Suppose the price of a stock today is $510.$ The strike price of call and put

options trading on the stock is $490.$ These options mature in $6$ months. The

discount risk$-$free rate is $6\%$ per half annum. The risk$-$neutral binomial analysis

with a time step of $3$ months for a European call option produces an option

value of $53\mathrm{.}3.$ The downward factor for the lattice model of the stock$-$price

process is $0\mathrm{.}928.$ Identify $1)$ the risk$-$neutral probabilities of the binomial tree,

and $2)$ the bi$-$monthly volatility of stock returns. Show your calculations.