How to calculate implied volatility smile of basket using correlations?

For a basket, the realized volatility can be calculated using:

$$\sqrt{\sigma_1^2 + \sigma_2^2 + 2 \sigma_1 \sigma_2 \rho}$$

If I have the volatility surface of two underlyings S1,S2 (strike space).

And for each point I calculate the vols using above formula, how accurate is the approximation? I can extend this to multiple assets using simple cholesky transformation.

Correlation used is historical correlation, and not implied correlation.

• Please make sure to take some time to do some formatting next time. I almost closed it as it was barely understandable. Please refrain from using abbreviations and use mathematical notation if possible. – SRKX Mar 3 '15 at 7:15
• What are your $\sigma_1$ and $\sigma_2$? The implied volatilities of two options for the same strike but on different underlyings? And you're trying to estimate the implied volatility of a basket with one of each option? – SRKX Mar 3 '15 at 7:17
• yes they are IV's for different underlyings same strike. Yes I am trying to estimate IV. I am assuming implied correlation stays constant – user139258 Mar 4 '15 at 7:37