Simulating Correlated Asset Prices with Geometric Brownian Motion

Summary

The document asks how to simulate two positively correlated time series when one is a stock modeled with geometric Brownian motion (GBM). It describes estimating GBM inputs from historical stock data and reports a Pearson correlation of 0.5 between the two observed series. The central question is how to incorporate dependence when generating sample paths for both series and whether the second series should also be modeled as a GBM.

It does not provide a simulation method or resolve whether GBM is appropriate for the second series. The distinction between correlating price levels and correlating returns is left open, as are the choice of dependence model and estimation of its parameters. The reported historical correlation alone does not specify the joint dynamics or guarantee that simulated paths will preserve the relationship. The text is therefore a useful statement of a modeling problem, but it offers no empirical assessment beyond the cited correlation and no practical results from a simulation.

Key ideas

• The document considers joint simulation of a stock price and another positively correlated series.
• It proposes fitting GBM inputs for the stock from historical data.
• It reports a Pearson correlation of 0.5 between the historical series.
• It asks how to model the second series and incorporate dependence in simulated paths.
• It does not establish that the second series follows GBM or supply a simulation procedure.

Tags

# Simulating two correlated time series using GBM


# Simulating two correlated time series using GBM












My situation is the following: I have two time series TS1 and TS2, whereas TS1 is a stock price. According to literature, TS2 is positively correlated to TS1. Furthermore, since TS1 is a stock price, it can be modelled to follow a Geometric Brownian Motion. I have the histrocial data of both series available ranging back a few years.

My goal is to simulate sample paths of both series in Python, taking into account the correlation. Since I have the historical data of TS1 available, simulating TS1 in Python is no problem. I can get the input parameters for the GBM simulation from the historical data and then create multiple GBM sample paths. My question is, how do I proceed to simulate TS2?

I already used the Pearson correlation coefficient between the historical data of both time series and I got a value of 0.5 which confirms the suggestion in literature. As a result, I guess I should model TS2 to also follow a GBM. But how do I take into account the correlation between both time series in my simulation? How can I find out what the correlation exactly is between both time series, since the Pearson correlation coefficient only tells me that there exists a correlation.

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