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.