Constant Elasticity Of Variance Model

Black-Scholes fixes volatility at a single number no matter how the stock has moved. Real stocks don't behave that way: when a company's shares fall, the firm gets more leveraged relative to its remaining equity, and the same stock tends to become jumpier. Black-Scholes has no mechanism for that. The Constant Elasticity of Variance (CEV) model adds exactly one parameter that lets volatility rise as the price falls, without abandoning a closed, tractable formula.

A rubber band that stiffens as it stretches

Stretch an ordinary rubber band and it resists roughly proportionally — that's Black-Scholes, where the pull (volatility) is fixed regardless of how far you've pulled. Now imagine a rubber band that gets noticeably harder to stretch further the more slack has already been pulled out of it. CEV builds that stock-dependent stiffness directly into the model: as the stock price falls, one parameter setting lets volatility rise, mimicking the real-world "leverage effect" where falling equity prices go hand in hand with rising jumpiness.

$$dS_t = \mu S_t\, dt + \sigma\, S_t^{\beta}\, dW_t$$

In plain English: this looks like ordinary geometric Brownian motion, except the volatility term is $\sigma S_t^{\beta-1}$ times the stock, where $\beta$ (a number typically less than 1 for equities) controls how volatility responds to the price level. Set $\beta=1$ and $\sigma S^{\beta-1}=\sigma$, a constant — you get plain Black-Scholes back exactly. Set $\beta<1$ and volatility rises as $S$ falls, because $S^{\beta-1}$ grows as $S$ shrinks (a negative exponent).

Worked example 1: watching volatility respond to price. Take $\sigma=2$ and $\beta=0.5$ (so $\sigma S^{\beta-1}=\sigma S^{-0.5}=\sigma/\sqrt S$). At $S=100$ (a $100 stock): local volatility is $2/\sqrt{100}=2/10=0.20$, i.e. 20%. Now let the stock fall to $S=64$ ($64): local volatility becomes $2/\sqrt{64}=2/8=0.25$, i.e. 25%. A 36% drop in the stock mechanically raised volatility from 20% to 25% — a 25% relative jump in vol — with no separate assumption bolted on; it falls straight out of the single exponent $\beta$.

Worked example 2: matching a target skew. Suppose a desk observes that a 90-strike put on a $100 stock needs 25% implied volatility to match its market price, while the 100-strike option needs 20%. Under CEV with $\beta-1 = \ln(0.25/0.20)/\ln(90/100) = \ln(1.25)/\ln(0.9) = 0.2231/(-0.1054) = -2.117$, so $\beta \approx -1.117$. That's a much steeper elasticity than the $\beta=0.5$ used above — a reminder that fitting CEV to real equity skews often needs $\beta$ well below the textbook "leverage effect" range, which is one reason desks graduate to fuller local-volatility or stochastic-volatility models for anything beyond a rough first fit.

Tilt the surface and notice how a downward-sloping skew — richer puts, cheaper calls — is exactly the shape CEV with $\beta<1$ produces mechanically, without needing a separate volatility process at all.

What this means in practice

CEV sits between Black-Scholes and the full local-volatility framework of Local Volatility and Dupire's Formula: it's a one-parameter model, so it's easy to calibrate and has (mostly) closed-form option prices, but it can't independently fit an arbitrary skew shape the way a full local-volatility surface can. It's most useful as a quick, interpretable sanity check on skew direction, and as the historical ancestor of the CEV-based interest-rate models still used on rates desks today.