Mr N and Sons both wish to buy stocks in H$.$ They dont have enough money right now, so they are considering buying either forwards or options on the stocks, both with a term of $4$ years.The stock price at time $0$ is $ $10$ with standard deviation of $12\%$ per annum. The stockdoes not pay any dividend. The continuously compounded risk$-$free rate of interest is $5\%$ perannum.

i$.$ Calculate the $4$ year forward price on one stock. $[1]$

ii$.$ Calculate the price at time $0$ of a $4$ year call option on one stock with a strike price of $ $12\mathrm{.}21.[3]$

Mr N$\text{'}$s sons, A and R enters into one forward contract, while Mr N buys one call option. At time $4$ the stock is worth $ $12.$

iii. Calculate the accumulated profit or loss at time $4$ for A and R$.[1]$

iv$.$ Calculate the accumulated profit or loss at time $4$ for Mr N$.[2]$

v$.$ Explain why Mr N makes a loss despite having an option that does not force him to buy the stock. $[2]$

vi$.$ Calculate the range of stock prices at time $4$ which would leave Mr Ngwere better off than his Sons. $[3]$

Stochastic models of security prices.

vii. Outline eight differences between a fundamental model and a statistical model $[4]$