标题： 两个股票期权的竞赛：Black-Scholes 预测 显示英文标题

标题： Two stock options at the races: Black-Scholes forecasts

摘要： 假设一个人购买了两只非常相似的股票，并想知道经过一段时间T后，其中一只股票会对整体资产做出多少贡献，当然，期望它大约是总和的1/2。 在这里，我们根据经典的布莱克-斯科尔斯（BS）模型来研究这个问题，关注随机变量w = a_T^{(1)}/(a_T^{(1)} + a_T^{(2)})的概率密度函数P(w)的演变，其中a_T^{(1)} 和a_T^{(2)}是由两个完全相同的BS随机方程产生的两种（无论是欧式还是亚式）期权的值。 我们证明，在BS模型范围内，P(w)的行为与基于常识的预期惊人地不同。 对于欧式期权，P(w)总是会经历一个转变（当T接近某个临界值时），从单峰形式变为双峰形式，最可能的值接近0和1，而且引人注目的是，w=1/2是最不可能的值。 这意味着两个期权之间的对称性自发地被打破，只有一个完全主导总和。 对于路径依赖的亚式期权，我们观察到同样的异常行为，但仅限于一定参数范围之内。 在此范围之外，P(w)始终是一个以w=1/2为最大值的钟形函数。 显示英文摘要

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.