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I can calculate the "Implied Beta" using implied volatility for the option stock, and implied volatility for the market (VIX). Is there any way to calculate also the correlation without performing a statistical analysis on historical data.

$\beta = ({\rho \sigma_{aapl}\ \sigma_{sp500}})/{\sigma_{sp500}} $

I know about the "Implied correlation" but this is the average correlation for all the stocks with the S&P500. I would the correlation for that particular stock

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The simple answer is no. You need historical data to backuo the implied correlation.

A smart way to do it is to use Buss and Vilkov (2009) methodology.

Denote the risk-neutral correlation between each pair of stocks: $\rho_{ij,t}^Q$. The presence of the correlation premia led Buss and Vilkov (2009) to estimate the risk-neutral correlation by making: \begin{equation} \rho_{ij,t}^Q=\rho_{ij,t}^P-\alpha_t(1-\rho_{ij,t}^P) \end{equation}

Combining the above equation with the identifying restriction that equates the the observed implied variance of the market index $(\sigma_{i,t}^Q)^2$ with the calculated implied variance of a portfolio of all market index constituents $i =1,...,N$: \begin{equation} (\sigma_{i,t}^Q)^2=\sum_{i=1}^{N}\sum_{j=1}^{N}w_iw_j\sigma^Q_{i,t}\sigma^Q_{j,t}\rho^Q_{ij,t} \end{equation} one can solve for $\alpha$ and consequently get $\rho_{ij,t}^Q$.

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