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 asset $1($which pays no dividends prior to date

and financial derivatives written on asset $1,$ and two default$-$free

coupon bonds $(C_A,F_A,T_A)$ and $(C_B,F_B,T_B).$ Moreover, no arbitrage

opportunities can be found at date $0.$

For iin$\{A,B\},$ let $P_i(t,a_t)$ denote the price of coupon bond $(C_i,F_i,T_i)$

at date $t{T}_i$ in event $a_t,$ after the date$-t$ coupon payments have been

made. Furthermore, for any traded asset $j,$

$p_j(t,a_t)$ denotes the price for asset $j$ at date $t$ in event $a_i$;

$G_j(t,a_t)$ denotes the forward contract created at date $t$ in event

$a_t$ for delivery of one unit of asset $j$ at date $2($recall that $G_j(t,a_t)$

also denotes the specified forward price$)$;

$H_j(t,a_t)$ denotes the futures contract created at date $t$ in event $a_t$

for delivery of one unit of asset $j$ at date $2($recall that $H_j(t,a_t)$

also denotes the specified futures price$).$

We are given the following information:

$(C_A,F_A,T_A)=(25,500,2),(C_B,F_B,T_B)=(135,500,2).$

$p_1(0,)=800,H_1(0,)=1,080,p_1(1,E)=880,p_1(1,E^c)=1,760.$

Mr$.$ C is endowed with nothing but $50,000$ in cash and $500$ units

of coupon bond $A$ at date $1$ in event $E,$ and he chooses to turn

these endowments into $500$ units of bond $B$ at date $1$ in event $E.$

Mr$.$ D is endowed with nothing but $44,000$ in cash and $500$ units

of coupon bond $A$ at date $1$ in event $E^c,$ and he chooses to turn

these endowments into $500$ units of bond $B$ at date $1$ in event $E^c.$

Now, solve the following problems.

$($i$)$ Mr$.$ F is endowed with $48,400$ dollars at date $2.$ He only wants to

consume a sure amount at date $1.$ Then the maximum amount that

Mr$.$ F can consume at date $1i$

$($ii$)$ Mr$.$ J is endowed with $660,000$ dollars at date $1.$ He only wants

to consume at date $0.$ Then the maximum amount that Mr$.$ J can

consume at date $0$ is

$($iii$)$ Mr$.$ G is longing $2$ units of bond $B$ and shorting $2$ units of bond

A at date $0,$ and he will clear these positions at date $1.$ This bond$-$

trading strategy generates date$-1$ cash flow No$.10$ in event $E$

and date$-1$ cash flow $.....$ event $E^c.$