Correlated Stochastic Processes

Question

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?

Answer 1

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.