Heston Model

Definition

The Heston (1993) model is a continuous-time stochastic-volatility option-pricing model in which the instantaneous variance $V_t$ follows a mean-reverting square-root (Cox-Ingersoll-Ross) diffusion while the asset price follows a geometric diffusion driven by $\sqrt{V_t}$. Its two defining features are (i) a diffusion term on variance proportional to $\sqrt{V_t}$, which keeps variance non-negative and gives the process an affine structure, and (ii) a non-zero correlation $\rho$ between the return and variance innovations, which captures the empirical leverage effect. Because the log-price's conditional characteristic function is exponential-affine and available in closed form, European options admit a semi-closed-form price via Fourier inversion — making Heston the tractable workhorse of the affine stochastic-volatility class.

Key Ideas

• Square-root variance dynamics. $dV_t = \kappa(\theta - V_t)\,dt + \sigma_v\sqrt{V_t}\,dW_t^V$: mean-reversion speed $\kappa$, long-run variance $\theta$, and vol-of-vol $\sigma_v$. The $\sqrt{V_t}$ diffusion is what distinguishes Heston from the GARCH-diffusion limit, whose vol-of-vol is $\propto V$ (see GARCH and BEKK-GARCH).
• Leverage via correlated Brownian motions. $\mathrm{corr}(dW^V, dW^{\text{return}}) = \rho$, typically $\rho < 0$, generating the asymmetric put-skew that zero-correlation models (e.g. the arithmetic Ornstein-Uhlenbeck model of Stein-Stein) cannot produce.
• Affine / characteristic-function pricing. The conditional characteristic function of the log-price is exponential-affine in $V_t$, so option prices follow from a one-dimensional Fourier inversion rather than simulation.
• Non-central chi-square transition law. The square-root process has a known non-central $\chi^2$ transition density; the moment generating function of average variance can be read directly off the CIR bond-pricing formula (Stochastic Volatility Option Pricing).
• Nested in richer models. Heston is the SV core of the SVSI-J family, which adds stochastic interest rates and Poisson jumps (Bakshi-Cao-Chen (1997) Empirical Performance of Alternative Option Pricing Models).

How It Works

Under the physical measure the joint dynamics are $$dS_t/S_t = \mu\,dt + \sqrt{V_t}\,dW_t^S, \qquad dV_t = \kappa(\theta - V_t)\,dt + \sigma_v\sqrt{V_t}\,dW_t^V, \qquad \mathrm{corr}(dW^S, dW^V) = \rho.$$ The variance is non-negative and, under the Feller condition $2\kappa\theta \ge \sigma_v^2$, strictly positive. The crucial analytical property is that the log-price $x_t = \ln S_t$ has a conditional characteristic function of exponential-affine form, $$E\big[e^{iu\,x_T} \mid x_t, V_t\big] = \exp\!\big(A(u,\tau) + B(u,\tau)\,V_t + iu\,x_t\big), \qquad \tau = T - t,$$ where $A$ and $B$ solve a system of Riccati ordinary differential equations with closed-form solutions. A European call is then recovered by Fourier inversion as $C = S_t\,\Pi_1 - K e^{-r\tau}\,\Pi_2$, with the risk-neutral exercise probabilities $\Pi_1, \Pi_2$ obtained from integrals of the characteristic function — the "semi-closed-form" solution. In the uncorrelated special case ($\rho = 0$) the same prices follow from the moment generating function of average variance $AV = T^{-1}\!\int_0^T V_s\,ds$, which for the square-root model equals $\exp(N^*(T) + M^*(T)V_0)$ read off the CIR bond price (Stochastic Volatility Option Pricing); the general correlated case requires Heston's characteristic-function route because $S_T \mid AV$ is no longer lognormal. The convexity of the Black-Scholes Option Pricing price in variance, combined with the dispersion of $AV$, produces the U-shaped volatility smile analytically.

Why It Matters

Heston is the reference stochastic-volatility model in derivatives pricing because it delivers non-negative variance, a leverage-driven skew, and near-closed-form option prices simultaneously. Empirically, incorporating stochastic volatility is the first-order improvement over Black-Scholes in pricing and hedging S&P 500 options, reducing absolute pricing errors by 25-70%; adding stochastic interest rates or jumps yields only second-order gains (Bakshi-Cao-Chen (1997) Empirical Performance of Alternative Option Pricing Models). The model's two structural parameters — the leverage $\rho < 0$ and the volatility risk premium $\lambda_v < 0$ — jointly and analytically explain a cluster of return-volatility "puzzles": the ambiguous sign of the volatility-feedback slope, the stronger leverage asymmetry of implied than Realized Volatility, and the downward bias of implied-volatility forecasts (Bollerslev-Zhou (2006) Volatility Puzzles: A Simple Framework for Gauging Return-Volatility Regressions). As the canonical single-factor affine model it also anchors the broader affine jump-diffusion pricing framework (Sundaresan (2000) Continuous-Time Methods in Finance: A Review and an Assessment).

Open Questions

• Smile term structure. Pure Heston SV matches the volatility smile across strikes at a given maturity but not the rate at which the smile flattens with maturity; adding return jumps only partially corrects this (Sundaresan (2000) Continuous-Time Methods in Finance: A Review and an Assessment).
• Parameter inconsistency. Option-implied $\rho$ and vol-of-vol $\sigma_v$ are several times larger in magnitude than their time-series estimates, indicating the single-factor square-root structure remains misspecified (Bakshi-Cao-Chen (1997) Empirical Performance of Alternative Option Pricing Models).
• Feller condition. Whether $2\kappa\theta \ge \sigma_v^2$ holds at empirically calibrated parameters (variance touching zero) affects simulation schemes and boundary behaviour.
• One-factor limitation. A single variance factor cannot jointly fit the level, slope, and curvature of the implied-vol surface, motivating multi-factor and rough-volatility extensions.

Related

• Stochastic Volatility
• Jump-Diffusion Model
• GARCH and BEKK-GARCH
• Black-Scholes Option Pricing
• Realized Volatility
• Diffusion Process
• Risk-Return Tradeoff
• Fourier Transform Option Pricing
• GARCH Option Pricing