What makes the Heston model different from local vol?

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AcadiFi · Accepted Answer

Heston treats volatility as a second random factor with its own mean-reverting dynamics. Unlike local vol, Heston has realistic forward smile dynamics...

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Miroslav Chenovski at Kyiv Options Institute uses Heston for long-dated FX and equity exotics. Heston's five parameters — V0, theta, kappa, xi, rho — control:
- V0: current variance
- theta: long-run variance
- kappa: mean-reversion speed (half-life = ln(2)/kappa)
- xi: volatility-of-volatility
- rho: correlation between spot and vol (typically -0.7 for equity)

Heston's edge:
1. Forward smile preserves shape — calibrated skew persists
2. Vol-of-vol generates realistic volga
3. Negative rho captures equity leverage effect (vol up when price down)
4. Semi-closed-form vanilla pricing via characteristic function and Fourier inversion

Weaknesses:
- Can't perfectly fit very short-dated smiles without extreme parameters
- Feller condition (2*kappa*theta > xi^2) often violated, allowing V=0
- Calibration is non-convex; local minima trap practitioners

Modern desks use the Heston-LV hybrid: stochastic vol dynamics from Heston, plus a local vol 'leverage function' that forces exact vanilla calibration. This gives the best of both — smile is perfectly matched at time zero, and smile evolves realistically. Miroslav's pricing engine calibrates Heston parameters once per day, then fits the leverage surface to polish vanilla prices.

What makes the Heston model different from local vol?

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