= Solution
One alternative is a <stochastic volatility model>, such as the <Heston model>:
$$
dS_t=\mu S_tdt+\sqrt{v_t}S_tdW_t^{(1)},\qquad
dv_t=\kappa(\theta-v_t)dt+\xi\sqrt{v_t}\,dW_t^{(2)},\qquad
d\langle W^{(1)},W^{(2)}\rangle_t=\rho\,dt.
$$
The parameters satisfy $\kappa,\theta,\xi>0$ and $|\rho|\le1$, and $v_0\ge0$. The square-root <Itô diffusion> has a nonnegative solution; $2\kappa\theta\ge\xi^2$ is a standard sufficient condition for the positive initial <variance> not to hit zero. Mean reversion of $v_t$ allows persistent high- and low-volatility episodes, and negative $\rho$ can produce the empirical association between falling prices and rising volatility. Mixtures of conditional return distributions produce richer tails and option <implied volatility> shapes than constant-volatility <geometric Brownian motion>.
For data showing <volatility clustering> and option smiles, this is often a more useful fit. It is not universally superior: more parameters create estimation and calibration uncertainty, the model is more expensive to use, and its continuous price paths still exclude price jumps. With two independent noise directions and only a <stock> and a <bank account>, the market is generally incomplete, so a volatility <risk premium> or an extra traded instrument is needed for unique <contingent claim> pricing. \b[The alternative improves flexibility at the cost of calibration and hedging complexity.]
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