Every lesson in this unit priced options with a single volatility . Open an equity-index or FX options chain and that assumption fails immediately: the you invert from a 90% put differs from the on an ATM call with the same maturity. The smile (or skew) is not a minor correction — it is the market's verdict that constant-volatility GBM is wrong. This closing note names the empirical facts, explains implied vol as a quotation convention, sketches the modelling ladder without re-deriving Dupire, and hands off to the advanced derivatives unit where surface arbitrage, calibration, and smile-aware hedging are developed in full.
Learning objectives. By the end of this note you should be able to:
- describe the equity skew and FX smile patterns and relate them to tail and crash risk;
- define the implied-volatility surface and total variance ;
- state why a model calibrated only to today's vanillas can still misprice path-dependent exotics;
- place local vol, stochastic vol, and jumps on the modelling ladder and point to the unit where each is treated;
- extract one implied vol from a market price (bisection) on a non-flat surface.
In this lesson
- 4.1 What the market shows
- 4.2 Implied volatility — recap and surface
- 4.3 The modelling ladder (bridge, not derivation)
- 4.4 Where the foundational unit ends
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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