State-Dependent Correlation Between the S&P 500 and the VIX

It is well known that the correlation between the S&P 500 and the VIX index is negative. In fact, arbitrage and hedging strategies have been designed around this relationship, for example, hedging vega risks with delta.

However, practitioners also recognize that this correlation can break down and even turn positive. Reference [1] addresses this issue by examining the time-varying nature of the correlation. The paper first posits and demonstrates that volatility-of-volatility (VoV) risk is distinct from volatility (Vol) risk, and then classifies the market into four regimes:

1. Low VOL risk and low VOV risk (LVOL–LVOV)
2. High VOL risk and low VOV risk (HVOL–LVOV)
3. Low VOL risk and high VOV risk (LVOL–HVOV)
4. High VOL risk and high VOV risk (HVOL–HVOV).

The authors pointed out,

We develop theoretical hypotheses on the correlations between the S&P 500 and VIX based on the VFE and two types of risk in the S&P 500–VIX pair: (1) VOL risk in the S&P 500 (i.e., the underlying asset) and (2) VOV risk in VIX (i.e., its VOL product). According to our hypothesis, the correlations between the S&P 500 and VIX do not remain constant and change across various VOL and VOV risk states. We developed a regime‐switching model with state‐contingent correlations to test our hypotheses.

Our empirical results are consistent with the following notions. First, shocks that drive high VOL risk are not necessarily identical to those that drive VOV risk and thus VOL and VOV do not respond to shocks in the same way. These results imply a four‐state VOL system: high/low VOL risk and high/low VOV risk pairs. Second, the minimum inverse correlation between the S&P 500 and VIX is observed under a high VOL–low VOV risk state. In contrast, the maximum inverse correlation between the S&P 500 and VIX occurs under a high VOL–high VOV risk state. Third, the proposed state‐contingent correlations prove more effective in portfolio construction than conventional time‐dependent correlations.

In short, the paper develops a regime-switching model to show that the relationship between the S&P 500 and the VIX varies across distinct volatility (VOL) and volatility-of-volatility (VOV) risk states. It identifies a four-state system and demonstrates that correlation strength depends on the joint VOL–VOV regime.

This is an important contribution, as it quantifies the dynamic relationship between the S&P 500 and the VIX, thereby providing useful guidance for refining hedging and arbitrage strategies.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Leon Li, Carl R. Chen, Volatility Risk and Volatility-of-Volatility Risk: State-Dependent Correlations Between VIX and the S&P 500 Stock Index and Hedging Effectiveness, Journal of Futures Markets, 2025; 1–20

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