Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } 1$ (with probability $p$ )or down by $d 0$ between the first and second periods.Which of the following is most accurate? A) $Q = Q ^ { D }$ Always. B) $Q \leq Q ^ { D }$ Always. C) $Q \geq Q ^ { D }$ Always. D)Depending on the parameters $\mathcal { U }$ , $d$ ,and $D$ ,both $Q Q ^ { D }$ And $Q \leq Q ^ { D }$ Are possible.

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 $R = \$ 1$ at the end of the period. Let $Q$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there are no dividends;and let $Q ^ { D }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there is a dividend of size $D > 0$ between the first and second periods.Which of the following is most accurate?
A) $Q = Q ^ { D }$
Always.
B) $Q \leq Q ^ { D }$
Always.
C) $Q \geq Q ^ { D }$
Always.
D)Depending on the parameters $\mathcal { U }$
,
 $d$
,and
 $D$
,both
 $Q > Q ^ { D }$
And
 $Q \leq Q ^ { D }$
Are possible.

Quizplus · Accepted Answer

In a binomial tree model, the introduction of a dividend (D > 0) effectively reduces the price of the underlying asset, making it less likely for the call option to finish in-the-money compared to a scenario without dividends. Therefore, the risk-neutral probability of finishing in-the-money without dividends (Q) is always greater than or equal to the probability with dividends (Q^D), as dividends decrease the stock price, making it harder for the option to be in-the-money.

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 $\mathcal { U } > 1$