In this paper The Interplay between Stochastic Volatility and Correlations in Equity Autocallables by Alvise De Col, Patrick Kuppinger (2017) https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3228065, it mentioned

I am confused by the last equation, how did the paper get $$dW_{s1} dW_{v2} = \rho_s \rho_{sv} dt$$, given that

$$dW_{s1} dW_{v1}=dW_{s2} dW_{v2}=\rho_{sv}dt$$

and $$dW_{s1} dW_{s2} = \rho_s dt$$,

the $$dW_{s1} dW_{v2}$$ should be either

$$[\rho_s \rho_{sv} - \sqrt{1-\rho_s^2}\sqrt{1-\rho_{sv}^2} ] dt$$ or

$$[\rho_s \rho_{sv} + \sqrt{1-\rho_s^2}\sqrt{1-\rho_{sv}^2} ] dt$$

Any help are appreciated.

Btw, thanks @noob2 for editing, it's much easier to read now.

The usual ansatz for these kind of setups is to find those components of a Cholesky decomposition of the correlation matrix of your stochastic drivers $$dW_{S_1}, dW_{S_2}, dW_{V_1}, dW_{V_2}$$ such that all conditions are fulfilled.

Let us assume a 4x4 correlation matrix $$R$$ that we decompose using Cholesky to

$$L(R) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ a & d & 0 & 0 \\ b & e & g & 0 \\ c & f & h & i \end{pmatrix}$$
such that $$LL^T=R$$, i.e.

$$LL^T = \begin{pmatrix} 1 & a & b & c \\ . & a^2+b^2 & ab+de & ac+df \\ . &. & b^2+e^2+g^2 & bc+ef+gh \\ .& . & . & c^2+f^2+h^2+i^2 \end{pmatrix}$$

Let us now identify the rows / columns with $$S_1$$, $$S_2$$, $$V_1$$, $$V_2$$, and put in all assumptions from your text plus the usual assumptions regarding the diagonals of the correlation matrix:

• $$\mathrm{E}(dW_{S_1}dW{S_2})=a=\rho_Sdt$$
• $$\mathrm{E}(dW_{S_1}dW{V_1})=b=\rho_Vdt$$
• $$\mathrm{E}(dW_{S_1}dW{V_2})=c=\rho_S\rho_Vdt$$
• $$\mathrm{E}(dW_{S_2}dW_{V_1})=ab+de=\rho_S\rho_Vdt$$
• $$\mathrm{E}(dW_{S_1}dW_{S_1})=1dt$$
• $$\mathrm{E}(dW_{S_2}dW_{S_2})=a^2+b^2=1dt$$
• $$\mathrm{E}(dW_{V_1}dW_{V_1})=b^2+e^2+g^2=1dt$$
• $$\mathrm{E}(dW_{V_2}dW_{V_2})=c^2+f^2+h^2+i^2=1dt$$

You may then proceed to solve for all variables. Close inspection shows that there's one additional degree of freedom:

• $$\mathrm{E}(dW_{V_1}dW_{V_2})=bc+ef+gh=Adt$$

With these ingredients, you can quite simply and iteratively solve for $$a,b,c,d,e,f,g,h,i$$ and obtain a correlation matrix fulfilling all conditions, i.e.

$$R=\mathrm{E} \begin{pmatrix} dW_{S_1}dW_{S_1} & dW_{S_1}dW_{S_2} & dW_{S_1}dW_{V_1} & dW_{S_1}dW_{V_2}\\ dW_{S_1}dW_{S_2} & dW_{S_2}dW_{S_2} & dW_{S_2}dW_{V_1} & dW_{S_2}dW_{V_2}\\ dW_{S_1}dW_{V_1} & dW_{S_2}dW_{V_1} & dW_{V_1}dW_{V_1} & dW_{V_1}dW_{V_2}\\ dW_{S_1}dW_{V_2} & dW_{S_2}dW_{V_2} & dW_{V_1}dW_{V_2} & dW_{V_2}dW_{V_2} \end{pmatrix}= \begin{pmatrix} 1 & \rho_S & \rho_V & \rho_S\rho_V \\ \rho_S & 1 & \rho_S\rho_V & \rho_V \\ \rho_V & \rho_S\rho_V & 1 & A \\ \rho_S\rho_V & \rho_V & A & 1 \end{pmatrix}dt$$

You may think of (the vector of) your correlated stochastic drivers as a linear transformation of uncorrelated stochastic drivers $$d\tilde{W}_i$$, transformed by the lower Cholesky:

$$dW=Ld\tilde{W}$$ and thus

\begin{align} \mathrm{E}\left(dW\left(dW\right)^T\right)&=L\mathrm{E}\left(d\tilde{W}\left(d\tilde{W}\right)^T\right)L^T\\ &=L\mathrm{I}L^T\\ &=LL^T\\ &=Rdt \end{align}

where $$\mathrm{I}$$ is the identity matrix.