$($a$)$The current price of a non$-$dividend paying stock is $\backslash (\backslash$$ $245\backslash ).$ Assume that European options on the stock exist and the minimum unit to trade is one option. You buy one share of the stock. At the same time, you write a European call option on the stock with the strike price of $\backslash (\backslash$$ $260\backslash )$ and buy a European put option with the strike price of $\backslash (\backslash$$ $230\backslash ).$ The current option premiums of the call and the put are $\backslash (\backslash$$ $2\mathrm{.}10\backslash )$ and $\backslash (\backslash$$ $5\mathrm{.}60\backslash ),$ respectively. The two options expire at time T$.$ Draw the payoff diagram of the above strategy. $($A payoff diagram ignores all option premiums today.$)$ Label the diagram with numerical values. Carefully justify what you draw.

$($b$)$ Suppose that the stock price at time T turns out to be $\backslash (\backslash$$ $258\backslash ).$ How much is the payoff of the above strategy in $($a$)?$ How much is the profit of the above strategy in $($a$)?$

$$ If you modify the strategy by selling two $\backslash (\backslash$$ $260\backslash )$ call options $($instead of one$),$ other aspects of the strategy remain the same: buy one share of the stock and buy the $\backslash (\backslash$$ $230\backslash )$ put. Draw the payoff diagram $($and no justification required here$).$ What are the maximum payoff and the maximum profit of this modified strategy?