Please solve this question without chatgpt. DO NOT COPY AND PASTE ANOTHER SIMILAR ONE. ELSE I WILL DOWNVOTE. A $$knock$-$out barrier option$($call or put$)$ is an exotic path dependent option, which may only be exercised at maturity if the price of the underlying never hits a pre$-$agreed barrier price H throughout the life of the option. Symmetrically, a $$knock$-$in barrier option$$ may only be exercised at maturity if the price of the underlying hits the barrier price H$.$

Consider call options with strike $100$ and $3-$month maturity when the underlying spot price is $95.$

$($a$)$ Should the price of barrier options be more, less or equally as expensive as plain vanilla options with the same characteristics? Justify your answer.

$($b$)$ Suppose the price of a plain vanilla call with these characteristics is currently $2.$

i$.$ What is the value of a knock$-$out call with barrier H $=97?$

ii$.$ What is the value of a knock$-$in call with barrier H $=97?$

iii. If the value of a knock$-$out call with barrier H $=90$ is $1\mathrm{.}90,$ what is the value of a knock$-$in call with the same barrier? Justify your answer.

$($c$)$ Assume that the underlying is a non$-$dividend paying stock and that the continuously compounded risk$-$free rate of interest is $5\%.$ With a $3-$period binomial tree calculate the value of a knock$-$out call with barrier H $=90,$ assuming that the price of the stock may increase or decrease by $10$ at each step.

Hint: Although part $($a$)$ and $($b$)$ do not require a tree, it helps understand the exotic option.