In the one$-$period binomial model, the stock price of Apple Inc. $($AAPL$)$ is $$768$ in the

up move and $$512$ in the down move. Also, the one$-$period risk$-$free rate rf $2\%($per annum;

NOT rc$),$ time to maturity $=1$ year, and the current stock price S$(0)=$$$640.$ One unit of asset$-$

or$-$nothing call $($ANC$)$ option pays the option holder one unit of asset or its equivalent cash

value if ST at expiration is greater than the exercise price of $$640,$ and zero otherwise.

$($a$)(2\%$; $2\%)$ Find the martingale probability for the up and down move using the stock price

as numeraire, respectively. Note: You may use EQ$[$B$($T$)/$S$($T$)|$F$(0)]=$B$(0)/$S$(0),$ where

B$($T$)=1+$rf and B$(0)=1.$ F$(0)$ represents the current information set $($for instance, we know the

current stock price is S$(0)).$

$($b$)(2\%)$ Find the current price of the ANC option using stock$-$measure. That is$,$

EQ$[$ANC$($T$)/$S$($T$)|$F$(0)]=$ANC$(0)/$S$(0).$

$($c$)(3\%$; $3\%)$ Find the martingale probability for the up and down move using the savings

account as numeraire, respectively. Note: You may use EP$[$S$($T$)/$B$($T$)|$F$(0)]=$S$(0)/$B$(0),$ where

B$($T$)=1+$rf and B$(0)=1.$ F$(0)$ represents the current information set.

$($d$)(3\%)$ Calculate the expected value EP$[$ANC$($T$)/$B$($T$)|$F$(0)].$