Exercise $3\mathrm{.}4.$ This problem refers to the model of Example $3\mathrm{.}1\mathrm{.}2,$ whose

Radon$-$Nikody'm process $Z_n$ appears in Figure $3\mathrm{.}2\mathrm{.}1.$

$($i$)$ Compute the state price densities

${}_3(HHH).$

${}_3(HHT)={}_3(HTH)={}_3(THH).$

${}_3(HTT)={}_3(THT)={}_3(TTH),$

${}_3(TTT)$

explicitly:

$($ii$)$ Use the numbers computed in $($i$)$ in formula $(3\mathrm{.}1\mathrm{.}10)$ to find the time$-$zero

price of the Asian option of Exercise $1\mathrm{.}8$ of Chapter $1.$ You should get

$(4,4)$ computed in part $($ii$)$ of that exercise.

$($iii$)$ Compute also the state price densities ${}_2(HT)={}_2(TH).$

$($iv$)$ Use the risk$-$neutral pricing formula $(3\mathrm{.}2\mathrm{.}6)$ in the form

$V_2(HT)=\frac{1}{{}_2(HT)}{E}_2[{}_3{V}_3](HT),$

$V_2(TH)=\frac{1}{{}_2(TH)}{E}_2[{}_3{V}_3](TH)$

to compute $V_2(HT)$ and $V_2(TH).$ You should get $V_2(HT)=v_2(4,16)$ and

$V_2(TH)=v_2(4,10),$ where $v_2(s,y)$ was computed in part $($ii$)$ of Exercise

$1\mathrm{.}8$ of Chapter $1.$ Note that $V_2(HT)V_2(TH).$