$(15$ points$)$ A stock price is currently $$100.$ Over each of the next two three$-$month periods

it is expected to go up by $10\%$ with probability $55\%$ or down by $10\%$ with probability $45\%.$

The risk$-$free interest rate is $1\%$ per annum with continuous compounding. Use a two period

binomial tree to answer the parts below. Be sure to draw and clearly label the binomial tree.

Hint: $e^{-0\mathrm{.}01*\frac{1}{4}}(0\mathrm{.}45*16)$~~$7\mathrm{.}18,e^{-0\mathrm{.}01*\frac{1}{4}}(0\mathrm{.}45*17)$~~$7\mathrm{.}63,$ and $e^{-0\mathrm{.}01*\frac{1}{4}}(0\mathrm{.}55*17)$~~$9\mathrm{.}33$

$($a$)$ Estimate the value of a half$-$year European put option with a strike price of $$98.$

$($b$)$ Estimate the value of a half$-$year American put option with a strike price of $$98.$