Utilise Excel pleaseQuestion $5$

$($a$)$ The current market price of a three$-$month European put option on a

non$-$dividend paying stock with a strike price of $43$ is $5\mathrm{.}50.$ The stock price

is $45\mathrm{.}00,$ and the risk$-$free interest rate is $6\mathrm{.}5\%.$

i$.$ If a three$-$month call option with the same strike price is currently

selling for $3\mathrm{.}50,$ what opportunities are there for an arbitrageur? How

can he exploit this arbitrage? Explain your answer and show all

workings.

ii$.$ Would the market prices listed above still provide an arbitrage

opportunity if the stock price was $40\mathrm{.}77$ per share in one month?

Explain your answer and show all workings.

$($b$)$ What is the probability that a European call option on a stock with an

exercise price of $53\mathrm{.}00$ and a maturity date in twelve months will be

exercised? The current stock price is $46\mathrm{.}00,$ the interest rate is $7\mathrm{.}65\%$ per

annum, and stock return volatility is $31\mathrm{.}00\%$ per annum. Explain your

answer and show all workings.

$($c$)$ A stock is currently selling for $70\mathrm{.}00.$ A put option has a maturity of $2$

years, and during this time, the stock price is expected to increase by $35\%$

or to decrease by $25\%$ in each year. The annual risk$-$free interest rate is

$6\mathrm{.}0\%$ and the strike price is $65\mathrm{.}00.$ The put option is currently selling for

$6\mathrm{.}00.$ Is the option more likely an American or European put option? Use a

$2-$step binomial option pricing model to determine the answer.