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Correlated Geometric Brownian Motion Prices and Returns

Article Quant Q&A · Author: Willart
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Summary

The document explains how correlation between two stocks’ Brownian shocks relates to correlation between their simulated price levels under correlated geometric Brownian motion (GBM). It gives formulas for each asset’s mean and variance, their joint expectation and covariance, and the resulting time-dependent price correlation. The method is to correlate the return shocks using the Brownian correlation parameter, rather than trying to impose a separate correlation directly on price levels.

The answer concludes that return correlation serves as a practical proxy for stock-price correlation. The relationship depends on both assets’ volatilities and the time horizon, so the level correlation is not numerically identical to the shock correlation in general. The discussion assumes GBM dynamics and specified drift and volatility parameters; it does not explain how to estimate those parameters reliably, and it cautions that historical data cannot reveal a definitive “true” correlation.

Key ideas

  • Under correlated GBM, price-level correlation follows from the correlation between the assets’ Brownian shocks.
  • The resulting price correlation depends on the time horizon and both assets’ volatilities.
  • Correlated return shocks can be used to generate correlated price paths.
  • The formulas rely on GBM assumptions and do not settle how historical correlation should be estimated.

Tags

EquitiesStatisticsBacktesting
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# How to simulate correlated stock prices (not returns)


# How to simulate correlated stock prices (not returns)












Suppose we have two stocks following GBMs. Drift and volatility are calculated based on historical data. Furthermore the stocks are assumed to be correlated (i.e. they move together, if stock 1 goes up, stock does also). How can I actually simulate their future stock prices so that they stay together? For return I would use Cholesky Decomposition, but since I'm looking directly at price levels and not returns I doubt that would be correct.

## Answer by Kurt G. (score 2)

https://quant.stackexchange.com/a/68149

For two GMBs $$ S_i(t)=S_i(0)e^{(r-\delta_i)t+\sigma_i W_i(t)-\sigma_i^2 t/2}\,,\quad i=1,2 $$ with ${\rm Corr}[W_1(t),W_2(t)]=\rho\,t$ we have \begin{align} \mathbb E\left[S_i(t)\right]&=S_i(0)e^{(r-\delta_i)t}\,,\\[2mm] \mathbb E\left[S_i^2(t)\right]&=S_i^2(0)e^{2(r-\delta_i)t+\sigma_i^2t}\,,\\[2mm] {\rm Var}\left[S_i(t)\right]&=S_i^2(0)e^{2(r-\delta_i)t}\left(e^{\sigma_i^2t}-1\right)\,,\\[2mm] \mathbb E\left[S_1(t)S_2(t)\right]&=S_1(0)S_2(0)e^{(2r-\delta_1-\delta_2)t+\sigma_1\sigma_2\rho\,t}\,,\\[2mm] {\rm Cov}\left[S_1(t),S_2(t)\right]&=S_1(0)S_2(0)e^{(2r-\delta_1-\delta_2)t}\left(e^{\sigma_1\sigma_2\rho\,t}-1\right)\,. \end{align} Therefore, $$ \boxed{{\rm Corr}\left[S_1(t),S_2(t)\right]=\frac{e^{\sigma_1\,\sigma_2\,\rho\,t}-1}{\sqrt{e^{\sigma_1^2t}-1}\sqrt{e^{\sigma_2^2t}-1}}\,.} $$ Obviously, the correlation of $S_1$ and $S_2$ is increasing/decreasing/positive/negative when $\rho$ is increasing/decreasing/positive/negative. In other words: for all practical purposes the correlation of the returns, $\rho$ is as good as the correlation of the stocks. I am of the opinion that there is no time series of historical data that will tell you what the true correlation is.

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