Generalizing Geometric Brownian Motion

Peter Carr and Zhibai Zhang

Papers from arXiv.org

Abstract: 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.

Date: 2018-09
New Economics Papers: this item is included in nep-rmg
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