Stochastic Volatility and the Heston Model

Local volatility fits every vanilla but gets the dynamics of the smile wrong because volatility in it is a deterministic function of spot. Stochastic volatility takes the honest step: make volatility its own random process, driven by its own Brownian motion, correlated with the stock. The Heston model is the canonical example, realistic smile dynamics, and, uniquely, a semi-closed-form solution that made it the industry standard. This is the model that gets the behaviour of vol right even if it fits the surface less perfectly than local vol.

The Heston SDEs

Heston models the variance $v_t$ (not the vol) as a mean-reverting square-root (CIR) process, correlated with the stock:

$$\begin{aligned} dS_t &= r S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^{S},\\ dv_t &= \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^{v},\\ dW_t^{S}\,dW_t^{v} &= \rho\,dt. \end{aligned}$$

Each parameter has a direct, tradeable interpretation:

• $\theta$, long-run variance: the level variance mean-reverts to; sets the far-maturity ATM vol.
• $\kappa$, mean-reversion speed: how fast variance pulls back to $\theta$; sets how quickly the term structure of vol flattens.
• $\xi$, vol-of-vol: the volatility of variance; controls the curvature (butterfly) of the smile. $\xi = 0$ collapses Heston to Black-Scholes.
• $\rho$, spot-vol correlation: controls the slope (skew). Negative $\rho$ (equities) makes vol rise as spot falls, fattening the left tail and producing the downward skew.
• $v_0$, initial variance: anchors near-dated ATM vol.

The square-root diffusion $\xi\sqrt{v_t}$ keeps variance non-negative and makes its increments larger when variance is high, volatility clustering. The Feller condition $2\kappa\theta \ge \xi^2$ guarantees $v_t > 0$ (strictly stays off zero); when violated, variance can touch zero, which matters for numerical schemes.

Why stochastic vol reproduces the smile

Mixing lognormals over a random variance produces a leptokurtic (fat-tailed) terminal distribution, a symmetric smile even with $\rho = 0$, because averaging Black-Scholes prices over volatility states is convex in vol (positive volga). Turning on $\rho < 0$ correlates big down-moves with vol spikes, skewing the distribution left and tilting the smile into the observed equity skew. Cleanly: vol-of-vol $\xi$ builds the smile's curvature; correlation $\rho$ builds its slope. Unlike local vol, when spot moves the smile moves in a realistic, floating way, so forward skew and barrier/cliquet prices come out sensible, the reason SV is preferred for exotics despite a looser fit.

The semi-closed-form solution

Heston's fame rests on tractability. Although there is no Black-Scholes-style formula, the characteristic function of $\ln S_T$ is known in closed form. Writing the call as the Black-Scholes-like decomposition

$C = S_0 P_1 - K e^{-rT} P_2,$

the exercise probabilities $P_1, P_2$ (the analogues of $N(d_1), N(d_2)$) are recovered by Fourier inversion of the characteristic function:

$P_j = \frac12 + \frac{1}{\pi}\int_0^\infty \operatorname{Re}\!\left[\frac{e^{-i\phi\ln K}\,f_j(\phi)}{i\phi}\right]d\phi,$

where each $f_j(\phi) = \exp\!\big(C_j(\phi,T) + D_j(\phi,T)v_0 + i\phi\ln S_0\big)$ and $C_j, D_j$ solve a pair of Riccati ODEs with explicit solutions. So pricing reduces to a single numerical integral, fast and stable, the property that made Heston the workhorse for calibration (which requires thousands of repricings). Carr-Madan's FFT approach evaluates the whole strike vector in one transform.

Why SV over LV

The comparison is the heart of the topic:

	Local vol	Stochastic vol (Heston)
Fits today's vanillas	Exactly	Approximately (5 params)
Smile dynamics	Wrong (flattens)	Realistic (floats)
Forward skew / cliquets	Mispriced	Reasonable
Vol-of-vol / VIX options	Impossible ($\sigma$ deterministic)	Native
Completeness / hedging	Complete	Incomplete (extra vol risk)

Local vol wins on static fit; Heston wins on dynamics and on being able to price products that depend on the randomness of vol. In practice desks run local-stochastic vol (LSV): a Heston-like stochastic backbone multiplied by a local-vol "leverage function" calibrated to reproduce vanillas exactly, best of both.

Worked example

Calibrate Heston to an equity index: typically $v_0 \approx \theta \approx (18\%)^2 = 0.032$, $\kappa \approx 2$ (variance half-life $\ln 2/\kappa \approx 4$ months), $\xi \approx 0.4$, $\rho \approx -0.7$. Check Feller: $2\kappa\theta = 2(2)(0.032) = 0.128$ vs $\xi^2 = 0.16$, violated, common for equities, so variance can graze zero and a full-truncation Euler or the QE scheme is needed for Monte Carlo. The strong negative $\rho = -0.7$ delivers the steep index skew; the $\xi = 0.4$ vol-of-vol gives the wing curvature.

What breaks in practice

• Feller violation and simulation bias. With $2\kappa\theta < \xi^2$, naive Euler discretization of the square-root process goes negative and biases prices; Andersen's QE or the exact Broadie-Kaya scheme is required.
• Imperfect fit and calibration instability. Five parameters cannot match a whole surface exactly, especially the short-dated wings, and calibration can be ill-posed (different parameter sets fit similarly), regularization or LSV is used.
• One vol factor. A single stochastic variance struggles to fit both the short and long ends of the term structure at once; multi-factor (double-Heston) or adding jumps (Bates) is common.
• No jumps. Pure diffusion still under-prices very short-dated wings, where a jump component (Bates = Heston + Merton jumps) is needed.

In interviews

Write the two SDEs from memory and state what each parameter does: $\theta$ long-run variance, $\kappa$ mean-reversion speed, $\xi$ vol-of-vol (curvature), $\rho$ correlation (skew), and the CIR variance ensuring positivity with the Feller condition $2\kappa\theta\ge\xi^2$. Explain why stochastic vol beats local vol: right smile dynamics (floating vs flattening) so it prices forward-start and barrier exotics properly, and it can price options on vol itself. Know that Heston is famous because its characteristic function is closed-form, so pricing is one Fourier integral, enabling fast calibration. A sharp follow-up is the LSV compromise. This model sits directly behind variance swaps and the VIX.