# 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

## 1 Answer

The formula works for total variance, not "strike specific" variance that you need to construct basket vol surface from components, because single historical correlation (or correlation matrix) just does not provide enough information to uniquely reconstruct expected distribution of basket returns (unless for a trivial case where all components are gaussian, which is not what you're asking). The best you can do is to approximate.

Given the information you provided ( components surface + historical correlation/correlation matrix ) I would suggest using gaussian copula. Copula is a way to "combine" distributions of components into distribution of basket. Check out "Pricing Basket Options With Skew" http://wilmott.com/pdfs/100826qu.pdf Section 4.1 for a detailed description of such algorithm.