We write $(x{)}_{+}=$max$\{0,x\}.$ Throughout we assume that the numeraire asset is

constant and equal to $1,$ so $S=$hat$(S).$

Let $T=1$ and F$\_0=\{,\}.$ In particular $S_{0}in{R}^d.$ We define

Given $vin{R}^d,$ a Basket or Index option with weights $v$ is the financial

instrument whose payoff is the random variable $f^v(S)$:$=v*S_1.$

Given $KinR$ a Call option on asset i with strike $K$ is the financial instrument whose payoff is the random variable $f_{i,K}^{\text{c}\text{a}\text{l}\text{l}}$:$=(S_1^i-K{)}_{+}.$

Given $KinR$ a Put option on asset i with strike $K$ is the financial instrument

whose payoff is the random variable $f_{i,K}^{\text{p}\text{u}\text{t}}$:$=(K-S_1^i{)}_{+}.$

$($a$)$ Find the set of prices of $f^v.$

$($b$)$ Establish the put$-$call parity

$f_{i,K}^{\text{c}\text{a}\text{l}\text{l}}-f_{i,K}^{\text{p}\text{u}\text{t}}=S_1^i-K,$

and obtain with its help the relationship between the set of prices of the call

option and the set of prices of the put option.

$()$ If $f_{i,K}^{\text{c}\text{a}\text{l}\text{l}}$ is replicable, is then also $f_{i,K}^{\text{p}\text{u}\text{t}}$ replicable? If so$,$ what is the relationship between the replicating trading strategy of

$f_{i,K}^{\text{p}\text{u}\text{t}}$ and the replicating strategy of $f_{i,K}^{\text{c}\text{a}\text{l}\text{l}}?$