You boss at StockAnalyst Inc. asks you to explore the pricing of options. You look at a stock that does not pay any dividend and that is currently traded at S$0=60.$ You estimate that the volatility of this stock is $10\%$ per annum. You find an at$-$the$-$money put option on this stock with a $6-$month maturity that trades at $$3.$ The continuously compounded $($annualized$)$ risk$-$free rate is $5\%.$

$(1)$ Considering a one$-$period binomial model, compute the risk$-$neutral probability that the stock price will go up$.$

$(2)$ Compute the price of a put option with the same characteristics as the traded one, based on your information.

$(3)$ Compute the risk$-$free position $($B$)$ needed to create a replication portfolio of a put option that is similar to the traded one.

$(4)$ Using the put$-$call parity and the observed put price $($$$3),$ what should be the price of the corresponding European call?