Let stock follow the GBM

$dS(t)=rSdt+S(t)dW(t)$

Choose some reasonable values for interest rate $r$ and volatility $,$ the initial stock price $S(0),$ for example $S(0)=100.$ Let strike of European put option be $K=0\mathrm{.}95*S(0)$ and the expiry of the option is $T=2$ years. ${?}^1$

a$)$ Calculate the value of the option using the BS formula. $(1$p$)$

b$)$ Check it by simulations: create at least $1000$ simulations of stock price at horizon $T,$ and calculate the payoff

max$(K-S^{(i)}(T),0)$

where $S^{(i)}(T)$ is the final stock price at time $T=2$ in the simulation $i.$ The average across simulations discounted with the risk free rate $r$ should be close to the analytical value $(2p).$ We call it $MC$ value of the put option.