A European call option with strike $$22$ and expiry in $90$ days is

currently available for $$2\mathrm{.}40.$ The underlying asset has current value

S $(0)=20.$ The yearly continuously compounding interest rate is

$4\%.$

$($a$)$ Calculate the daily implied volatility.

$($b$)$ An Arrow$-$Debreu security has the same underlying asset and

time to expiry as the European call, and in a six$-$step model

pays $$1$ at state j $=3.$ Use a binomial tree to calculate the

premium of this Arrow$-$Debreu security.

$($c$)$ Which state price is equal to the Arrow$-$Debreu security pre$-$

mium? Use a formula to calculate this state price and show

that it is equal to your solution in part $($b$).$