Estimating Stock–Bond Implied Correlation from Option Volatilities
Summary
The document explains how to infer implied correlation between a stock and a zero-coupon bond from option-implied volatilities. The required inputs are volatility quotes for options on each asset and on a portfolio containing both; the portfolio option may need to be arranged over the counter. The suggested calculation derives correlation from the three volatilities, using the relationship between basket variance and the assets’ individual variances.
The discussion recommends matching at-the-money-forward options with the same expiry. It notes that currency correlations may be easier to estimate because options on currency pairs can provide the combined-asset volatility. The method depends on suitable, comparable market quotes and the assumptions behind the variance relationship. The source cautions that OTC prices can be difficult to interpret, so an estimate derived from them should be treated carefully. It offers a method rather than empirical evidence about the accuracy or usefulness of any particular estimate.
Key ideas
- Implied correlation can be inferred from option volatilities on two assets and a portfolio containing both.
- The basket option may require an over-the-counter quote when no exchange-traded product is available.
- Use at-the-money-forward quotes with matching expiries to improve comparability.
- OTC prices may be difficult to interpret and warrant caution.
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Full text
# Implied Correlation using market quotes # Implied Correlation using market quotes Is there a way to retrieve the implied correlation between stock price and zero coupon bonds? ## Answer by Richi Wa (score 2) https://quant.stackexchange.com/a/11380 as in the question about average/implied correlation to do this in a straight way (in a perfect world where all this were given) you would need: immplied vol for the stock, implied vol for the bond and implied vol for an option on a portfolio/basket that contains both assets. If you get a quote for the latter then this sounds possible. I have never seen an exchange traded product with a basket of stocks and bonds - so you would need something OTC. Using such OTC prices I would be careful do derive further conclusions. ## Answer by Matt Wolf (score 2) https://quant.stackexchange.com/a/11395 As Richard pointed out you will need the implied volatility (iV) for the option on the stock, the iV of an option with the zero bond as underlying and the iV of an option on the stock and zero bond combined (most likely an OTC derivative). You can then easily derive the implied correlation : impliedCorrel = (pow(iV[stock],2) + pow(iV[zBond],2) - pow(iV[otcDeriv],2)) / (2*iV[stock]*iV[zBond]) (sorry I am not good at laTeX, and credit to Richard for having first pointed out the implied volatility of the basket option) Implied correlations can be more easily derived for currencies, where you would just replace the iV of the otc derivative by the iV of the cross currency option contract (most use otc contracts, others futures options on the currencies). In either way, make sure you pick the implied vol of the at-the-money-forward and matching expiries for a more accurate result.
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