Consider an economy with perfect financial markets that extends for

three dates $(t=0,1,2)$ with $4$ date$-2$ states of nature $({}_1,{}_2,{}_3,{}_4).$

The common information structure for investors is as follows. At $t=0,$

investors know that the true state is an element of $=\{{}_1,{}_2,{}_3,{}_4\}.$

At $t=1,$ investors know whether the true state is an element of $E=$

$\{{}_1,{}_2\}$ or an element of $E^c=\{{}_3,{}_4\}.$ At $t=2,$ investors know

exactly which among ${}_1,{}_2,{}_3,{}_4$ is the true state. It is known that

markets are dynamically complete, and there are many assets traded

at date $0,$ including assets $1-4($which pay no dividends before $2)$ and

some financial derivatives written on $1$ unit of asset $1,$ zero$-$coupon

bonds maturing at dates $1$ and $2,$ and so on$.$ Moreover, no arbitrage

opportunities exist at any time.

The following table gives some $($but not all$)$ information about the

prices of traded assets:

In the above table, for $j=1,2,3,4,p_j(t,a_t)$ is the price of asset $j$ at

the time$-$event node $(t,a_t).$ Let $H(t,a_t)$ and $G(t,a_t)$ be the futures

price and forward price stated in a contract created at the time$-$event

node $(t,a_t)$ for delivery of $1$ unit of asset $1$ at date $2.$ You are told that

$G(1,E^c)=258.$ Also, let $B(0,)$ be the date$-0$ price of the default$-$free

zero$-$coupon bond maturing at date $$ with face value equal to one dol$-$

lar.

$($i$)$ Then $H(0,)=$ Consider an economy with perfect financial markets that extends for

three dates $(t=0,1,2)$ with $4$ date$-2$ states of nature $({}_1,{}_2,{}_3,{}_4).$

The common information structure for investors is as follows. At $t=0,$

investors know that the true state is an element of $=\{{}_1,{}_2,{}_3,{}_4\}.$

At $t=1,$ investors know whether the true state is an element of $E=$

$\{{}_1,{}_2\}$ or an element of $E^c=\{{}_3,{}_4\}.$ At $t=2,$ investors know

exactly which among ${}_1,{}_2,{}_3,{}_4$ is the true state. It is known that

markets are dynamically complete, and there are many assets traded

at date $0,$ including assets $1-4($which pay no dividends before $2)$ and

some financial derivatives written on $1$ unit of asset $1,$ zero$-$coupon

bonds maturing at dates $1$ and $2,$ and so on$.$ Moreover, no arbitrage

opportunities exist at any time.

The following table gives some $($but not all$)$ information about the

prices of traded assets:

In the above table, for $j=1,2,3,4,p_j(t,a_t)$ is the price of asset $j$ at

the time$-$event node $(t,a_t).$ Let $H(t,a_t)$ and $G(t,a_t)$ be the futures

price and forward price stated in a contract created at the time$-$event

node $(t,a_t)$ for delivery of $1$ unit of asset $1$ at date $2.$ You are told that

$G(1,E^c)=258.$ Also, let $B(0,)$ be the date$-0$ price of the default$-$free

zero$-$coupon bond maturing at date $$ with face value equal to one dol$-$

lar.

$($i$)$ Then $H(0,)=$

$($ii$)$ Consider a coupon bond $(C,F,T)=(36,1080,2).$ This coupon

bond has a date$-0$ price equal to

No$.2$

$($iii$)B(0,1)=$

and $B(0,2)=$

$($iv$)$ Mr$.$ N knows at date $0$ that he is endowed with $180,000$ dollars at

date $1,$ and he wants to consume a sure amount at date $2$ by trading

only at date $0.$ Then the maximum amount that $Mr.N$ can consume

at date $2$ is

$($v$)$ The date$-2$ price of asset $1$ is such that $p_1(2,{}_2)=$

and $p_1(2,{}_4)=,$ No$.7.$