Let variance itself follow CIR, correlate it with the spot, and the option price still has a closed-form characteristic function. The smile shape and the negative skew fall out of two SDEs and four parameters.
If you hedge an OTM put using Black-Scholes at the at-the-money vol, you will lose money systematically — the put is more expensive than BS predicts. That gap is the smile, and it exists because equity vol moves randomly and spikes when the market falls. Heston (1993) is the first model to capture this in closed form: let vol follow its own diffusion, correlate it negatively with the spot, and the equity skew falls out of the joint dynamics. Crucially, the European option price still has an exact formula — a one-dimensional Fourier inversion of the characteristic function — so calibration stays fast. Three parameters buy the whole smile: $\kappa$ (how fast vol mean-reverts), $\sigma_v$ (vol-of-vol, which fattens the wings), and $\rho$ (leverage, which tilts the smile so OTM puts cost more than calls).
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Worked example
Price a 1-year European call on a non-dividend stock under Heston with $S_0 = 100$, $K = 100$, $r = 5\%$, $v_0 = 0.04$ (so initial vol is 20%), $\kappa = 2$, $\theta = 0.04$, $\sigma_v = 0.30$, $\rho = -0.7$. Compare to the Black-Scholes price at $\sigma = \sqrt{v_0} = 20\%$, and read off the implied vol of the Heston price.
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Reflection
Heston introduces three new parameters ($\kappa, \sigma_v, \rho$) on top of BS's one ($\sigma$). Why is $\rho$ specifically the parameter that controls *skew* (left-right asymmetry of the smile) rather than $\sigma_v$? Where in the joint $(S, v)$ dynamics does the asymmetry come from?