All the models developed so far — local vol, Heston, SABR — specify the dynamics of the underlying, and from those dynamics, one derives the implied vol surface. A different approach inverts the logic: parameterise the dynamics of the implied vol surface directly. The practical motivation is delta hedging: in a sticky-delta world, the hedge ratio for an option depends not only on the current delta but also on how the surface will move when the spot moves — the "vol of vol" in the Greeks sense.
This note covers three ideas from Gatheral: Derman's empirical skew rule (a simple parametric model for how the smile moves), the no-arbitrage constraints on surface dynamics (why arbitrary smile movements create static arbitrage), and the digital cliquet as the product that most cleanly prices the forward vol dynamics.
Learning objectives. By the end of this note you should be able to:
- state Derman's sticky-delta and sticky-strike rules and their implied delta adjustments;
- explain what "skew stickiness" means and how to estimate it from historical SPX data;
- describe the digital cliquet payoff and explain why it is pure forward-vol exposure.
In this lesson
- 4.1 Derman's skew stickiness ratio
- 4.2 Derman's empirical rule
- 4.3 Constraints on surface dynamics from no-arbitrage
- 4.4 The digital cliquet
Before you start
- Long-Expiry and Extreme-Strike Asymptotics — Lee's Moment Formula
- Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Advanced Derivatives and Exotic Options syllabus.