Variance Risk Premium (VRP)

• Variance Risk Premium (VRP) is the difference between risk-neutral and physical expected volatility, capturing the compensation investors demand for bearing volatility risk.
• It is estimated using model-based techniques like Realized GARCH and model-free methods based on variance swaps, which relate observed market data to risk factors.
• VRP influences option pricing and portfolio construction by accounting for investor risk aversion, regime shifts, and macroeconomic dynamics across markets.

The variance risk premium (VRP) is a central concept at the intersection of volatility modeling, asset pricing, and derivative markets. It quantifies the premium investors require to bear future volatility risk, measured as the discrepancy between risk-neutral and physical expectations of future variance or volatility. The VRP is simultaneously a pricing kernel phenomenon, a traded asset characteristic (especially for variance and volatility swap markets), and a crucial variable for state-of-the-art option pricing and risk management models. Its magnitude, drivers, and dynamic properties are tightly linked to investor risk aversion, market regimes, and the structure of stochastic discount factors that map real-world probabilities to risk-adjusted (risk-neutral) measures.

1. Formal Definition

The VRP is defined in terms of expectations under the physical measure ($\mathbb{P}$) and the risk-neutral measure ($\mathbb{Q}$). In the volatility modeling context, for a given future horizon from $t$ to $t+T$, the VRP is: $$\mathrm{VRP}_t = \mathbb{E}_t^{\mathbb{Q}}\big[\sigma_{t+1:t+T}\big] - \mathbb{E}_t^{\mathbb{P}}\big[\sigma_{t+1:t+T}\big]$$ where $\sigma_{t+1:t+T}$ denotes realized or model-implied volatility over $T$ periods. In the special case of one-step-ahead variance,

$$\mathrm{VRP}_t = \mathbb{E}_t^{\mathbb{Q}}\big[ \sqrt{h_{t+1}} \big] - \mathbb{E}_t^{\mathbb{P}}\big[ \sqrt{h_{t+1}} \big]$$

For variance swaps and model-free measurement, the 30-day VRP can be constructed via: $$\mathrm{VRP}_t^\mathrm{model} = 100 \left( \sqrt{\frac{252}{22}\sum_{k=1}^{22}\mathbb{E}_t^\mathbb{Q}[h_{t+k}]} - \sqrt{\frac{252}{22}\sum_{k=1}^{22}\mathbb{E}_t^\mathbb{P}[h_{t+k}]} \right)$$ and the market-implied version by subtracting rolling realized variance from the observed VIX (Hansen et al., 2021, Rauch et al., 2016).

2. Economic Interpretation and Pricing Kernel Link

Economically, the VRP represents the compensation required by investors for exposure to non-hedgeable stochastic variance. The discrepancy between $\mathbb{Q}$ and $\mathbb{P}$ arises from the convexity of the pricing kernel with respect to market returns or realized volatility. In the pricing kernel representation,

$PK(r) = \frac{q(r)}{p(r)}$

where $p(r)$ and $q(r)$ are physical and risk-neutral return densities, a U-shaped or W-shaped kernel reflects elevated aversion to moves in the tails, and mathematically is associated with a positive VRP (Almeida et al., 2024). The premium emerges because option (especially out-of-the-money) prices embed a higher probability of extreme variance than observed empirically.

The stochastic discount factor (SDF) approach, particularly when it is exponentially affine in both return and volatility shocks (as in Realized GARCH and related models), enables explicit characterization of the volatility risk loading: $$M_{t+1} = \exp(-\lambda\,z_{t+1} - \xi\,u_{t+1} - \tfrac{1}{2} [\lambda^2 + \xi^2])$$ Here, $\lambda$ and $\xi$ separately load on equity and volatility shocks, providing tractable decompositions of the VRP (Hansen et al., 2021, Tong et al., 2021).

3. Model-Based and Model-Free Measurement

Two main strands of VRP estimation dominate the literature:

• Model-based approaches: Use a specified joint process for returns and volatility (e.g., Realized GARCH, Heston–Nandi GARCH, Markov-switching Realized GARCH) under both $\mathbb{P}$ and $\mathbb{Q}$, with a misspecified or calibrated SDF to extract the VRP from model-implied expectations (Hansen et al., 2021, Tong et al., 2021, Hansen et al., 2022).
• Model-free (swap-based) approaches: Construct the VRP as the difference between expected realized variance (physically observed) and the fair value of a variance or log-variance swap (risk-neutral, inferred from option prices). The latter requires robust numerical integration and, with Neuberger’s Discretisation-Invariant characteristics, eliminates path-dependency and monitoring errors: $$\mathrm{VRP}_t = \mathbb{E}_t^\mathbb{P}\left[ \sum_{i=1}^n (\Delta \log S_i)^2 \right] - \mathbb{E}_t^\mathbb{Q}\left[ \sum_{i=1}^n (\Delta \log S_i)^2 \right]$$ or, for log-variance payoffs,

$$\phi(\Delta \log S) = 2 \left( e^{\Delta \log S} - 1 - \Delta \log S \right)$$

which admits exact, partition-invariant replication (Rauch et al., 2016).

These approaches can be mapped to observed time series (e.g., S&P 500, Bitcoin), using high-frequency returns, VIX, and option panels for accurate estimation.

4. Dynamic Properties and Regime Dependence

The dynamics of the VRP are inherently time-varying and regime-dependent. Key findings include:

• Persistence and regime-switching: The VRP time series is highly persistent but less so than raw volatility or VIX. Markov-switching models reveal strong persistence within regimes (regime autocorrelation > 0.99), and the VRP level responds sharply to transitions between "low" and "high" volatility states (Tong et al., 2021).
• Decomposition and risk factors: Empirical decompositions reveal that in models with dual shocks (return and volatility), almost all variation in the VRP is attributable to the volatility shock ($\xi$), with leverage effects providing minimal contribution (Hansen et al., 2021).
• Equity vs. crypto markets: In the S&P 500, the VRP is larger during high-volatility regimes, reflecting pro-cyclical risk aversion. In the Bitcoin market, the VRP peaks in low-volatility states, indicating that premium for variance protection is more pronounced when realized volatility is subdued, plausibly due to "complacency risk" (Almeida et al., 2024).

Table 1 illustrates typical regime-specific VRP levels in the Bitcoin market:

Regime	VRP (annualized, %) (density)	VRP (annualized, %) (BVIX)
Overall	7	14
High-volatility	4	12
Low-volatility	10	17

(Almeida et al., 2024)

5. VRP Determinants and Factor Exposures

Multifactor regressions show the VRP responds asymmetrically to return shocks and is related, albeit weakly, to size and growth factors:

• Negative excess returns raise the VRP more than positive returns depress it ($\hat{\beta}_{ER^2}>0$).
• Small-cap and low-growth stocks slightly increase the VRP.
• There is only moderate correlation between VRP and tail risk premia (skewness, kurtosis), and these latter are primarily driven by momentum, not by VRP determinants (Rauch et al., 2016).

Empirical results for the S&P 500 show VRP sensitivity:

• Full sample (daily): $\hat{\beta}_{ER} \approx -0.61$, $\hat{\beta}_{ER^2} \approx +0.14$, $\hat{\beta}_{\mathrm{Size}} \approx +0.06$, $\hat{\beta}_{\mathrm{Growth}} \approx -0.09$.
• Crisis periods: effects become more pronounced.

In option pricing models with time-varying volatility risk aversion (DHNG), the variance risk ratio $\eta_t = h_{t+1}^*/h_{t+1}$ is highly persistent ($\phi \approx 0.99$), and dynamically tracks market sentiment, uncertainty, and macroeconomic forecasts (Hansen et al., 2022).

6. Implications for Pricing, Hedging, and Portfolio Construction

• Option pricing accuracy: Incorporating dual-shock or time-varying risk aversion models (Realized GARCH, DHNG, MS-RG) materially reduces option pricing errors (e.g., >50% reduction in VIX/option RMSE in the S&P 500), improves fit to observed pricing kernels, and enables richer modeling of risk-neutral skew/kurtosis (Hansen et al., 2021, Hansen et al., 2022, Tong et al., 2021).
• Portfolio applications: Discretization-invariant moment swaps enable investors to separate exposure to variance from tail risks (skew/kurtosis), supporting improved risk targeting and hedging under diverse market conditions (Rauch et al., 2016).
• Market characteristics: In most developed equity markets, the VRP is on average positive (4.8% per annum in S&P 500), though it can be negative in certain states or asset classes (Hansen et al., 2021, Tong et al., 2021, Almeida et al., 2024).
• Macro linkages: The VRP, or more precisely, time-varying measures like the variance risk ratio ($\eta_t$), are significantly associated with macro uncertainty, economic sentiment, and survey-based measures, as well as contemporaneous realized volatility (Hansen et al., 2022).

7. Methodological Advances and Research Frontier

Recent methodological developments include:

• The use of realized measures (e.g., realized variance from high-frequency data) within GARCH-type models to separate risk premia on equity and volatility risk (Hansen et al., 2021).
• State-dependent and time-varying pricing kernels that allow for regime switches, providing better fit to empirical option returns (Tong et al., 2021).
• Score-driven and AR-based filtering for extracting dynamic volatility risk aversion from VIX and option surfaces (Hansen et al., 2022).
• Clustering and density-estimation strategies to partition market regimes (e.g., volatile vs. tranquil) in both crypto and equity markets, revealing distinct inverted or pro-cyclical volatility risk premia (Almeida et al., 2024).

Across model classes, the dual-shock structure, regime dependence, and incorporation of higher-moment risks are consistently empirically supported. Implementation best practices include using investable swap series (matching sampling to swap monitoring), ensuring arbitrage-free option surface smoothing, and robust estimation techniques for time-varying model parameters (Rauch et al., 2016, Hansen et al., 2022).

The VRP thus functions as a barometer of both market risk appetite and underlying volatility structure, with broad implications for derivative pricing, asset allocation, and risk transfer markets.