# Correlation between Two Factor Gaussian Shortrate Model and Black Scholes Model

I want to implement a two factor Gaussian Shortrate Model \begin{align} r(t) & = x(t) + y(t) + \phi(t), \\ dx(t) & = -ax(t)dt + \sigma dB_1 (t), \\ dy(t) & = -by(t)dt + \eta dB_2(t), \end{align} with correlation $$dB_2(t)dB_2(t) = \rho_{12}dt$$ and do have the price process of an asset $$S(t)$$ given bei the Black Scholes equation \begin{align} dS(t) = S(t) \big[(r(t)-y)dt + \mu (t) dB_S (t) \big], \qquad y>0, \,\ S(0)=0, \end{align} so that you have \begin{align} S(t) = e^{\int_0^t (r(s)-y)ds - 0.5 \cdot \int_0^t \mu (s)^2 ds + \int_0^t \mu(s) dB_S(s)} . \end{align}

My Question is: How do i have to choose the correlations \begin{align} dB_1(t)dB_S(t) &= \rho_{1S} dt, \\ dB_2(t) dB_S(t) &= \rho_{2S} dt, \end{align} so that i can get a predetermined correlation between $$dB_S(t)$$ and the sum $$d(B_1(t) +B_2(t))$$, i.e. i want to habe \begin{align} d(B_1(t) +B_2(t))dS(t) = \rho dt \end{align}.
• For simplicity, you can assume that $\rho_{2S}=0$. – Gordon May 28 '19 at 17:13