A binomial tree setting has an up-move of $u 1$ (with probability $p = 0.75$ )and a down move of $d \frac { 1 } { 2 }$ ) B)Is $

Question

A binomial tree setting has an up-move of $u > 1$ (with probability $p = 0.75$ )and a down move of $d < 1$ (with probability $1 - p = 0.25$ ),with $u d = 1$ .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 up-move A)Is $> \frac { 1 } { 2 }$ ) B)Is $< \frac { 1 } { 2 }$ ) C)Is equal to $\frac { 1 } { 2 }$ ) D)Is equal to $p$ )

Quizplus · Accepted Answer

CIn a risk-neutral world, the expected return on the asset must equal the risk-free rate, which is 0 in this case. Given $u > 1$, $d < 1$, and $ud = 1$, the risk-neutral probability $q$ of an up-move is calculated using the formula $q = \frac{1-d}{u-d}$. Since $u > 1$ and $d < 1$ with $ud = 1$, it implies $d = \frac{1}{u}$, leading to $q = \frac{1 - \frac{1}{u}}{u - \frac{1}{u}}$. This simplifies to $q = \frac{u - 1}{u - \frac{1}{u}}$, which is greater than $\frac{1}{2}$ because $u > 1$. Therefore, the risk-neutral probability of an up-move is greater than $\frac{1}{2}$ and is not necessarily equal to the actual probability $p = 0.75$, making choices A and C correct.

A Binomial Tree Setting Has an Up-Move Of u>1u > 1u>1

A Binomial Tree Setting Has an Up-Move Of $u > 1$