Suppose N $=3.$ Consider a contract that pays at termination V$3=$ S$3$ min$\{$S$0,$ S$1,$ S$2,$ S$3\}.$

$($a$)$ Use the backward recursive formula to evaluate V$0,$ V$1,$ V$2$

$($b$)$ Review Example $1\mathrm{.}3\mathrm{.}2$ and Lab $1$ Example. Define;

Y$0=$S$0$

Y$1=$min$\{$S$0,$S$1\}$

Y$2=$min$\{$S$0,$S$1,$S$2\}$

Y$3=$min$\{$S$0,$S$1,$S$2,$S$3\}$

It is known that each Vn is a function in terms of Sn and Yn$.$ For example, let v$3($s$,$y$)=$ s$$y$,$ then V$3=$ v$3($s$,$y$)$ when S$3=$ s and Y$3=$ y$.$ Write down the formulas for v$2,$ v$1,$ v$0.$

$($d$)$n is a function of Sn and Yn; the function is written as $\backslash$delta n$.$ Give an explicit formula for $\backslash$delta n$.$