$3$:

$C=SN(d_1)-E{e}^{-rt}N(d_2)$

$d_1=\frac{ln(\frac{S}{E})+(r+\frac{{}^2}{2})t}{\sqrt[\phantom{2}]{t}}$

$d_2=d_1-\sqrt[\phantom{2}]{t}$

a$)$ Use the Black$-$Scholes option pricing model to value a call option on a

share with the information as specified above.

b$)$ What would be the value of an European put option on the same share

with the same exercise price and expiry date as for the call?

c$)$ An investor owns $1,500$ shares. What action should she take to hedge

her holding using call options? How many options would be required,

and what would be the total value of these options?

d$)$ Assuming the investor were to hedge her holding using put options, how

many options would be required, and what would be the total value of

these options?

e$)$ Assume the share price changes $($today$,$ i$.$e$.,$ assume no change in time

to expiry$)$ to $85p.$ How much will the investor have gained or lost if she

hedged using

i$)$ Call options?

ii$)$ Put options?

f$)$ Why was the position not perfectly hedged?