Portfolio of sum of two Bachelier processes

Question

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 1

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: $dP = \sigma_A dW_A + \sigma_B(\rho dW_A + \sqrt{1-\rho^2}dW_Z)$