Let the forward be FtF_t and the volatility αt\alpha_t. The model sets dFt=αtFtβdWt(1)dF_t = \alpha_t F_t^{\beta} dW^{(1)}_t and dαt=ναtdWt(2)d\alpha_t = \nu \alpha_t dW^{(2)}_t, with correlation ρ\rho between the two Brownian motions. The four parameters are α\alpha (the initial volatility), β\beta, ρ\rho, and ν\nu (the volatility of volatility). The name is an acronym of Stochastic Alpha Beta Rho1.

Consider a hypothetical example in which the forward is 2.00%, the maturity is 1 year, and α=0.03\alpha = 0.03, β=0.5\beta = 0.5, ρ=−0.3\rho = -0.3, and ν=0.4\nu = 0.4. The implied volatility from the approximation formula is 21.40% at a strike of 2.00% (ATM) and 19.53% at a strike of 2.50%2.

The approximation formula is not an exact solution. When the forward is negative, the model cannot be used except when β=0\beta = 0, so a shift is added1.

Notes

  1. 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 pages 62–63) and section 4.5 (printed page 65). ↩ ↩2

  2. Hagan, Kumar, Lesniewski, Woodward, "Managing Smile Risk" (Wilmott magazine, 2002), equations (2.15a)–(2.18) (printed pages 88–89). The example values are recomputed from the formulas. ↩