Monte Carlo

Very similar to historical simulation although the main difference is how the first step of the algorithm is calculated.
The return (or price) of an asset at a given time in the future is generated/estimated using a random number.
Also known as Stochastic Simulation
This is the most flexible method
It involves simulating the random price behaviour of financial assets using a computer.
Each simulation gives a possible value for the portfolio at the end of the period.
After a sufficient number of simulations, the simulated distribution of portfolio values (from which the VAR estimates can be inferred) should converge to the unknown true distribution.
linear positions (ie not options) require fewer iteations
This is extremely intensive and sometimes the cost may outweigh the benefit
This method can be used to value complex derivatives.

This is the only way to realistically model several underlyings that have a correlation that is not linear.

Use Mathematics > Cholesky Decomposition to produce a lower triangular matrix.

More suitable when portfolios contain options
Assume that returns exhibit a normal distribution
The minimum number of iterations depends on the portfolio
This method has been used since 1977

3 steps in generating a VAR estimate

Advantages
1) Allows you to choose data sets individually for each variable.
For example some parameters (such as volatility) may be best estimated with more recent data while other parameters may require longer data periods
The weighting of data can also be chosen individually
2) Missing data for one variable does not have to be missing from other variables
3) Can incorporate factors which have no historical data
4) Volatilities and correlations can be estimated using different statistical techniques such as implied volatility, GARCH

Monte Carlo simulations generate an estimated asset price
Model risk - the risk of loss arising from the failure of a model to accurately match reality

The precision of an estimated parameter is proportional to the inverse of the square root of the number of iterations
Monte Carlo simulations tend to use a normal distribution but other distributions can be used
To oversome the problem of correlation between assets a procedure called Choleskly Decomposition is used

The Monte Carlo method for calculating VAR is more suitable when portfolios contain options
Assume that returns exhibit a normal distribution
The minimum number of iterations depends on the portfolio
There are three steps to generating a VAR estimate
1) generate scenarios
2) calculate the value of the portfolio for each scenario
3) reorder the results
This approach involves generating a large number of different price scenarios to value the assets over a range of possible market conditions
The portfolio is then revalued using all the price scenarios
Finally the portfolio revaluations are ranked to select the required level of confidence for the VAR calculation

The advantage of the Monte Carlo approach is that it can be used to value options and more complex derivatives accurately