Independent Binary Equity Trees Cannot Represent Correlated Returns

Summary

The document considers how to calibrate separate two-branch price trees for multiple equities to match both their expected returns and cross-asset correlations. The accepted response points out a structural limitation: when each asset’s up-or-down transition is independent, the resulting tree construction cannot generate nonzero correlation between the assets. Solving the stated constraints under that setup yields zero correlation.

This is a useful warning when building discrete models for derivatives on several underlyings. Matching each asset’s risk-neutral expected return does not, by itself, provide a mechanism for matching dependence. A model intended to price products exposed to correlated assets needs a joint transition structure or another way to introduce dependence. The document does not specify such a construction, derive a general multinomial calibration, or resolve the computational concern about Monte Carlo for high-variance products. Its takeaway is limited to the independence flaw in the proposed separate trees.

Key ideas

• Separate binary trees with independent transitions produce zero cross-asset correlation.
• Matching each asset’s expected return does not ensure that the model captures dependence.
• A joint model needs a mechanism that links the assets’ transitions.
• The document identifies a limitation but does not provide a calibrated alternative or pricing method.

Tags

# calibrating two (or X) equity diffusion trees


# 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) ?

## Answer by user25844 (score 1, accepted)

https://quant.stackexchange.com/a/41073

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 ?

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