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.

$-$Can you please draw the binomial option pricing models$-$