If the volatility of a non-dividend-paying stock is 20% per annum and a risk-free rate is 5% per annum,which of the following is closest to the Cox,Ross,Rubinstein parameter p for a tree with a three-month time step?
A) 0.50
B) 0.54
C) 0.58
D) 0.62

Question

Quizplus · Accepted Answer

The Cox, Ross, and Rubinstein (CRR) model parameter $p$ can be calculated using the formula: $$p = \frac{e^{r\Delta t} - d}{u - d}$$ where $r$ is the risk-free rate, $\Delta t$ is the time step (in years), $u$ and $d$ are the up and down factors respectively. The up and down factors can be calculated using the volatility $\sigma$ as follows: $$u = e^{\sigma\sqrt{\Delta t}}$$ $$d = e^{-\sigma\sqrt{\Delta t}}$$ Given $\sigma = 20\% = 0.20$ per annum, $r = 5\% = 0.05$ per annum, and a three-month time step ($\Delta t = \frac{3}{12} = 0.25$ years), we can calculate $u$ and $d$: $$u = e^{0.20\sqrt{0.25}} = e^{0.20 \times 0.5} = e^{0.1}$$ $$d = e^{-0.20\sqrt{0.25}} = e^{-0.1}$$ Now, calculating $p$: $$p = \frac{e^{0.05 \times 0.25} - e^{-0.1}}{e^{0.1} - e^{-0.1}}$$ $$p = \frac{e^{0.0125} - e^{-0.1}}{e^{0.1} - e^{-0.1}}$$ Numerically solving this gives a value close to 0.54, making option B the closest choice.

If the Volatility of a Non-Dividend-Paying Stock Is 20% Per