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CEV Model

CEV (Constant Elasticity of Variance) is the simplest model that produces skew. It is the backbone inside SABR -- set vol-of-vol to zero in SABR and you get CEV. One parameter controls everything.

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One parameter: beta

Beta controls how vol scales with the underlying price. Lower beta = more skew. That is the entire model.

Explore Beta​

Drag the slider to see how the smile changes as beta moves from lognormal (flat) to normal (steep skew). The dashed blue line always shows the Black-Scholes reference (beta = 1) so you can see the skew that CEV adds.

CEV Smile Explorer

Traditional rates assumption. Vol rises as price drops, creating moderate put skew.
10%20%30%758595ATM105115125StrikeImplied Vol (%)CEV (β=0.5)Black-Scholes (β=1)
β (backbone)0.50
0 = normal, 0.5 = square root, 1 = lognormal (Black-Scholes)

Drag β down to see how skew appears. The dashed blue line shows the flat Black-Scholes smile for reference.

What beta does​

  • beta = 1 (lognormal): Percentage moves stay constant. A 50 stock and a 500 stock both move 2% per day. This is Black-Scholes -- perfectly flat smile, no skew.
  • beta = 0.5 (square root): A middle ground. Implied vol rises as price drops, but not as aggressively as the normal model. The traditional assumption in rates markets.
  • beta = 0 (normal): Dollar moves stay constant. A 1moveisa1 move is a 1 move regardless of price level. Vol (as a percentage) explodes as price drops -- maximum skew. ATM vol stays constant while OTM put vol rises sharply.

Strengths and Limitations​

Strength
What it means for you
One parameter
Nothing to overfit. Beta encodes a single assumption about how vol relates to price.
Natural skew
Lower beta automatically creates put skew -- no extra fitting needed.
Foundation for SABR
Understanding CEV gives you intuition for what the beta parameter does inside SABR.
Limitation
What it means for you
No smile curvature
CEV produces skew (tilt) but not smile (curvature). Both wings do not lift -- you need vol-of-vol (as in SABR) for that.
Static
It is a local vol model. It describes what vol does right now, not how vol itself might change randomly.
Never used alone
CEV is always part of SABR or another model. Nobody calibrates CEV by itself for trading.
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A building block, not a trading model

CEV tells you what beta does inside SABR, which is a trading model. If beta confuses you in SABR, come back here. For delta and vega hedging, you need a model that also captures term structure.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test your understanding before moving on.

Q: What happens to the vol smile as you lower beta from 1 toward 0?
Q: Why can't CEV produce a vol smile (curvature in both wings)?
Q: If you set nu = 0 in SABR, what model do you get?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition​

Learn CEV from scratchInteractive lesson · no prerequisites

This lesson starts from the idea that volatility can depend on price level, then shows how beta creates skew and how CEV sits between Black-Scholes, the normal model, and SABR.

See also:

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