Asian options, like barrier options are path dependent. When exercised, an Asian

option is dependent on some average of underlying asset prices over a fixed period

of time. How the underlying asset prices are averaged can vary; for example, you

could average over asset prices from t $=0$ to t $=$ T $,$ or you could average over asset

prices from t $=$ T $/2$ to t $=$ T $.$ The averaging would be specified in the Asian option

contract.

$($a$)$ Say the expiry values of an Asian call option with strike $$10$ are determined from

the average of the underlying asset prices over the life of the option. For a two$-$

step model with underlying asset prices determined from crr notation, S $=10,$

u $=1\mathrm{.}2$ and d $=1/$u$,$ find all expiry values of this Asian option.

$($b$)$ Use the binomial pricing formula to determine the premium of the Asian option

considered in part $($a$),$ where this option cannot be executed prior to expiry. The

return over each step is R $=1\mathrm{.}1.$

$($c$)$ What is the premium of the vanilla European call otherwise identical to the Asian

option considered in parts $($a$)$ and $($b$)?$ Discuss the differences in the European

and Asian premium values.