A local volatility model replaces Black-Scholes' constant with a deterministic surface calibrated so that the model prices every European option on the surface exactly. Dupire (1994) and Derman-Kani (1994) independently showed that any admissible implied vol surface uniquely determines such a surface, and the previous topic derived the key identity: local variance equals the risk-neutral conditional expectation of instantaneous variance given the terminal stock level. This topic develops the local vol model fully — its derivation, calibration to market data, implementation via PDE and Monte Carlo, and the crucial critique that its forward smile is wrong.
By the end of this topic you should be able to:
- derive the Dupire PDE and local vol formula from European call prices;
- implement a local vol surface in a finite difference scheme;
- use Andersen's interpolation method to extract a smooth local vol function from sparse market quotes;
- explain why local vol produces a flat forward smile and what the consequence is for exotic pricing.
In this lesson
- Contents
- The big picture
The full lesson is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Advanced Derivatives and Exotic Options syllabus.