$1.$ An up$-$and$-$in European call has strike price K $=20,$ barrier price B $=30.$ In Cox$$

Ross$$Rubinstein notation the underlying asset price is defined by S $=20,$ u $=1\mathrm{.}2,$

d $=1/$u $.$ The return over each time step is R $=1\mathrm{.}1.$

$($a$)$ Construct a four$-$step binomial pricing tree for the vanilla version of this European

call.

$($b$)$ Adjust your four$-$step binomial pricing tree from part $($a$)$ for the up$-$and$-$in Eu$-$

ropean call. Discuss the complication at node $(4,3).$

$($c$)$ Say an up$-$and$-$out European call is available with the same underlying asset,

strike, time to expiry and barrier. Construct a four$-$step binomial pricing tree for

this call.

$($d$)$ Show that in$-$out parity holds for the call values calculated in parts $($a$)($c$).$

$($e$)$ Now consider American versions of the three calls considered above and compare

them to their otherwise equivalent European calls.