# calibrating two (or X) equity diffusion trees

I have two equities S1 and S2.

Each one follows the following tree evolution :

$$S_1 \rightarrow \left \{ \begin{matrix} S_1 (1+u_1) & \text{with probability } p_1 \\ S_1 (1-d_1) & \text{with probability } 1-p_1 \\ \end{matrix} \right .$$

$$S_2 \rightarrow \left \{ \begin{matrix} S_2 (1+u_2) & \text{with probability } p_2 \\ S_2 (1-d_2) & \text{with probability } 1-p_2 \\ \end{matrix} \right .$$

I would like to calibrate $p_1$ and $p_2$ to fit the conventional risk-free rates.

We can show that having $\pi = (r_i - d_i)/(u_i - d_i)$ yields an expected return of $r$.

However, we this methodology is not able to calibrate considering a potential correlation between my equities.

Do you know how trees are calibrated or used in order to verify a second constraint:

$corr(S_i , S_j) = c(i,j)$

Should I conclude that I need a $(X(X+1)/2 )$-nomial tree if I have $X$ correlated equities and a risk-free rate expectation constraint (X constraints of expected return + X(X-1)/2 correlation constraints) ?

These tree are uncorellable. Solving the equations will lead to $c(i,j) = 0$. It kind of makes sense since the probabilities of migration are independant.

I wonder what solutions exists to price products where underlyings are correlated and have such an high variance that Monte-Carlo approach is not considerable ?