Bollerslev-Zhou (2006) Volatility Puzzles: A Simple Framework for Gauging Return-Volatility Regressions

Summary

Bollerslev and Zhou build a closed-form theoretical framework on the Heston (1993) one-factor affine stochastic volatility model that reconciles three apparently contradictory families of empirical findings about return-volatility regressions. The paper shows that all three "puzzles" — the ambiguous sign of the contemporaneous volatility feedback effect, the stronger leverage asymmetry in implied than in realized volatility, and the systematic downward bias of implied-volatility forecasts — follow analytically from just two structural parameters: the instantaneous leverage $\rho < 0$ and the negative volatility risk premium $\lambda_v < 0$. A Monte Carlo study establishes that 5-minute realized volatility is essentially error-free for monthly regressions, whereas daily-squared-return-based realized volatility introduces large, sample-size-invariant biases. Empirical results on S&P500 monthly data (January 1990–February 2002) confirm all three propositions.

Key Claims

• Proposition 1 (Volatility feedback): The population slope in a realized-vol–return regression is $\beta = \lambda_s + \rho\kappa/\sigma$, which can be negative even when the true risk premium $\lambda_s > 0$, because $\rho < 0$. The implied-vol slope is $\beta^* = \lambda_s \cdot (a_\Delta/a^*_\Delta)$, which is always positive but less than $\lambda_s$; the downward bias arises because $\kappa^* = \kappa + \lambda_v < \kappa$ (slower risk-neutral mean-reversion). S&P500 estimates: $\hat\beta = -0.53$ (realized), $\hat\beta^* = +0.24$ (implied).
• Proposition 2 (Leverage asymmetry): Implied vol always responds more asymmetrically to past returns than realized vol: $\delta^* < \delta < 0$. The negative vol risk premium amplifies the implied leverage coefficient through $a^*_\Delta$. Long asymmetry regressions: realized leverage is insignificant; implied leverage is highly significant ($\hat\delta^* \approx -0.31$ to $-0.55$).
• Proposition 3 (Implied-vol forecasting bias): The population slope in the implied-on-realized regression is exactly $\varphi_1 = a_\Delta/a^*_\Delta < 1$. This is structural, not a sign of inefficiency: slower risk-neutral mean-reversion makes implied vol systematically overshoot future realized vol. Empirical: $\hat\varphi_1 = 0.68$ (std dev form), $0.41$ (variance form).
• Monte Carlo (Table 2): Replacing integrated variance with 5-minute realized vol has negligible effect on any of the three regression slopes at monthly frequency. Daily-squared-return-based realized vol introduces large, persistent biases — notably, the volatility feedback slope is biased toward zero by ${\sim}0.08$ and this bias barely shrinks from $T=150$ to $T=600$.
• VIX (Chicago Board Options Exchange volatility index) measure: The paper uses the new CBOE model-free VIX (Britten-Jones–Neuberger 2000 approach) as the empirical proxy for $E^*_t[\int V_s\,ds]$. The implied vol averages 5.64% annualized (std form) vs. 3.43% for realized vol — the persistent gap is a direct implication of negative $\lambda_v$.

Concepts Introduced or Extended

• Stochastic Volatility
• Realized Volatility
• Risk-Return Tradeoff

Entities Mentioned

• Tim Bollerslev
• Hao Zhou

Quotes

"Whereas the volatility feedback effect as measured by the sign of the correlation between contemporaneous return and realized volatility depends importantly on the underlying structural model parameters, the correlation between return and implied volatility is unambiguously positive for all reasonable parameter configurations." (p. 124)

"The asymmetric response of current volatility to lagged negative and positive returns, typically referred to as the leverage effect, is always stronger for implied than realized volatility." (p. 124)

"Implied volatilities generally provide downward biased forecasts of subsequent realized volatilities." (p. 124)

"Only if the underlying leverage coefficient is zero ($\rho = 0$) will the regression be unbiased for estimating $\lambda_s$." (p. 129)

My Take

The paper is a methodological "everything follows from the structural model" contribution of considerable elegance: three exact closed-form propositions, three empirical puzzles, one mechanism. The key insight — that $\rho$ and $\lambda_v$ jointly drive all three biases — is non-obvious and actionable: before attributing any feedback/leverage/forecasting anomaly to market structure or behavioral effects, one should verify it is not mechanically implied by these two parameters. The limitation is the one-factor no-jump affine structure; the authors acknowledge it but show it suffices for monthly S&P500 data. A subtle tension: the model-free VIX used empirically is derived without assuming Heston, while the theoretical implied volatility (IV) is Heston-specific. The monthly horizon and S&P500 focus also constrain generalizability. The Monte Carlo finding on daily vs. 5-minute realized volatility (RV) is practically important and underappreciated.