Suppose that a non$-$dividend paying stock XYZ is currently traded at $$120.$ The price will either go up by

$20\%$ of will decrease by $10\%.$ The risk$-$free rate is $5\%$ per annum.

a$)$ Find the price of a one$-$year European Call option with the exercise price of $$123$ using

replicating portfolio approach. $(15$ points$)$

b$)$ Derive the risk$-$neutral probabilities and find the price of a Put option with an exercise price of

$(15$ points$)$

c$)$ What is the fair price on a Call option with the same exercise price as in $($b$)?(10$ points$)$

d$)$ Why do the prices of two call options differ in questions $($a$)$ and $($c$)?$ Explain why you expect one

of them to be more expensive than the other. $(10$ points$)$