$19.$E$.6^{?}$ There are two dates: At date $1$ there are three states; at date $0$ there is trade in assets.

There are two basic assets whose return vectors in current dollars are

$r_1=(64,16,4)$ and $r_2=(0,0,1).$

The market prices of these assets are $q_1=32$ and $q_2=1,$ respectively. In the following you

are asked to price by arbitrage a variety of derived assets.

$($a$)$ Suppose that one unit of a derived asset is described as "One unit of this asset confers

the right to buy one unit of asset $1$ at $75\%$ of its spot value in period $1($after the state of the

world occurs$)."$ Write the return vector of this asset and price it$.$

$($b$)$ The situation is the same as in $($a$)$ except that the asset is modified to read "One

unit of this asset confers the right to buy one unit of asset $1$ at $75\%$ of its spot value in period

$1($after the state of the world occurs$)$ provided the spot value is at least $10."$

$($c$)$ Suppose that the asset is as in $($b$)$ except that $"$at least ${10}^n$ is replaced by $"$at least $19."$

Write down the return vector and argue that this asset cannot be priced by arbitrage with the

available primary assets.

$($d$)$ How would the analysis in $($c$)$ differ if we had in addition a riskless asset with a price

equal to $1?($You do not need to compute the price explicitly.$)$

$($e$)$ Suppose that now the asset is further complicated to read "One unit of this asset confers,

at the choosing of the holder, either $1$ dollar in period $1$ or the right to buy one unit of asset

$1$ at $75\%$ of its spot value in period $1($after the state of the world occurs$)."$ Write the return

vector of this asset and price it$.$

$($f$)$ The situation is the same as in $($e$)$ except that the asset is modified to read "One unit

of this asset confers, at the choosing of the holder, either $1$ dollar in period $1$ or the right to

buy one unit of asset $1$ at $75\%$ of its spot value in period $1($after the state of the world occurs$)$

provided this value is at least $10.$