Problem $5(30$ points$)$

You work for a derivative trading desk and your task is to explore any mispricing opportunity in the option market. You look at the stock Pear and Co$,$ which does not pay any dividend. This stock is currently traded at S$0=60$ and the volatility of its return is $\backslash$sigma $=10\%$ per annum. You find an at$-$the$-$money $($ATM$)$ call option on this stock with $6-$month maturity that trades at $$3.$ The effective $($annualized$)$ risk$-$free rate is $5\mathrm{.}127\%.$

$($A$)$Compute the corresponding continuously compounded risk$-$free rate. $(5$ points$)$

$($B$)$Considering a one$-$period binomial model, compute the future prices of the stock. $(5$points$)$

$($C$)$Compute A and B to create a replication portfolio. $(5$ points$)$

$($D$)$Compute the price of the call option and determine whether there exists an arbitrage opportunity. $(5$ points$)$

$($E$)$Explain which transactions you would use to exploit a potential arbitrage opportunity. $(5$ points$)$

$($F$)$ Using the put$-$call parity and the observed call price, what should be the price of the corresponding European put?$(5$ points$)$

G$)$Based on observed option prices, calculate the cost of implementing a $6-$month straddle$(5$ points$)?$