The implied-volatility smile (Note 1) is the market telling us that constant- GBM is wrong. This note climbs the ladder of fixes — term-structure vol, Dupire local volatility, the Heston stochastic-variance model, and the SLV hybrid that the FX industry actually runs — explaining at each rung what problem the model solves and what new problem it creates. The Heston model gets the full treatment: why it exists (it reproduces all four stylised facts of volatility), what each of its five parameters does to the smile (the calibration intuition), how to price under it (semi-closed form, 2D PDE with ADI, or Monte Carlo with the full-truncation scheme), and how to calibrate it to a market surface.
Learning objectives. By the end of this note you should be able to:
- distinguish instantaneous, realised and implied volatility and state the four stylised facts;
- explain the local-vol vs stochastic-vol trade-off (fit vs dynamics) and where SLV sits;
- write the Heston SDEs and map each parameter to its smile effect;
- describe the three pricing routes (characteristic function, ADI PDE, full-truncation MC);
- set up the calibration objective and sensible initial guesses;
- explain the SLV leverage function and the Gyöngy mimicking argument behind it.
In this lesson
- 6.1 Volatility: empirical facts
- 6.2 The modelling ladder
- 6.3 The Heston model
- 6.4 Simulating Heston
- 6.5 Calibration
- 6.6 Stochastic-local volatility (SLV) — the industry hybrid
- 6.7 Jumps — the other extension
The full lesson, with worked examples and exercise solutions, is for members.
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