Shares of firm XYZ are currently traded at $$100\mathrm{.}00.$ Over the next two years, each year, the price can move up by $31\mathrm{.}35\%$ or down by $9\mathrm{.}51\%,$ with equal probability. The risk$-$free interest rate is fixed at $6\mathrm{.}32\%$ per year $($simple annual rate$)$ for all maturities. XYZ pays no dividends.

Consider an exotic option on XYZ$.$ This option matures at t$=2$ and can be exercised at most once: at time t$=1,$ or at time t$=2.$ If held until t$=2,$ the option pays S$2$ K if exercised, and nothing otherwise. Assume K is $$100\mathrm{.}00.$

At time t$=1,$ the holder of the option can choose one of the following three actions:

$1.$ Do nothing and wait until the next period

$2.$ Exercise the option at time t$=1$ and collect S$2-$ K

$3.$ Cancel the option contract and collect $$20\mathrm{.}00$ at time t$=1$

Use the above to answer the following parts $($A$)($C$).$

Compute the price of the exotic option at time t$=0$