Consider a one$-$period model with only two possible end$-$of$-$period

states. Three assets are traded in an arbitrage$-$free market. Asset $1$ is

a risk$-$free asset with a price of $1$ and an end$-$of$-$period dividend of $R^f$

$,$ the risk$-$free gross rate of return. Asset $2$ has a price of $S$ and offers

a dividend of $uS$ in state $1$ and $dS$ in state $2.$

a$)$ Show that in order to avoid arbitrage, it must be the case that

$K{C}_u=$max$(uS-K,0)C_d=$max$(dS-K,0){}_1{}_2{}_1=\frac{1}{R^f}\frac{u{C}_d-d{C}_u}{u-d},{}_2=\frac{C_u-C_d}{(u-d)S}C=\frac{1}{R^f}(q{C}_u+(1-q)C_d)q=\frac{R^f-d}{u-d}d$

Exercises

Asset $3is$ a call option $on$ asset $2$ with $an$ exercise price $ofK.$ Thus

the dividend $of$ Asset $3$ will $be{C}_u=$max$(uS-K,0)in$ state $1$ and

$C_d=$max$(dS-K,0)in$ state $2.$

$b$