We are at the beginning of year $1.$ F operates a store that will have a value at the end of year $1$ equal to m$1\mathrm{.}2$ with risk$-$neutral probability of $0\mathrm{.}4$ or m$0\mathrm{.}4$ with risk$-$neutral probability of $0\mathrm{.}6.$

F considers buying additional commercial space at the beginning of year $1$ to be able to launch a second store at the beginning of year $2.$ The cost of the operation at the beginning of year $1$ of m$0\mathrm{.}9.$ The value of that second store at the beginning of year $2$ would be the same the value of the first store at that same time. If F buys space at the beginning of year $1,$ but eventually decides not to open a second store at the beginning of year $2,$ it can sell that space at that time for m$0\mathrm{.}6.$ The risk$-$free rate is rf $=2\%.$

$($a$)$ Suppose F does buy space at the beginning of year $1.$ What will F do at the beginning of year $2$ if the value of the first store is m$1\mathrm{.}2$ million? Same question if the value is $0\mathrm{.}4$ million. $(1$ points$)$

$($b$)$ Should F buy commercial space at the beginning of year $1?(1$ points$)$

$($c$)$ F is willing to pay more for commercial space for a second store than the market value of the first store, because the second store has a put value embedded into it$.$ What$$s the value of this put option? $(1$ points$)$