Q$1)(12$ marks$)$

The price of a non$-$dividend paying stock at time $1,S_1,$ is related to the price at time $0,S_0,$ as

follows:

$S_1=u{S}_0$ with probability $p$ and $S_1=d{S}_0$ with probability $(1-p).$ The continuously

compounded rate of return on a risk$-$free asset is $r.$

$($i$)$ Derive an expression for the replicating portfolio for a European call option written

on the stock that expires at time $1$ and has a strike price of $k,$ where $QqQ*[7]qpd{S}_0.[5]$

$(ii)$ Show that the price $of$ the option $in(i)$ can $be$ written $as$ the discounted expected

payoff under a probability measure $Q.$ Hence find $an$ expression for the probability, $q,$

$ofan$ upward move $in$ the stock price under $Q*[7]$

$(iii)$ Explain the relationship between the $Q$ probability measure $in(ii)$ and the real world

probability measure. Explain what relationship you would expect $q$ and $pto$ have $if$ all

investors are $(a)$ risk averse, $(b)$ risk seeking, $or(c)$ risk$-$neutral. $[5]$