In the Black model, volatility is a constant. A stochastic volatility model gives the volatility its own Brownian motion. The SABR model sets a correlation ρ\rho with the movement of the underlying asset. Markets have both calm and turbulent periods, and the idea reflects the fact that volatility is not constant1.

A Ministry of Finance paper lists three extensions for representing the volatility smile: making volatility a function of something such as the underlying asset's price, giving volatility its own stochastic dynamics, and adding a jump term. Stochastic volatility is the second. In bond market practice, volatility is usually treated as a non-random variable, and the SABR model is one example of stochastic volatility used in practice2.

Models differ in how volatility moves. In the SABR model, volatility moves without mean reversion (dαt=ναt dWtd\alpha_t = \nu \alpha_t \, dW_t). The meaning of the parameters differs by model, so values are not compared directly with those of other models.

Notes

  1. Hagan, Kumar, Lesniewski, Woodward, "Managing Smile Risk" (Wilmott magazine, 2002), section 1 (printed page 84; in Black's model the volatility is in theory a constant) and section 2.3 (printed page 88; the authors give calm and turbulent periods as the reason for treating volatility as a random function). ↩

  2. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions: models) (in Japanese), Finance, October 2021, section 4.2 (printed page 62). ↩