Local volatility fits the surface but generates wrong dynamics. The Heston model (1993) takes a different approach: it models the source of the smile — a stochastic variance process — explicitly. The volatility is no longer a deterministic function of the spot; it has its own random evolution, driven by a second Brownian motion correlated with the stock. This correlation is the mechanism for the skew; the vol-of-vol is the mechanism for the curvature.
The Heston model is the industry benchmark for equity derivatives. It has five interpretable parameters, a semi-analytic pricing formula for Europeans (via characteristic function inversion), and a tractable simulation scheme. Its weakness is also its clarity: the forward smile it generates does not match the market's short-dated skew, which motivates the jump-diffusion and rough-vol extensions covered in later topics.
By the end of this topic you should be able to:
- state the Heston SDE system and interpret each of the five parameters in terms of its effect on the vol surface;
- verify the Feller condition and explain why it matters;
- derive the characteristic function of using the affine structure of the Heston model;
- price European options by Fourier inversion of the characteristic function;
- calibrate the five parameters to market quotes using the ATM approximation as a seed;
- simulate the Heston process via the Milstein scheme, full-truncation, and the exact scheme.
In this lesson
- Contents
- The big picture
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