英文标题：
《Generalizing Geometric Brownian Motion》
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作者：
Peter Carr and Zhibai Zhang
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最新提交年份：
2018
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英文摘要：
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.
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中文摘要：
为了将标准布朗运动$Z$转化为正过程，几何布朗运动（GBM）$e ^{\\beta Z\\u t}、\\beta>0$被广泛使用。我们通过引入不对称参数$\\ alpha \\ geq 0$来推广这一正过程，该参数描述了当过程达到新低时的瞬时波动率。对于我们的新流程，$\\ beta$是价格任意高时的瞬时波动率。我们的推广保留了GBM的正性、恒定比例漂移和可跟踪性，同时将瞬时波动率表示为$\\α$和$\\β$的随机加权平均值。该过程的运行最小值和相对缩尺也是可分析的。让$\\ alpha=\\ beta$，我们的正过程简化为几何布朗运动。通过在新过程中添加一个跳转到默认值，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以风险中性的方式驱动的，我们对几种衍生产品进行定价，包括香草、屏障和回望期权。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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一级分类：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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