1 the inputs are the annualized m s and s0 that must be

Question

Create a Brownian Motion Process for Stock Returns Using Monte Carlo Simulations in Excel

1. The inputs are the annualized m, s, and S0 that must be calibrated to an index (say, SX5E). Because of the choice of discretization in (4.71), one should use simple historical returns (as opposed to log),

S₀ will be the last available price for the index in question.

2. Use the prescription in (4.72) to generate 65,000 paths for 21 time steps.

3. Calibration and Generation Check. For the 21^st day return, the Mean, Stdev, and Kurt should be close to the inputs, with Kurt = 3 for Brownian Motion. The 21^st day return will be

Here ?t=. Therefore,

The level of convergence between the simulated μ^MC and σ^MC and the input ones is determined by the quality of the random numbers and the number of paths generated. A detailed discussion on MC simulations and convergence can be found in Glasserman [2003].

4. For the 21^st MC return, one can calculate the 99% and 99.9% VaR and CVaR as described in Chapter 3.