KUSMI TEA share price is currently $$130.$ It is known that at the end of six months it will be either $$140$ or $$120.$ The risk$-$free rate of interest with continuous compounding is $7\mathrm{.}00\%$ per annum.

Required:

I. Calculate the value of a six$-$month European put option on the stock with an exercise price of $$132.$ Required to design a one$-$step tree and include the prices of the stock and the value of the option on every node.

II$.$ Verify that no$-$arbitrage arguments $($Calculate and apply $($delta$)$ Delta"$)$ and risk$-$neutral valuation arguments $($Calculate and apply probability $"$p$")$ give the same answers in assessing the put price.

Required the calculation by binomial tree.

III. What will be the initial riskless portfolio?

IV$.$ Applying "put$-$call parity" find the value of a European call with the same terms $($strike$,$ risk$-$free rate and maturity$)$ of the previous European put? Required the calculation.

V$.$ How does a boost in the risk$-$free rate will affect both, the put and the call? Explain your answer.