The current price of a stock $($underlying asset$)$ is selling for $95$ and has a standard deviation of reg$\%$ per annum, where reg is $19$ times $10.$ Standard deviation is $90\%$ and interest rates are $10\%$ p$.$a$.$

Answer the following questions $($show all the details of your calculation, explain intermediate results, and show your results with $4$ decimal places$).$

a$)$ Using a one$-$step binomial option pricing model, calculate the following:

i$.$ The price of a European put option with a strike price of $100$ maturing in six months.

ii$.$ The price of an American put option with a strike price of $100$ maturing in six months.

b$)$ Using a two$-$step binomial option pricing model, calculate the following:

i$.$ The price of a European put option with a strike price of $100$ maturing in six months.

ii$.$ The price of an American put option with a strike price of $100$ maturing in six months.

c$)$ Compare and critically discuss the results from parts $($a$)$ and $($b$)($tip: does the price changes when the size of the steps changes and why, which one of the two prices you think is more reliable and why, what was the problem when pricing an American$-$style option$).$