A binomial tree with one-month time steps is used to value an index option.The interest rate is 3% per annum and the dividend yield is 1% per annum.The volatility of the index is 16%.What is the probability of an up movement?
A) 0.4704
B) 0.5065
C) 0.5592
D) 0.5833

Question

Quizplus · Accepted Answer

The probability of an up movement in a binomial tree model can be calculated using the formula: $p = \frac{e^{(r - q) \Delta t} - d}{u - d}$, where $r$ is the risk-free interest rate, $q$ is the dividend yield, $\Delta t$ is the time step (in this case, 1 month or 1/12 year), $u$ and $d$ are the up and down factors respectively. The up and down factors can be calculated using the volatility ($\sigma$) with formulas $u = e^{\sigma \sqrt{\Delta t}}$ and $d = \frac{1}{u}$. Given $r = 0.03$, $q = 0.01$, and $\sigma = 0.16$, we first calculate $u$ and $d$, then use them to find $p$.Given $\sigma = 0.16$ and $\Delta t = 1/12$, $u = e^{0.16\sqrt{1/12}}$ and $d = 1/u$. Plugging these into the formula for $p$, and considering $e^{(r - q) \Delta t}$ with $r = 0.03$ and $q = 0.01$, we can solve for $p$. The correct calculation leads to a probability close to 0.5065, which matches choice B.

A Binomial Tree with One-Month Time Steps Is Used to Value