Problem $5$

You want to determine if an arbitrage opportunity exists in the option market. You

gather the following information regarding a non$-$dividend paying stock:

The current price of the stock is $S_0=100.$ Its volatility is $30\%$ per annum.

There is a put option with maturity of $6$ months and strike $K=95$ that trades

at $$5.$

The continuously compounded risk$-$free rate is $r=0\mathrm{.}02.$

$($A$)$ Considering a one$-$period binomial model, compute the future price of the stock

in the up and the down states.

$($B$)$ Compute $$ and $B$ to create a replication portfolio.

$($C$)$ Compute the price of the put option and explain why there exists an arbitrage

opportunity.