# Correlated Stochastic Processes

Let say, I have 2 stochastic processes: \begin{align} dS_1 &= \left( r - q_1 \right)S_1 dt + \sigma_1 S_1 dW_1 \\ dS_2 &= \left( r - q_2 \right)S_2 dt + \sigma_2 S_2 dW_2 \end{align} The correlation between these 2 processes is $$\rho$$. Now I define 2 new processes as: \begin{align} x_1 = \sigma_1 \log S_2 + \sigma_2 \log S_1 \\ x_2 = \sigma_1 \log S_2 - \sigma_2 \log S_1 \end{align}

As per Hull's book, these processes $$x_1, x_2$$ are uncorrelated with standard deviation $$\sigma_1 \sigma_2 \sqrt{2 \left( 1+\rho \right)}$$ and $$\sigma_1 \sigma_2 \sqrt{2 \left( 1-\rho \right)}$$ respectively.

How can I show this result?

• What have you tried? Also, the correlation $\rho$ is between the Brownian Motions $W_1$ and $W_2$ correct? Commented Nov 3, 2020 at 9:52
• I am able to derive the $dt$ term, but not the volatility part. Apologies if this is very trivial question. Any pointer is appreciated. Commented Nov 3, 2020 at 9:54
• If $S_i$ is log-normal, then $\log S_i$ is normal. Have you tried computing the covariance between $x_1$ and $x_2$? Commented Nov 3, 2020 at 9:56

I have the impression the expressions in your question miss the time term $$t$$, though this does not change much. Define for $$i,j\in\{1,2\}$$: \begin{align} s_i(t)&=\log S_i(t) \\ y_{i,j}(t)&=\sigma_is_j(t) \end{align} Then: \begin{align} V(y_{i,j}(t))&=\sigma_i^2V(s_j(t)) \\ &=\sigma_i^2V(\sigma_jW_j(t)) \\ &=\sigma_i^2\sigma_j^2t \\ &=V(y_{j,i}(t)) \end{align} Moreover: \begin{align} C(y_{1,2}(t),y_{2,1}(t))&=\sigma_1\sigma_2C(s_2(t),s_1(t)) \\ &=\sigma_1^2\sigma_2^2C(W_2(t),W_1(t)) \\ &=\sigma_1^2\sigma_2^2\rho t \end{align} Hence: \begin{align} &V(x_1(t))=\sigma_1^2\sigma^2_2t+\sigma_2^2\sigma_1^2t+2\sigma^2_1\sigma^2_2\rho t=\sigma_1^2\sigma_2^22(1+\rho)t \\ &V(x_2(t))=\sigma_2^2\sigma^2_1t+\sigma_1^2\sigma_2^2t-2\sigma^2_1\sigma^2_2\rho t=\sigma_1^2\sigma_2^22(1-\rho)t \end{align} Finally, by bi-linearity and symmetry of covariance: \begin{align} C(x_1(t),x_2(t))&=C(y_{1,2}(t)+y_{2,1}(t),y_{1,2}(t)-y_{2,1}(t)) \\ &=V(y_{1,2}(t))-C(y_{1,2}(t),y_{2,1}(t))+C(y_{2,1}(t),y_{1,2}(t))-V(y_{2,1}(t)) \\ &=V(y_{1,2}(t))-V(y_{2,1}(t)) \\ &=0 \end{align} $$x_1$$ and $$x_2$$ are uncorrelated $$-$$ note that for normally-distributed random variables, null correlation also implies independence.
• Many thanks. Is there any way on how to generalise this for n initial correlated stochastic processed for $S_t$? Commented Nov 3, 2020 at 21:22
• Generally, you can express a specific covariance structure in your asset process as $CdW$ with $C$ the lower cholesky decomposition of the covariant empire matrix. I think that you may be able to use its inverse $C^{-1}$ to weight the assets in order to de-correlate the assets. Commented Nov 3, 2020 at 22:04