  For option pricing models and heavy-tailed distributions, this study proposes
a continuous-time stochastic volatility model based on an arithmetic Brownian
motion: a one-parameter extension of the normal stochastic alpha-beta-rho
(SABR) model. Using two generalized Bougerol's identities in the literature,
the study shows that our model has a closed-form Monte-Carlo simulation scheme
and that the transition probability for one special case follows Johnson's
$S_U$ distribution---a popular heavy-tailed distribution originally proposed
without stochastic process. It is argued that the $S_U$ distribution serves as
an analytically superior alternative to the normal SABR model because the two
distributions are empirically similar.
