In local volatility models, volatility depends on both time and spot. It is therefore not static, as it changes with $t$ and $S_t$, but it is not an independent random variable (unlike stochastic volatility models).

CEV (constant elasticity of 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

Dupire local vol

Under the risk-neutral measure, the spot follows:

\[dS_t = (r - q)\,S_t\,dt + \sigma_{loc}(S_t, t)\,S_t\,dW_t\]

$r$ is the risk-free rate and $q$ the dividend yield. The local vol $\sigma_{loc}(S, t)$ is given by Dupire’s formula, so that the model reprices every vanilla on the implied vol surface.