Suppose that there is a European put option written on a stock. This option has $3$ months to maturity. It has an exercise price of $15.$ Currently the stock on which this option is written has a price of $16$ and the put has a premium of $0\mathrm{.}5$ per share

a$)$ What is the current intrinsic value and the time value of that option?

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b$)$ Suppose that expected volatility of the stock and the interest rates do not change but stock price is down to $14$ after $1$ month. Will there be a change in the intrinsic value and time value of that option?

c$)$ Do an investor who holds this option until maturity make a profit or loss if the price of the underlying stock is $14\mathrm{.}5$ at the maturity date?