A stock is currently priced at $$25.$ In $6$ months it will be either $$26$ or $$30.$ The risk$-$free rate is $12\%$ per annum with continuous compounding.

$($a$)$The notes discuss that No Arbitrage implies the up and down factors satisfy certain conditions. What are they and verify them in this example.

$($b$)$What is the price of a European put option expiring in $6$ months with strike price $$28.$

$1$

$($c$)$ If your portfolio is long this European put, how many stocks should your portfolio have to be risk$-$free?

$($d$)$ If your portfolio is short three of these European puts, and long a bond which pays $$100$ in $6$ months, how many stocks should your portfolio have to be risk$-$free?