Modeling the Sum of Correlated Bachelier Processes

Summary

The document considers a portfolio formed by adding two assets whose prices follow Bachelier processes, each with its own volatility and Brownian driver. The question is how to represent the portfolio dynamics when the drivers have correlation. The answer clarifies that correlation is already present if the Brownian motions are constructed as correlated; summing the asset increments therefore carries that dependence into the portfolio.

For an explicit representation, one Brownian increment can be written as a correlated component of the first increment plus an independent component scaled by the square root of one minus the squared correlation. Substituting this decomposition into the portfolio increment makes the shared and independent sources of risk visible. The answer also points to Cholesky decomposition as a general approach for multiple correlated drivers. The treatment assumes constant volatilities and a specified correlation structure, and does not discuss drift, rebalancing, or nonlinear portfolio exposures.

Key ideas

• The sum of asset increments retains the correlation already encoded in their Brownian drivers.
• A correlated Brownian motion can be decomposed into a shared component and an independent component.
• Substitution of that decomposition separates common and asset-specific sources of portfolio risk.
• Cholesky decomposition extends the construction to multiple correlated processes.

Tags

# Portfolio of sum of two Bachelier processes


# Portfolio of sum of two Bachelier processes












Suppose you construct a portfolio of two stocks, whose values $A$ and $B$ are modelled as a Bachelier process: $$dA = \sigma_A dW_A(t) \text{ and } dB = \sigma_B d W_B(t).$$ Each of the stock prices is driven by a different Brownian motion with correlation $\rho$. The value of the portfolio is $P = A + B$. I want to model this portfolio; so I started like this: $$dP =dA + dB = \sigma_A dW_A(t) + \sigma_B d W_B(t),$$ however, I feel like you can include the correlation somehow, but I don't know how. Any ideas?

Thanks in advance.

## Answer by mbison (score 5, accepted)

https://quant.stackexchange.com/a/22547

Since $dW_A$ and $dW_B$ are already correlated as per the way you construct it, your portfolio being the sum of the two is already correlated.

If you want it very explicitity written out, then you could rewrite $dW_B = \rho dW_A + \sqrt{1-\rho^2}dW_Z$ where $dW_Z$ is independent of $dW_A$. More generally (higher dimensions) you can use Cholesky.

Now with this decomp your portfolio dynamics are: $dP = \sigma_A dW_A + \sigma_B(\rho dW_A + \sqrt{1-\rho^2}dW_Z)$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.