Module 10 · Advanced
SABR & the volatility smile
Black's formula (last module) assumes a singlevolatility. But the market doesn't: if you back out the implied vol at every strike, it isn't flat — it forms a smile (or a lopsided skew). Out-of-the-money options trade richer than a single-vol model says, because real markets have fat tails and jumps.
SABR (Stochastic Alpha–Beta–Rho) is the market-standard model for that shape, with four intuitive knobs:
- α — the overall level of volatility (moves the whole smile up/down).
- β — the “backbone,” how ATM vol moves as the rate moves (0 = normal, 1 = lognormal).
- ρ — correlation between the rate and its vol; negative ρ tilts the smile into a skew (higher vol at low strikes — the classic rates shape).
- ν — vol-of-vol; how bouncy volatility itself is. More ν = more curvature (a deeper smile).
Drag the knobs and watch the smile reshape in real time.
🎛 SABR smile
ATM implied volatility: 15.1%
Black's formula assumes one volatility, but the market prices a different implied vol at every strike — the smile. SABR models it with four knobs: α sets the overall vol level, β the backbone, ρ tilts it (negative ρ → higher vol at low strikes, the classic rates skew), and ν (vol-of-vol) adds curvature — more ν, more smile. Drag ρ negative and ν up to see the skewed smile appear. This is how desks quote swaption vols across strikes. Educational tool — not investment advice.
Things to try
- • Set ν = 0 — the smile flattens to nearly a line. ν is what creates the curvature.
- • Drag ρ negative — the smile tilts into a downside skew (low strikes richer). Positive ρ tilts it the other way.
- • Raise α — the whole smile lifts (more vol everywhere).