Q$1.$

Consider a three$-$period binomial tree model for a stock price process $S_t,$ under which the stock price either rises by $18\%$ or falls by $15\%$ each month. No dividends are payable.

The continuously compounded risk$-$free rate is $25\%$ per month.

Let $So=$ R$85.$

Consider a European put option on this stock, with maturity in three months $($i$.$e$.$ at time $t=3)$ and strike price R $90.$

$($i$)$ Calculate the price of this put option at time $t=0.[4]$

$($ii$)$ Calculate the risk$-$neutral probability that the put option expires out$-$of$-$the money.

$[3]$

$($iii$)$ Assess whether the probability calculated in part $($ii$)$ would be higher or lower under the real$-$world probability measure. $[$No further calculation is required.$][3]$