Excel
02 Dec 2008
May
22
Monte Carlo: Correlated random variables
by David Harper, CFA, FRM, CIPM
To simulate a single asset (stock) that moves with geometric brownian motion (GBM), we specify a model that has two components: drift and shock. The drift is expected return, it is linear. If only for drift, the stock would move in a straight line. But the drift is "interrupted" by a random shock. The shock is a function of volatility and randomness. In the discrete-time formula below, the shock component is the product of volatility (sigma), the random shock (epsilon) and the square root of the time interval:
The random error (epsilon) is merely randomizing the volatility. It must be a standard normal random variable. By normal, we mean it is normally distributed. By standard, we mean its mean is zero and the standard deviation is one. (since it is normally distributed, we are lucky enough to only require these two parameters to describe it). So, by multiplying volatility by a standard normal random variable, we are randomizing the asset's volatility.
What if we have two assets in a portfolio? Or alternatively, this could model a single risk factor in a portfolio and we may want to model another. We could specify the same exact model again. But this assumes they are independent; by independent, we mean they have no correlation. This is often unrealistic.
How do we model two random variables (e.g., assets, risk factors) by GMB when they are correlated? As we see from the Jorion reading, we merely transform the random (epsilon) variable in the second series. If eta[1] and eta [2] are uncorrelated random variables, we transform them into correlated (epsilon[1] and epsilon[2]) variables:
To see this is action, look at the spreadsheet below. You can open your own read/write version here.
Comments
Please go back and take a look at the output of your spreadsheet. Your formula above should replace rho with square root of rho. The actual correlation your spreadsheet is does not match the input correlation, but is instead the square root of the input correlation.
Dear Lee, dear David,
where could I find more about simulation of two correlated assets? I’m writing a diploma thesis about hedging option on asset A with a correlated asset B. What Jorion reading do you mean? What section in it? Please reply soon. Thanks in advance!
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