Two stock options at the races: Black-Scholes forecasts

TL;DR

This paper investigates the distribution of relative contributions of two identical stock options under the Black-Scholes model, revealing unexpected symmetry-breaking behavior where one option tends to dominate over time.

Contribution

It demonstrates that within the Black-Scholes framework, the probability density function of the relative option contribution transitions from unimodal to bimodal, indicating spontaneous symmetry breaking.

Findings

European options show a transition to bimodal distribution as time approaches a threshold.

Asian options exhibit similar behavior only within certain parameter ranges.

In some cases, the probability peaks at the equal contribution point, w=1/2.

Abstract

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.