| Математика
Portfolio simulation

This is an article that will sum up all previous results made in:
1. Cholesky decomposition
2. Stock price simulation
3. Ito’s lemma and stock price

By applying Cholesky decomposition to the portfolio of 4 stocks(A,B,C and D), usnig Monte-Carlo simulation and taking into account Geometric Brownian Motion(GBM) we simulate our portfolio to see possible outcomes.
Let a correlation matrix be defined as follows:

Correlation matrix

The matrix is positive defined, therefore the Cholesky decomposition is used:

Cholesky matrix

Stock price movement formula’s inputs are volatility and mean. Also, current stock price will be taken into consideration:

Volatility, mean and stock price

Next step is to simulate random variables. Although we use RAND() function in Excel, other possible generators can be used to finitune the final results.
Standard normal cumulative distribution function also used to apply stock price normal distribution, so the final function is NORMSINV(RAND()) produces the following results:

random variables

Of cause, it is better to take as many as posiible steps to get better results, here is shown only first 5 steps.
These variables are used with Cholesky matrix to get correlated random variables:

correlated variables

Now,to clarify the results, all the values in column A are the same as in the previous picture. the values in the B column are calculated by using Cholesky matrix(the second raw) and random variable values for A and B, for example, -0.57696=0.8*(-0.87294)+0.6*0.202329.
Column C and D are calculated accordingly to the same algorithms but there are additional variable. For example, value D in the top right cell is:

Lognormal prices according to GBM formula and stock prices taken from Lognormal prices with EXP() formula are shown below:

simulated prices

Now we have complete algorithm to make Monte-Carlo simulation to estimate risk and possible outcomes of an investment portfolio:
1. Use a model, i.e. GBM for stock prices
2. Define correlation matrix calculated from stocks historical prices.
3. Generate random numbers
4. Estimate the output applying stock volatility, mean and Choletsky decomposition