Problem $1($Pricing a Convertible Bond$)(50$ points$)$

Renegade Financial Corporation has issued a pure discount convertible

bond with the following characteristics:

$$ The bond promises to pay in two periods $($say at t $=2)$ Z dollars if the

holders do not convert it into one share of the stock. There is no risk

of bankruptcy.

$$ Bondholders have the option of converting the bond into one share of

the stock in period $1($that is as t $=1$ which is the period before the bond

matures$).$ If the bond is converted, then the payoff is the $($random$)$ price

of the share.

$$ To repeat: the final payoff is either Z or the market price of one share

$($that we denote $$S$2$ to indicate that it is a random variable$).$ The deci$-$

sion whether to convert the bond into stock has to be made at time one.

At this time the price of a share is $$S$1.$

Assume that in this economy the following securities are traded in period

t $=0.$

$$ A one period bond $($price P B

$0(1)),$ a two period bond $($price P B

$0(2)),$

shares in this company $($price $$S$0),$ and calls on the stock of this company

for all possible strike prices $($price C$0($S$,$ K$)$ for all possible K$.)$

$$ In period t $=1$ the same bonds and the stock will be traded as well.

$1.(15$ points$)$ Determine the price at t $=1$ of a pure discount bond that

will pay one dollar at t $=2.$ Denote that price as P B

$1(1).$

$2.(35$ points$)$ Use arbitrage arguments to determine the price at t $=0$ of

the convertible bond.