Generalizing Geometric Brownian Motion

TL;DR

This paper introduces a generalized positive stochastic process extending Geometric Brownian Motion by incorporating an asymmetry parameter, allowing for more flexible modeling of volatility and enabling analytical pricing of derivatives.

Contribution

It proposes a new asymmetric process that generalizes GBM, capturing different volatility behaviors at lows and highs while maintaining analytical tractability.

Findings

The new process preserves positivity and tractability.

It allows for explicit pricing of vanilla, barrier, and lookback options.

The model includes a jump-to-default feature for risk management.

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.