Consider pricing European options on a stock with an initial price of $$87$ and a strike price of $$87.$ The options mature in $10$ months, and the risk$-$free rate of interest is $3\mathrm{.}50\%$ per annum. The volatility is $=0\mathrm{.}83.$ If a $20$ period binomial tree is to be used, then the risk$-$neutral probability of an up move, $q,$ in the stock price using the Cox$-$Ross$-$Rubinstien $($CRR$)$ solution

$0\mathrm{.}46203$

$-0\mathrm{.}00462$

$0\mathrm{.}85885$

$0\mathrm{.}54208$