In local volatility models, volatility depends on both time and spot.

CEV (constant elasticity variance)

\[\sigma(S_t) = a S_t^\beta\]

$a$ represents the volatility at today’s spot. $\beta$ is a parameter to fine-tune.

Depending on the value of $\beta$ (the elasticity parameter), the CEV model provides a more realistic description of market dynamics than the constant-volatility assumption. The parameter $\beta$ determines how volatility scales with the spot level. The power-function structure introduces a nonlinear dependence of volatility on spot, which allows the model to capture skew effects observed in the market.

$\beta = 1$ => Black–Scholes (linear in S)

$\beta < 1$ => volatility grows slower than linearly

$\beta > 1$ => volatility grows faster than linearly