$1)$ A share in Pear plc is priced at $12\mathrm{.}50.$ The volatility of the $($continuously

compounded$)$ rates of return on the shares is $38$ percent per year. The $3-$month

risk$-$free interest rate $($continuously compounded$)$ is $7$ percent per year.

European style call and put options with an expiry of $3$ months $(0\mathrm{.}25$ year$)$

and a strike price of $12\mathrm{.}00$ are available. No dividends are expected on Pear

plc shares during the life of these options.

a$)$ Using the Black Scholes model, find the current price of the call options.

b$)$ Using your answer from part $($a$),$ find the current price of the put options.

c$)$ If the market price of the put is $0\mathrm{.}55,$ is the implied volatility above or

below the historical volatility given in this question?

$2)$ What is the difference between historical volatility and implied volatility?

Which should be used in the Black$-$Scholes model, and why?

$3)$ Explain what is meant by realised volatility