Problem $2.$ Consider the following table of Call$/$Put premia for maturity $T=1$ year:

$\backslash$table$[[$Strike$,$Call Premium,Put Premium$],[95,6\mathrm{.}49,1\mathrm{.}78],[100,3\mathrm{.}76,3\mathrm{.}76],[105,1\mathrm{.}94,6\mathrm{.}65]]$

The current asset price is $S_0=94$ and the continuously compounded interest rate is $r=0\mathrm{.}05.$ Suppose you're long the stock with a cost basis of $$92.$

Draw the profit diagram at $T=1$ if in addition to being long the stock you buy a Put option with strike $K=95.$ On the same diagram, plot the profit if you instead buy a Put option with strike $K=100.$

Draw the profit diagram at $T=1$ if in addition to being long the stock you sell a Call option with strike $K=105($no Puts anymore$).$ On the same diagram, plot the profit if you instead sell a Call option with strike $K=100.$

Let us assume that the stock price after $1$ year has a Uniform distribution in the range $80,120.[$This is not a very realistic assumption, but makes the computation much easier$].$ Compute the expected profit for each of the $4$ strategies described above, as well as for the unhedged strategy of simply being long the stock. Which strategy yields the highest average profit?

Hint: since the underlying random variable is uniform, the expected profit is simply the average of your profit diagram in the range indicated, proportional to the area under the curve.