SABR from zero
1/5SABR gives vol its own process
In Black-Scholes, volatility is a constant. In the real world, vol moves around — and it moves with spot. SABR captures both of these facts.
The SABR model is a system of two coupled SDEs. The forward price F and the stochastic volatility σ evolve together:
dσ = α·σ·dW₂
corr(dW₁, dW₂) = ρ
Four parameters, each with a distinct market meaning. α is the vol-of-vol — it controls how aggressively the volatility itself fluctuates. β is the backbone — it determines whether the process behaves more like geometric Brownian motion (β=1) or arithmetic Brownian motion (β=0).ρ is the correlation between spot moves and vol moves — when spot drops, does vol go up? (In equity/crypto, yes: ρ < 0.)
The key insight: vol is not just unknown — it is random and correlated with the underlying. This single idea generates realistic smiles without needing a whole surface of parameters.
SABR was born in the rates world (Hagan, Kumar, Lesniewski, Woodward, 2002). Every swaption desk uses it to interpolate between quoted strikes. The reason is simple: four parameters per expiry, each maps to something observable, and you get an analytic formula for implied vol. No Monte Carlo needed for the smile.
β controls the backbone
The exponent β determines how the instantaneous vol scales with the level of the forward. It sets the character of the underlying process before vol-of-vol or correlation even enter the picture.
β = 1 (lognormal): Percentage moves are constant-sized. If BTC is at 60k, a 1% move is $600. If BTC is at 30k, a 1% move is $300. The dollar volatility scales with price. This is the classic GBM assumption.
β = 0 (normal): Dollar moves are constant-sized. Whether the rate is at 2% or 5%, the daily standard deviation in basis points is the same. This is common in rates markets.
β = 0.5 (CIR-like): A compromise. Vol scales with the square root of price. Popular for crypto and FX, where neither extreme fits perfectly.
Slide β below and watch the three reference smiles. At β=1, the smile is relatively symmetric in log-moneyness. At β=0, the skew profile changes dramatically. The backbone determines how the smile shifts when spot moves — this is how β links to sticky-strike vs sticky-delta behaviour.
In practice, β is often fixed rather than fitted. Rates desks typically use β=0.5 or β=0. Equity and crypto desks often use β=1. The reason:β is hard to disentangle from ρ in a single-expiry calibration. Fixing β and letting the other three parameters absorb the smile is standard practice.
Hagan's approximation
The reason SABR took over rates trading: Hagan et al. derived a closed-form approximation for the Black-Scholes implied vol as a function of strike. No PDE solving, no simulation — just a formula.
The stacked bars below decompose the implied vol at each strike into three additive contributions. The green base is the ATM vol level (what you would get with ρ=0 andν=0 — pure CEV). The orange layer is the first-order skew correction from ρ. The blue layer is the convexity correction from ν (vol-of-vol).
At-the-money, the skew and convexity corrections are roughly zero — the base dominates. In the wings, the corrections grow. Adjust the sliders to see how each parameter controls its corresponding layer.
Notice how the orange skew bars flip sign: they are positive on one side and negative on the other (when ρ ≠ 0). The blue convexity bars are always positive in the wings, adding premium to both deep puts and deep calls.
ρ and ν shape the smile
Once β and α set the backbone and overall vol level, the smile shape is controlled by two parameters: ρ (correlation) tilts the smile, and ν (vol-of-vol) bends it.
ρ is the skew dial. When ρ < 0, spot drops come with vol rises — puts become more expensive than calls. When ρ > 0, the opposite: calls are richer. Atρ = 0, the smile is symmetric (given β=1 or viewing in log-moneyness).
ν is the curvature dial. Higher vol-of-vol means vol itself is more volatile, which makes both wings more expensive. The smile gets wider and the kurtosis of the terminal distribution increases. At ν = 0, there is no smile at all — you are back to a pure CEV model.
The two panels below isolate each effect. Left: fix ν, slide ρ. Right: fix ρ, slide ν. The dashed line is the reference (ρ=0 or ν=0).
ν = 0.40: Moderate vol-of-vol: visible curvature in the wings.
This separation is powerful for intuition but imperfect in practice. ρ and ν are not fully orthogonal — changing one shifts the optimal value of the other during calibration. But the mental model holds: ρ rotates the smile, ν inflates it.
Calibration and pitfalls
SABR calibration means finding (α, ρ, ν) that make the model smile match observed market IVs — with βtypically fixed. Below, try to fit the model to synthetic market data by hand.
The orange circles are "market" implied vols. The green curve is your SABR model. The vertical lines show residuals — the gap between model and market at each strike. Drag the sliders to minimise the SSE (sum of squared errors). A good calibration gets the residuals close to zero everywhere, not just at ATM.
A few things practitioners learn quickly:
The Hagan approximation blows up in the wings. For deep OTM options (say, 10-delta puts on a 2Y swaption), the Hagan formula can produce implied vols that go negative or spike to absurd levels. This is the notorious "wing explosion" problem. Solutions include the arbitrage-free SABR formulation (Hagan-Lesniewski-Woodward 2014) or exact PDE-based approaches.
Negative rates broke the standard model. With β > 0, the forward F must be positive. When interest rates went negative (EUR, JPY, CHF), desks switched to shifted SABR: apply the model to (F + shift) where the shift makes the effective forward positive.
For crypto, β is usually fixed at 0.5 or 1.0. Crypto vol surfaces have extreme skew and fat tails. β=1 (lognormal) is the most common choice since crypto prices cannot go negative. Some desks use β=0.5 for a better fit in the wings.
SABR is per-expiry, not a surface model. Each expiry gets its own (α, ρ, ν) calibration. The model says nothing about how these parameters evolve across expiries. For term-structure consistency, you need additional constraints or a different framework (like SSVI or local-stochastic vol).
Where to go next:
SVI Parameterization — a surface-level model with calendar-spread arbitrage guarantees
Local Volatility — a complementary approach: deterministic vol that exactly matches all vanillas
Interpolation Methods — all smile/surface methods compared