Create in excel (or R or a program of your choice) a Geometric Brownian Motion (GBM) Monte Carlo simulation with the following parameters: S₀=10, risk-free rate=2%, drift=mu=5%, sigma=7%, dt=1day. Each simulation of S should be 360 days long. Run 300 simulations.

- Note that even though the stochastic equation is expressed as ds/s=... you will need to track and plot S=... Write down the equation used in the simulation process and the equation of S (if they are different).

- Note that the expression "drift=mu=5%" really means "drift=mu=5%/yr". Hence, once one can compute the daily drift

- Note that the expression "sigma=7%" really means "sigma=7%/yr". Hence, one can compute the daily standard deviation.

- Plot the results of a few simulations.

- compute E[S_T}, that is, the expected value of S_T

- compute E[S₀}, that is, the expected value of S₀. What is the relationship between E[S_T} and E[S₀}? Would the result be much different if the risk-free rate were stochastic, that is, changing at every time step?

If X is a normal random variable (mean=0, and standard deviation=1), is E[eX] = eE[X]?, or E[eX] / eE[X]? Why? Can you prove it using the Monte Carlo simulations you just developed.