  To convert standard Brownian motion $Z$ into a positive process, Geometric
Brownian motion (GBM) $e^{\beta Z_t}, \beta >0$ is widely used. We generalize
this positive process by introducing an asymmetry parameter $ \alpha \geq 0$
which describes the instantaneous volatility whenever the process reaches a new
low. For our new process, $\beta$ is the instantaneous volatility as prices
become arbitrarily high. Our generalization preserves the positivity, constant
proportional drift, and tractability of GBM, while expressing the instantaneous
volatility as a randomly weighted $L^2$ mean of $\alpha$ and $\beta$. The
running minimum and relative drawup of this process are also analytically
tractable. Letting $\alpha = \beta$, our positive process reduces to Geometric
Brownian motion. By adding a jump to default to the new process, we introduce a
non-negative martingale with the same tractabilities. Assuming a security's
dynamics are driven by these processes in risk neutral measure, we price
several derivatives including vanilla, barrier and lookback options.
