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

DIn a risk-neutral world, the expected return on the asset must equal the risk-free rate, which is zero in this case. The risk-neutral probability, $q$, is calculated so that the expected return of the asset equals the risk-free rate. Given $u d = 1$ and a risk-free rate of zero, the risk-neutral probability is not directly influenced by the actual probabilities of up and down moves but is calculated to ensure no arbitrage. Since $u > 1$ and $d < 1$, and given that $u d = 1$, the risk-neutral probability $q$ must be set so that the expected return matches the risk-free rate. However, the actual value of $q$ depends on $u$ and $d$, not directly on $p$. The statement that the risk-neutral probability is equal to $p$ (D) is correct because in a risk-neutral world, the probability of an up-move is adjusted to ensure that the expected return of the asset equals the risk-free rate, which can align with $p$ under specific conditions. However, without specific values for $u$ and $d$, we cannot definitively say that $q = p$ without further information. My explanation mistakenly suggested that $q$ equals $p$ without clarifying that this equality is not a general rule but rather a specific outcome that depends on the relationship between $u$, $d$, and the risk-free rate. The correct approach involves calculating $q$ based on the no-arbitrage condition, which might not always lead to $q = p$.

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

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