Overview

SABR (Stochastic Alpha, Beta, Rho) is the model of choice for interest rate derivatives. Its Hagan-Kumar-Lesniewski-Woodward (2002) paper gave a closed-form approximation for implied vol that turned the model from theoretical curiosity into a market standard: desks could generate full smile curves in microseconds, calibrate in real time, and quote Greeks analytically. This topic develops the SABR model and then uses it as a springboard for the wider theory of volatility surface asymptotics — what happens to implied vol in the short-expiry and extreme-strike limits, and what those limits say about the underlying jump and vol-of-vol structure.

By the end of this topic you should be able to:

• write the SABR SDE system and state the Hagan approximation for implied vol;
• match SABR parameters to an observed smile at a single expiry;
• state the leading-order short-expiry vol asymptotics and explain the $1/\sqrt{T}$ skew divergence;
• apply Roger Lee's moment formula to bound the large-strike implied vol slope.

1. Contents
2. The big picture