Why do we need correlated random variables in a Monte Carlo simulation?
Question: I don’t understand why a Monte Carlo simulation needs correlated random variables. Isn’t each simulation thread independent?
Background:
具體來說，我指的是 Malz 文本中第 319 頁上的以下範例（http://ca.wiley.com/WileyCDA/WileyTitle/productCd-0470481803.html).
He describes a Monte Carlo simulation with 1,000 simulation threads to calculate credit losses on a CDO with 100 underlying credits.
#1.
在模擬中，我們從 100 維聯合正態分佈中設置了一個包含 1,000 次繪製的矩陣。
#2.
我們為成對相關性 0、0.3、0.6、0.9 假設了 4 個單獨的假設
#3.
對於每個相關假設，將 1,000 個隨機法線的矩陣轉換為 1,000 個相關的隨機法線（當然仍然是 100 維法線）的矩陣。
我不明白為什麼我們需要將不相關的隨機法線矩陣轉換為相關的矩陣？不是每個模擬執行緒都獨立於前一個嗎？
當然，這意味著 100 個組件是成對相關的，但 1000 個抽籤是獨立的。
Correlated (simulated) features or variables are used when the source data has correlated features. If the original features in the source data are not correlated and are orthogonal, then there is no reason to use correlation when simulating. Many assets are correlated, mostly through volatility clustering and sentiment. Secular bull and bear markets can also cause many assets to correlate. If there was no correlation, then you could merely build a portfolio by randomly selecting the assets, and then weighting the amount owned by historical returns – which is never a good idea.
Bank portfolio stress testing involves use of copulas (simulations) to artificially introduce greater levels of correlation between portfolio assets to determine Variance at Risk, Expected Tail Loss, etc. Basically, when times are bad (e.g., subprime mortgage crisis), many assets begin to become strongly correlated. Thus, if there are large market corrections (loss), when many assets start correlating, the chances of greater loss are larger – because you are not only losing net worth via one asset, but losing a lot more via loss among many correlated assets. (a particular type of portfolio which can safeguard against high correlation during increased volatility with correlation is called a collar portfolio – which is a form of hedging). When correlation is also greater, there is less diversity – which is less sustainable.