Consider a two$-$period binomial model with the following inputs: $S_{t_0}=9$;

$T=1$ year; $u=1$;$25$;$d=0\mathrm{.}75$;$i=2\mathrm{.}5\%($compound interest$)$ on yearly basis. Assume that the underlying

asset pays no dividends. Solve the following problems:

$1)$ Find the no$-$arbitrage price of a European put option with strike price $K=10$ and maturity $1$

year.

$2)$ Verify that the European put price in point $1$ satisfies the Merton's constraints. Determine the

corresponding European call price using the put$-$call parity.

$3)$ Find the no$-$arbitrage price of an American put option with strike price $K=10$ and maturity $1$

year.

$($Show enough steps for all questions$).$