Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } 1$ (with probability $p$ )or down by $d q _ { 2 }$ ) B) $Q _ { 1 }

Question

Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } > 1$ (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,let $Q _ { 1 }$ be the risk-neutral probability of a one-period at-the-money call finishing in-the-money.and let $Q _ { 2 }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money.Which of the following is true? A) $Q _ { 1 } > q _ { 2 }$ ) B) $Q _ { 1 } < Q _ { 2 }$ ) C) $Q _ { 1 } = Q _ { 2 }$ ) D)Depending on the parameters $\mathcal { U }$ And $d$ ,more than one of the above is possible.

Quizplus · Accepted Answer

In a binomial tree model, the risk-neutral probability for an option to finish in-the-money can vary based on the parameters of the model, such as the up-factor ($\mathcal{U}$), down-factor ($d$), and the number of periods. Since the question does not specify the exact values of $\mathcal{U}$ and $d$, or how they relate to each other beyond $\mathcal{U} > 1$ and $d < 1$, the relationship between $Q_1$ and $Q_2$ can vary. For a one-period model, the risk-neutral probability is directly calculated based on the up and down movements and the risk-free rate. For a two-period model, the calculation involves more steps and the probabilities of different paths. Depending on the values of $\mathcal{U}$ and $d$, $Q_1$ could be greater than, less than, or equal to $Q_2$, making more than one of the statements potentially true under different conditions.

Consider a Binomial Tree Setting in Which in Each Period U>1\mathcal { U } > 1U>1

Consider a Binomial Tree Setting in Which in Each Period $\mathcal { U } > 1$