$($b$)$ Consider two options on the same stock and same time to maturity but with different strike prices. For option A$,$ the strike price $($K$1)$ is equal to $10$ USD, while for option B$,$ the strike price $($K$2)$ is equal to $9\mathrm{.}5$ USD. The current stock price $($S$)$ is equal to$10$ USD. There are no dividends and the risk$-$free is $3\%$ p$.$a$.$ In calculating the arbitrage$-$free option prices an investor$$s volatility estimate is $15\%$ p$.$a$.$ Yet option A trades for $0\mathrm{.}8$ USD and option B for $1$ USD.

i$.$ Compare the implied volatilities of both options A and B to the investor$$s estimate of $15\%$

ii$.$ ii$.$ Identify the optimal strategy in the two options. Using the investor's volatility estimate, derive the delta$-$neutral position of your call option portfolio.