$1.$ Consider a family of European call options on a non$-$dividend$-$paying stock, with

maturity T$,$ each option being identical except for its strike price. The current

value of the call with strike price K is denoted by C$($K$).$ There is a risk$-$free

asset with interest rate r $>=0.$

$($a$)$ Prove the following general relation using no$-$arbitrage arguments: K$2>=$

K$1$ implies K$2$ K$1>=$ C$($K$1)$ C$($K$2).$

$($b$)$ If you observe that the prices of the two options C$($K$1)$ and C$($K$2)$ satisfy

K$2$ K$1<$ C$($K$1)$ C$($K$2),$ construct a zero$-$cost strategy that corresponds to an arbitrage opportunity, and explain why this strategy leads to

arbitrage.