[量化金融] 比赛中的两个股票期权：布莱克-斯科尔斯预测 [推广有奖]

摘要翻译：
假设一个人买了两只非常相似的股票，想知道在一段时间T之后，其中一只股票会对总资产做出多少贡献，当然，他期望它应该是总资产的1/2左右。本文在经典的Black and Scholes(BS)模型中研究了这个问题，重点讨论了随机变量w=a_t^{（1）}/(a_t^{（1）}+a_t^{（2）}）的概率密度函数P(w)的演化，其中a_t^{（1）}和a_t^{（2）}是由两个完全相同的BS随机方程产生的两个（欧式或亚式）期权的值。我们表明，在BS模型的范围内，P(w)的行为出奇地不同于基于常识的期望。对于欧式期权，P(w)总是经历一个转变（当T接近某个阈值时），从单峰形式到双峰形式，最大概率值接近于0和1，而引人注目的是，w=1/2是最小概率值。这意味着两个选项之间的对称性自发地破裂，只有其中一个选项完全支配和。对于路径依赖的亚洲风格期权，我们观察到同样的异常行为，但仅在一定的参数范围内。在此范围之外，P(w)总是钟形函数，其最大值为w=1/2。
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英文标题：
《Two stock options at the races: Black-Scholes forecasts》
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作者：
Gleb Oshanin, Gregory Schehr
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最新提交年份：
2011
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：General Finance        一般财务
分类描述：Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
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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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英文摘要：
  Suppose one buys two very similar stocks and is curious about how much, after some time T, one of them will contribute to the overall asset, expecting, of course, that it should be around 1/2 of the sum. Here we examine this question within the classical Black and Scholes (BS) model, focusing on the evolution of the probability density function P(w) of a random variable w = a_T^{(1)}/(a_T^{(1)} + a_T^{(2)}) where a_T^{(1)} and a_T^{(2)} are the values of two (either European- or the Asian-style) options produced by two absolutely identical BS stochastic equations. We show that within the realm of the BS model the behavior of P(w) is surprisingly different from common-sense-based expectations. For the European-style options P(w) always undergoes a transition, (when T approaches a certain threshold value), from a unimodal to a bimodal form with the most probable values being close to 0 and 1, and, strikingly, w =1/2 being the least probable value. This signifies that the symmetry between two options spontaneously breaks and just one of them completely dominates the sum. For path-dependent Asian-style options we observe the same anomalous behavior, but only for a certain range of parameters. Outside of this range, P(w) is always a bell-shaped function with a maximum at w = 1/2.
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PDF链接：
https://arxiv.org/pdf/1005.1760