Consider the two$-$period binomial tree model for stock price S with S$0=4,$ u$=2,$ d$=1/2,$ N$=2.$ The risk$-$free interest rate is r$=0\mathrm{.}3$ so that $$1$ invested today grows to $$(1+$r$)$ at the end of first period and to $$(1+$r$)^2$ at the end of second period. Find the risk$-$neutral probabilities q and $1-$q of S going up and down, respectively. Compute the no$-$arbitrage price of put option with the strike price K$=3$ and expiry date T$=2.$ Find the number of shares at t$=0$ according to the replicating strategy for the put option above.