Consider a binomial tree setting in which in each period the price goes up by $u 1$ (with probability $p$ ) or down by $d

Question

Consider a binomial tree setting in which in each period the price goes up by $u > 1$ (with probability $p$ ) or down by $d < 1$ (with probability $1 - p$ ). The risk-free interest rate per time step is zero, so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting, the risk-neutral probability of an at-the-money two-period put finishing in-the-money is _____________ as that of a one-period at-the-money put finishing in-the-money.
A) At least as high.
B) At most as high.
C) Equal to.
D) More than one of the above is possible depending on the parameters.

Quizplus · Accepted Answer

In a binomial tree model, the risk-neutral probability of an option finishing in-the-money can vary based on the parameters of the model (u, d, p). For a two-period put, there are more paths the underlying asset can take, and some of these paths can lead to the put finishing in-the-money, even if it wouldn't in a one-period model. Thus, depending on the specific values of u, d, and p, the probability can be higher, lower, or the same, making more than one of the statements potentially correct.

Consider a Binomial Tree Setting in Which in Each Period u>1u > 1u>1

Consider a Binomial Tree Setting in Which in Each Period $u > 1$