Assume the following values when none is specified:

So $=100$

$\backslash$sigma $=20\%$ T$=1$ K $=100$ r $=5\%$

We assume the usual log$-$normal diffusion in the risk$-$neutral measure:

dSt $=$r$$dt$+\backslash$sigma $$dWtQ St

$1.$ Using the closed$-$form formula of ST $,$ use Monte$-$Carlo to give the price, delta, gamma and theta of the call option? Plot the values you get as a function of the number of simulations.

$2.$ Using a direct Euler schema, what is the price, delta, gamma and theta of the call option? Show the values you get using different discretization steps and different number of simulations.

$3.$ Using a Milstein schema, what is the price, delta, gamma and theta of the call option? Show the values you get using different discretization steps and different number of simulations.

$4.$ If you use the same discretization steps as well as the same number of simulations, which method performs the best?

$5.$ Redo the above questions with the added condition that you simulate ln$($St$)$ instead.