广义几何布朗运动
Generalizing Geometric Brownian Motion
摘要：
为了将标准的布朗运动$Z$转化为正过程，几何布朗运动（GBM）$e^{βZ_t}, β>0$被广泛应用。我们通过引入一个非对称性参数$ α\geq 0$来概括这个正过程，该参数描述了当过程达到一个新的低点时的瞬时波动率。对于我们的新过程，$β$是价格变得任意高时的瞬时波动率。我们的推广保留了GBM的正性、恒定比例漂移和易处理性，同时将瞬时波动率表示为$α$和$β$的随机加权$L^2$均值。该过程的运行最小值和相关参数也是易于分析的。利用$α= β$，我们的正过程简化为几何布朗运动。通过在新的过程中加入一个跳到违约的过程，我们引入了一个具有相同可处理性的非负鞅。假设证券的动态是由这些过程在风险中性测度，我们定价几个衍生品，包括香草，障碍和回望期权。
英文摘要：
To convert standard Brownian motion $Z$ into a positive process, Geometric Brownian motion (GBM) $e^{βZ_t}, β>0$ is widely used. We generalize this positive process by introducing an asymmetry parameter $ α\geq 0$ which describes the instantaneous volatility whenever the process reaches a new low. For our new process, $β$ 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 $α$ and $β$. The running minimum and relative drawup of this process are also analytically tractable. Letting $α= β$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
