Abstract
Strengthening of asset return dependence during the 2007–2008 credit crisis highlighted its dynamic and conditional nature. Option prices reflect the market assessment of how dependence between assets varies with price movements and time horizons, yielding the implied correlation surface. Return dependence increases in falling markets and makes correlation a priced risk factor, causing a spread between implied and actual correlation. Order flow pressure from hedging structured products also contributes to the spread. Prior to the crisis, the gap between implied and actual correlation motivated selling dependence between equities—dispersion trading. However, spiking dependence among stock returns during the crisis decimated correlation sellers. Selling atthemoney conditional correlation between NASDAQ100 components regains an attractive riskreturn profile during periods of strong bull market. This may be due to an increasing correlation risk premium caused by greater investor belief heterogeneity. The implied correlation surface enables the construction of strategies with exposures to dependence conditional on various market dynamics. In particular, a long correlation skew trade delivers attractive returns, while hedging the effects of volatility and mitigating exposure to the level of correlation. This suggests segmentation of the options market along the moneyness dimension. As a risk factor, a correlation skew trade is nearly orthogonal to the five Fama–French risk factors, as well as the momentum factor.
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Notes
 1.
According to Bennett (2014), a popular class of structured products that depend on correlation includes securities with payoffs determined by the best or worst performers among a basket of underlying assets (e.g. Altiplano, Everest, Himalayas). Short positions in such structured products confer negative correlation exposure to the vendors who then seek to hedge it in the options market.
 2.
Naïve implementation is defined by the use of index weights for option positions.
 3.
Wharton Research Data Services (2018c) “OptionMaterics”, wrds.wharton.upenn.edu.
 4.
Wharton Research Data Services (2018a) “Compustat”, wrds.wharton.upenn.edu.
 5.
Wharton Research Data Services (2018b) “CRSP”, wrds.wharton.upenn.edu.
 6.
 7.
Without loss of generality, current time is denoted by 0 and the notation is lightened by omitting it whenever no confusion can arise.
 8.
Averaging over a longer period introduces a greater probability of mixing in observations from different market environments.
 9.
While markets recognize the stochastic nature of correlation, few widelyrecognized option pricing models incorporate this crucial feature.
 10.
Since it is crucial to preserve the effect of different variances of the ICS component changes, SD is applied to the covariance matrix, and not to the correlation matrix.
 11.
Since implied volatilities are inputs for calculating implied correlation [see Eq. (1)], IVS construction algorithms may exert significant influence on results.
 12.
Dependence structures between changes of various financial quantities varies with the movements’ magnitude. This explains the effectiveness of nonGaussian copulas with positive tail dependence for modeling asset returns (see Diks et al. 2014).
 13.
Using straddles causes an immediate delta reduction of a position, while trading underlying assets removes the remaining exposures.
 14.
Correlation between assets is positive, and by its nature must be less than one.
 15.
If the goal is to obtain exposure to the implied correlation skew, while hedging against parallel shifts of the implied correlation surface (ICS), the relative size, \(\psi\), of the ATM leg must satisfy:
$$\begin{aligned} vega_{bkt}\left( OTM\right) \,\frac{\partial \sigma _{bkt}\left( OTM\right) }{\partial \rho }  \psi \, vega_{bkt}\left( ATM\right) \,\frac{\partial \sigma _{bkt}\left( ATM\right) }{\partial \rho } = 0, \end{aligned}$$(8)where OTM and ATM refer to parameters of outofthemoney and atthemoney options respectively. Correlation calendar and diagonal trades also hedge out parallel movements in the ICS, but have dispersion trades with different option maturities.
 16.
Generalization to using straddles and general scaling of a trade is straightforward.
 17.
The basis for spreadinduced cost calculation is half the difference between ask and bid prices. Spreadinduced cost estimate is based on spreads for held (long or short) quoted contracts, with extrapolation of costs for nonquoted positions. Since in a passive empirical study markets cannot be queried for quotes on held positions that did not trade, part of the modeled turnover is due to establishing equivalent positions in traded contracts. Such contracts are not treated as entirely new positions for the purpose of spreadinduced cost calculation, as this would exaggerate the actual rebalancing costs.
 18.
Shorting stocks requires margin, while long positions may use leverage. To avoid arbitrary assumptions, capital required for a delta hedge is not included in (13).
 19.
The pitfalls of stochastic volatility model are explored in Diavatopoulos and Sokolinskiy (forthcoming) and Lee and Sokolinskiy (2015).
 20.
A numerical study of the issue supports this conjecture.
 21.
Since vanna is \(\frac{\partial }{\partial \sigma \partial S}\), the corresponding change in factors is \(\varDelta \sigma \varDelta S\), where \(\sigma\) and S capture the magnitudes of parallel changes in volatilities and stock prices of assets comprising the basket.
 22.
In terms of implied volatility.
 23.
Data source: Kenneth French’s website, data for North American factors. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
 24.
The numbers in the table reflect daily returns.
 25.
Wharton Research Data Services (2018d), ”PastorStambaugh Liquidity Factors”, wrds.wharton.upenn.edu.
 26.
Results not included to conserve space.
 27.
Constant maturity swap.
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Acknowledgements
The work on this paper was mostly conducted while the author was a faculty member at Rutgers Business School—Newark and New Brunswick, prior to his employment by the Board of Governors of the Federal Reserve System. Only minor revisions were made during the author’s employment by the Board of Governors of the Federal Reserve System. The analysis and conclusions set forth are those of the author and do not indicate concurrence by other members of the research staff or the Board of Governors.
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The work on this paper was conducted almost entirely over the period during which Oleg Sokolinskiy was a faculty member at Rutgers University, Rutgers Business School  Newark and New Brunswick. The analysis and conclusions set forth are those of the author and do not indicate concurrence by other members of the research staff or the Board of Governors of the Federal Reserve System.
Appendices
Appendix 1: Implied volatility interpretation
Price of the underlying asset, \(S\left( t\right)\) evolves as
where \(\left\{ \sigma \left( t\right) \right\}\) is a random volatility process, and \(W^{\mathbb {Q}}\left( t\right)\) is the Brownian motion under a riskneutral probability measure \({\mathbb{Q}}\).
This section outlines the derivations and results presented in Gatheral (2006). From riskneutral pricing and the Black–Scholes partial differential equation (see equation (3.5) in Gatheral (2006)):
where \(\Gamma_{BS}\) is the BlackScholes gamma, as defined in Gatheral (2006), p27. Following Lee (2005) and Gatheral (2006), define the Radon–Nikodým derivative process:
Then, squared implied volatility is an integral over the expectations of random variance under measures \({\mathbb {V}}\left( t\right)\) (and not the riskneutral measure \({\mathbb {Q}}\); see equation (3.6) in Gatheral (2006)):
Using the Radon–Nikodým derivative process of Eq. (20) and iterated conditioning, Eq. (21) yields Eq. (4).
Appendix 2: Common alternative dispersion trade implementations
Since a parallel shift may have some heuristic appeal, it forms the basis for alternative implementations of a dispersion trade. Let a strategy have positions \(\left\{ p_i\right\} _{i=1}^{N}\). A parallel equal magnitude shift of all individual implied volatilities has the effect of
on a long correlation dispersion trade. Similarly, a parallel equal relative shift of all individual and basket implied volatilities has the effect of
on a long correlation dispersion trade. Other common ways of implementing a dispersion trade seek to hedge either parallel absolute or relative changes in all volatilities (also see Bennett (2014) for descriptions and illustrations of common implementations of correlation trades):

Vegaweighted implementation treats basket and individual volatilities as separate variables. Then, the effect of a parallel equal magnitude shift of all (individual and basket) implied volatilities is
$$\begin{aligned} vega_{bkt}  \sum _{i=1}^{N} p_i \, vega_i, \end{aligned}$$instead of a theoretically more sound Eq. (22). Consequently, vegaweighted dispersion trade implementation sets
$$\begin{aligned} p^{vega}_i = \frac{vega_{bkt}}{N \, vega_i}. \end{aligned}$$(24) 
Thetaweighted implementation also treats basket and individual volatilities as separate variables, but focuses on equal relative shift of all individual and basket implied volatilities. It assumes that such shifts would have the effect of:
$$\begin{aligned} vega_{bkt} \, \sigma _{bkt} \,  \sum _{i=1}^{N} p_i \, vega_i \, \sigma _{i} \,. \end{aligned}$$instead of the more accurate Eq. (23). As a result, thetaweighted dispersion trade implementation sets
$$\begin{aligned} p^{\theta }_i = \frac{vega_{bkt} \, \sigma _{bkt}}{N \, vega_i \, \sigma _i}. \end{aligned}$$(25)
Finally, a naïve approach consists of taking positions in proportion to index weights. While it features in research, it is theoretically unjustified and not popular with practitioners. The naïve approach will not receive further consideration in this paper. Table 8 summarizes practical implementations of correlation trades.
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Sokolinskiy, O. Conditional dependence in postcrisis markets: dispersion and correlation skew trades. Rev Quant Finan Acc 55, 389–426 (2020). https://doi.org/10.1007/s1115601900847y
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Keywords
 Implied correlation
 Correlation risk premium
 Conditional dependence
 Basket options
 Dispersion trading
 Market segmentation
 QQQ
JEL Classification
 G11
 G13