Say there are two periods: today $($date $0)$ and tomorrow $($date $1).$ Using the notation

introduced in Lecture $1$ there are $s=1,$dots,$S$ possible states of the world at date $1.$

There are $N$ traded assets, denoted by $n=1,$dots,$N.$ Let $x_{ns}$ denote the payoff of asset

$n$ in state $s.$ An Arrow$-$Debreu security for state $s$ pays off $1$ in that state, and $0$ in

every other state. The price of an Arrow$-$Debreu security is denoted by $q_s.$

As we mentioned in Lecture $1$ if there are as many assets as there are states of the

world then financial markets are $($effectively$)$ complete. For this one needs that assets

available are sufficiently different, since in this case one can always find a portfolio

of existing securities that pays the same as any AD security. Also it is needed that

negative values of $w$ are allowed. To find a portfolio that replicates AD$-$security $1$ we

need to solve for a vector of portfolio weights $w$ such that

$w^{\text{'}}x=\left[\begin{array}{cccc}{\displaystyle \underset{n=1}{\overset{N}{}}}{w}_n{x}_{n1}& ?& \text{d}\text{o}\text{t}\text{s}& {\displaystyle \underset{n=1}{\overset{N}{}}}{w}_n{x}_{nS}\end{array}\right]=\left[\begin{array}{cccc}1& ?& \text{d}\text{o}\text{t}\text{s}& 0\end{array}\right].$

in this case the solution is

$w^{\text{'}}=\left[\begin{array}{cccc}1& ?& \text{d}\text{o}\text{t}\text{s}& 0\end{array}\right]x^{-1}$

where $x^{-1}$ denotes, as usual, the inverse matrix. Obviously for this what is needed is

that the matrix $x$ is invertible

In this exercise we will find the replicating portfolios and we will use this to price new

assets.

$($a$)$ Suppose that $S=2,N=2,$ and that asset $1$ and asset $2$ have the same payoff in

state $1,$ i$.$e$.x_{11}=x_{21}.$ State necessary and sufficient conditions in terms of the

assets' payoffs in state $2,x_{12}$ and $x_{22},$ for the market to be $($effectively$)$ complete.

Briefly explain the underlying intuition.

$($b$)$ Again, suppose that $S=2,N=2.$ Asset $1$ pays off $3$ in state $1,$ and $1$ in state

Asset $2$ pays off $1$ in state $1,$ and $5$ in state $2.$ The price of asset $1$ is $0\mathrm{.}70,$

and the price of asset $2$ is $0\mathrm{.}90.$ Assume that there is no arbitrage.

i$.$ Find the replicating portfolios for the two Arrow$-$Debreu securities$.$

ii$.$ Find the prices $q_1$ and $q_2$ of the two Arrow$-$Debreu securities$.$

iii. Find the price of a new asset that pays off $2$ in state $1,$ and $1$ in state $2.$

iv$.$ Consider a risk$-$free bond that pays with certainty $1$ next period in all states.

Let us call the price of that bond $p_f.$ The risk$-$free interest rate of this economy

is defined as $\frac{1}{p_f}-1.$ Find the riskless interest rate of this economy.