Conditional dependence in post-crisis markets: dispersion and correlation skew trades

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 at-the-money conditional correlation between NASDAQ-100 components regains an attractive risk-return 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.

Appendices

Appendix 1: Implied volatility interpretation

Price of the underlying asset, $S\left( t\right)$ evolves as

$$\begin{aligned} dS\left( t\right) = \mu \left( t\right) S\left( t\right) dt + \sigma \left( t\right) S\left( t\right) dW^{\mathbb {Q}}\left( t\right) , \end{aligned}$$

(19)

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 risk-neutral probability measure ${\mathbb{Q}}$.

This section outlines the derivations and results presented in Gatheral (2006). From risk-neutral pricing and the Black–Scholes partial differential equation (see equation (3.5) in Gatheral (2006)):

$$\begin{aligned} \sigma _{IV}\left( K,T\right) ^2 = \frac{1}{T}\int _{0}^{T} \frac{ \text{ E }^{\mathbb {Q}} \left[ \sigma \left( t\right) ^2 S\left( t\right) ^2 \varGamma _{BS}\left( S\left( t\right) \right) | {\mathscr {F}}_0\right] }{\text{ E }^{\mathbb {Q}} \left[ S\left( t\right) ^2 \varGamma _{BS}\left( S\left( t\right) \right) | {\mathscr {F}}_0\right] } dt, \end{aligned}$$

where $\Gamma_{BS}$ is the Black-Scholes gamma, as defined in Gatheral (2006), p27. Following  Lee (2005) and Gatheral (2006), define the Radon–Nikodým derivative process:

$$\begin{aligned} \frac{d {\mathbb {V}}_t}{d {\mathbb {P}}} = \frac{S\left( t\right) ^2 \varGamma _{BS}\left( S\left( t\right) \right) }{\text{ E }^{\mathbb {Q}} \left[ S\left( t\right) ^2 \varGamma _{BS}\left( S\left( t\right) \right) | {\mathscr {F}}_0\right] }. \end{aligned}$$

(20)

Then, squared implied volatility is an integral over the expectations of random variance under measures ${\mathbb {V}}\left( t\right)$ (and not the risk-neutral measure ${\mathbb {Q}}$; see equation (3.6) in Gatheral (2006)):

$$\begin{aligned} \sigma _{IV}\left( K,T\right) ^2 = \frac{1}{T}\int _{0}^{T} \text{ E }^{{\mathbb {V}}\left( t\right) }\left[ \sigma \left( t\right) ^2\right] dt. \end{aligned}$$

(21)

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

$$\begin{aligned} vega_{bkt} \, \sum _{i=1}^{N} \frac{\partial \sigma _{bkt}}{\partial \sigma _{i}} - \sum _{i=1}^{N} p_i \, vega_i \end{aligned}$$

(22)

on a long correlation dispersion trade. Similarly, a parallel equal relative shift of all individual and basket implied volatilities has the effect of

$$\begin{aligned} vega_{bkt} \, \sum _{i=1}^{N} \frac{\partial \sigma _{bkt}}{\partial \sigma _{i}} \sigma _i - \sum _{i=1}^{N} p_i \, vega_i \, \sigma _i \end{aligned}$$

(23)

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):

• Vega-weighted 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, vega-weighted dispersion trade implementation sets

  $$\begin{aligned} p^{vega}_i = \frac{vega_{bkt}}{N \, vega_i}. \end{aligned}$$

  (24)

• Theta-weighted 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, theta-weighted 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.

Table 8 Alternative implementations of dispersion trades