CEV from zero
1/5One parameter controls the entire backbone
CEV is probably the simplest model that produces skew. One exponent -- β -- decides how the diffusion coefficient scales with the spot level. That is the whole trick.
In Black-Scholes, the spot SDE is dS = σ·S·dW. The noise term is proportional to S, so percentage volatility is constant. CEV generalises this to:
β = 0: you get the Bachelier / normal model. The diffusion is σ·dW -- additive noise, no price dependence at all.
0 < β < 1: something in between. The diffusion grows with S, but slower than proportionally.
Think of β as a dial on a mixing board. All the way right (β = 1) you get the lognormal world -- constant percentage wiggles. All the way left (β = 0) you get the normal world -- constant dollar wiggles. Everything in between is a blend. The model does not care about jumps, regimes, or stochastic vol. It just asks: how does the size of the random shock depend on the price level?
The percentage volatility under CEV is σ·Sβ−1. When β < 1, the exponent is negative, so percentage vol rises as S falls. That is the leverage effect, and it is the entire engine behind CEV skew. No extra parameters, no extra noise sources. Just the exponent.
β < 1 means vol goes up when spot goes down
This is the leverage effect. In equity and crypto markets, vol consistently rises when spot falls. CEV with β < 1 captures this mechanically, without needing a second stochastic factor.
If β = 0.5, the local vol function is σ·√S. When S drops from 100 to 50, the local vol does not drop proportionally -- it only drops by √(50/100) ≈ 0.71. But the spot fell by half. The percentage vol actually increases.
The effect is automatic and deterministic. There is no correlation parameter to tune, no second Brownian motion. The price-vol relationship is baked into the single exponent β.
This creates negative skew in implied volatility without any additional parameters. When the market drops, vol goes up mechanically, so OTM puts become more valuable. The put wing of the smile lifts.
The simulator above shows it plainly. Left panel: CEV price paths. When β < 1, paths that drop become visibly noisier -- wider swings at lower levels. Right panel: windowed realized vol plotted against the price level. The negative slope is the leverage effect.
Set β = 1 and the scatter plot flattens out. There is no price-vol dependence. That is the Black-Scholes world.
Set β > 1 and the relationship inverts: vol rises with price. This is unusual in practice, but it shows you the full range of the model.
The leverage effect is not just a model curiosity. It is observable in realized data for equities, credit, and crypto. When markets sell off, realized vol spikes. CEV says this is not because vol has its own random process -- it is because the diffusion coefficient mechanically depends on the price level. It is the cheapest possible explanation for skew.
The implied vol smile from CEV
CEV produces a specific implied vol shape controlled entirely by β. The shape is a tilt, not a U. CEV can do skew but it cannot produce a symmetric smile.
The mapping is straightforward:
β = 1: Flat smile. No skew, no curvature. This is Black-Scholes.
β < 1: Negative skew. The put wing is elevated, the call wing is depressed. The further β is below 1, the steeper the skew.
β > 1: Positive skew. Call wing rises, put wing drops. Rare in equity/crypto but possible in some commodity markets.
Critically, the smile from CEV is monotonic. It tilts one way or the other, but it does not have a U-shape. There is no mechanism for both wings to lift simultaneously, because there is no vol-of-vol or stochastic variance to generate symmetric wing enrichment.
The explorer above shows both pieces: the local vol function σ·Sβ on the left, and the resulting implied vol smile on the right. Drag βand watch them move together. The local vol slope directly drives the smile tilt.
At β = 1, the local vol function is a straight line through the origin (proportional to S). The smile is flat. As βdrops below 1, the local vol function curves downward at high S -- meaning the process becomes less volatile at higher prices. The smile tilts left.
CEV as SABR’s backbone
SABR’s forward equation is dF = σ·Fβ·dW₁. That is literally the CEV process. SABR just bolts on a second SDE for the vol parameter itself.
The full SABR system is:
Second line: σ is now stochastic. ν (vol-of-vol) controls how much σwanders. When ν = 0, σ is a constant and you are back in pure CEV.
Third line: the two Brownians are correlated. ρ adds an additional tilt on top of whatβ already provides.
So CEV is the deterministic foundation of SABR. The β exponent controls the backbone shape of the smile. SABR then adds stochastic vol on top: ν generates curvature (wing enrichment), and ρ adds an extra directional tilt.
In practice, rates desks often fix β at a conventional value (0.5 for rates, sometimes 0 or 1 depending on the regime) and then calibrate σ, ν, ρ to the observed smile. The backbone is chosen once; the stochastic overlay is fitted daily.
The comparison above makes it visual. The solid green curve is CEV alone -- a monotonic tilt. The dashed blue curve is SABR with the same β but nonzero ν. SABR adds the curvature that CEV cannot produce.
Set ν = 0 in the slider and watch the curves overlap perfectly. That confirms the relationship: SABR with zero vol-of-vol is exactly CEV. The backbone is shared.
When you calibrate SABR, the choice of β is not innocent. It determines how much of the observed skew is attributed to the backbone (price-dependent vol) versus the stochastic overlay (ρ tilt). Different βchoices lead to different ρ fits, which affects the forward dynamics and therefore the hedging behaviour. Understanding CEV on its own helps you understand what β is actually doing inside SABR.
Limits and uses
CEV is too simple for fitting real smiles. But it is the right mental model for understanding how price-dependent vol works, and it shows up inside every SABR calibration.
What CEV cannot do:
No curvature. Real smiles have both tilt and curvature -- put wings are steep, call wings are elevated. CEV produces a monotonic tilt but no U-shape. If you try to fit a real crypto smile with CEV alone, you will miss the wings entirely.
No term structure dynamics. CEV has no mean reversion, no vol clustering, no regime changes. The local vol function is static. Short-dated and long-dated smiles have the same shape, which contradicts observed term structure behaviour.
Absorption at zero. For β < 1, the process can reach zero and get absorbed. This is a technical headache for pricing and requires special boundary conditions.
What CEV is good for:
Teaching the leverage effect. If you want one model to explain why vol rises when spot falls, CEV is it. One parameter, one mechanism, clean intuition.
SABR backbone selection. When calibrating SABR, you choose β first. Understanding what CEV does on its own tells you what you are attributing to the backbone versus the stochastic overlay.
Quick skew approximations. The CEV implied vol expansion gives you an analytic relationship between β and skew steepness. If someone quotes you a skew number, you can back out the implied β in your head.
Normal vs lognormal debate. In rates markets, the choice between normal (β = 0) and lognormal (β = 1) quoting conventions is a live debate. CEV makes this a continuous spectrum rather than a binary choice.
CEV says: the size of the random shock depends on the price level, and β controls how. Everything else -- skew, leverage effect, SABR backbone -- follows from that single idea.
Where to go next:
SABR Model -- the stochastic vol extension that uses CEV as its backbone
SVI Parameterization -- direct smile fitting for production surfaces
Heston Model -- a different stochastic vol approach with mean-reverting variance
Interpolation Methods -- all methods compared