1
$\begingroup$

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) ?

$\endgroup$
0
$\begingroup$

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 ?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.