Consider a two$-$period Binomial Model. Suppose S$(0)=100,$u $=1\mathrm{.}5,$d $=0\mathrm{.}75,$t $=0\mathrm{.}5.$ The continuously compounded annualized risk$-$free rate $($r$)$ is $0\mathrm{.}3.$ Let the payoff of a $1-$year European put option with strike $105$ equals to max$\{$K $$S$(1),0\}.$

a$)$ Find the price of the European put option at time$-0,$ which is denoted as p$(0,$S$(0)).$

b$)$ Find the price of the European call option at time$-0,$ which is denoted as c$(0,$S$(0))$

c$)$ Verify that the put$-$call is true for the your answers in $4$a and $4$b

d$)$ Find the price of the American put option at time$-0$ with the same strike price. Do you early exercise? Compare your answer with $4$a and comment.

e$)$ Find the price of the American call option at time$-0$ with the same strike price. $($Hint$,$ an American call option with underlying asset that does not pay any divi dend will never early exercise. You can use this fact to intuitively explain $($no more than $3$ sentence$)$ to get the price of an American call$)$

f$)$ Suppose a derivative security that pays F$(1,$S$(1))=$ S$(1)105$ at maturity, what is the price of this derivative security at time$-0,$ i$.$e$.,$ find F$(0,$S$(0)).$ Assume that there is no early exercise. FINA$6542$ Quantitative Risk Management $3$

g$)$ Explain how you would use Monte Carlo simulation to verify the price of the security that you find in $4$f$.$ Note that for this question, the more detail your description, the better I can understand your logic. Bonus Question: Can you use Monte Carlo simulation to estimate the likelihood of early exercise for an American option?