# Variance Risk Premium: Implied Versus Expected Realized Variance

Define the variance risk premium under both sign conventions, compare risk-neutral and physical expectations, and understand why short-volatility returns include tail risk.

Canonical: https://wiki.fcontext.com/options/variance-risk-premium/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

The **variance risk premium (VRP)** is the price of bearing uncertainty about future return variance. A common seller-positive convention compares the option-implied, risk-neutral expectation of variance with the real-world, or physical, expectation over the same horizon:

`VRP_sell = Eᴽ[future variance] − Eᴾ[future variance]`

Positive `VRP_sell` means the variance level embedded in option prices exceeds the physical forecast. Some academic and trading sources reverse the subtraction and define `VRP = Eᴾ − Eᴽ`; the same economic state is then negative. State the sign convention before interpreting any chart.

VRP is measured in **variance**, not simply the difference between implied volatility and realized volatility. It is also an expected compensation, not a guaranteed return from selling options.

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## What the two expectations mean

`Eᴽ` is a risk-neutral expectation inferred from a broad set of option prices. “Risk-neutral” is a pricing measure: it incorporates how states are priced, not merely how often investors believe they will occur. Cboe's volatility-index methodology uses out-of-the-money options across strikes with inverse-strike-squared weights related to variance-swap replication. A single at-the-money IV is not a complete model-free variance estimate.

`Eᴾ` is a physical expectation of future realized variance. It is not directly observable before the horizon ends and must be forecast from historical data, a statistical model, surveys, or another declared method. Afterward, realized variance can be calculated from returns, but replacing the forecast with one realized outcome creates an **ex post realization**, not the original ex ante premium.

Option buyers may accept an implied variance above the central physical forecast because options transfer crash, jump, convexity, and liquidity risk. Sellers may earn compensation in ordinary periods while remaining exposed to losses concentrated in rare states. Supply, demand, dealer balance sheets, jumps, skew, and measurement choices all affect the observed spread; it is not a fixed law that every asset and horizon must have a positive seller-side VRP.

Use matched horizons, annualization, return definitions, sampling frequency, and price timestamps. Squaring an annualized volatility gives annualized variance only when conventions are consistent.

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## Matched-horizon example

Suppose a 30-day option strip implies annualized volatility of `24%`, while a documented physical forecast for the same 30 days is `18%`. Under the seller-positive convention:

`implied variance = 0.24² = 0.0576`

`expected realized variance = 0.18² = 0.0324`

`VRP_sell = 0.0576 − 0.0324 = 0.0252` annualized variance units.

The correct comparison is not `24% − 18% = 6%`; volatility and variance have different units. Under the reversed convention, the reported premium would be `−0.0252`.

Now suppose realized volatility over the next 30 days is `30%`, so realized variance is `0.30² = 0.0900`. The ex post variance difference for a position short realized variance relative to a `0.0576` strike is approximately `0.0576 − 0.0900 = −0.0324`, before notional scaling, financing, fees, and contract details. A premium estimated at entry did not prevent a loss when the tail state arrived.

An option strategy is not a pure variance swap. Delta, Gamma, skew, path, strike selection, discrete hedging, early exercise, and transaction costs can make its outcome differ materially from this variance comparison.

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## Risks and research controls

- Define whether the series is `Eᴽ−Eᴾ`, `Eᴾ−Eᴽ`, implied-minus-realized, or a tradable payoff; these are related but not interchangeable.
- Match horizon, annualization, calendar, return formula, sampling frequency, and option timestamp before subtracting estimates.
- Document how physical variance is forecast. Different windows and models can change both magnitude and sign.
- Use an option strip or a documented variance methodology; one contract's IV does not represent the whole risk-neutral distribution.
- Separate an ex ante premium from an ex post outcome. One calm or turbulent month does not identify a stable expected premium.
- Stress jumps, volatility clustering, skew steepening, correlation changes, Bid/Ask widening, and margin increases.
- Treat short-option proxies as nonlinear portfolios. Loss can be large or uncapped even when historical average returns appear positive.
- Account for discrete rebalancing, missing strikes, stale quotes, dividends, rates, exercise, settlement, and contract multiplier.
- Report distributions and drawdowns, not only averages. Rare losses can dominate a long series of small gains.

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## Common misconceptions

- **“VRP is IV minus future realized volatility.”** The formal object is a variance difference; volatility subtraction is a different statistic.
- **“VIX minus recent volatility is the exact premium.”** The horizons and expectations may not match, and recent realization is not a future physical forecast.
- **“Positive VRP makes short volatility safe.”** Compensation exists because the seller bears nonlinear, jump, liquidity, and tail risk.
- **“Risk-neutral means investors expect that exact variance.”** It is a pricing-measure expectation that includes state prices and risk preferences.
- **“The sign is universal.”** Published series often use opposite subtraction orders.
- **“A short straddle captures VRP cleanly.”** Its payoff also contains direction, skew, path, hedge, and execution exposure.
- **“A historical average must persist.”** Regimes, option demand, market structure, and estimation methods change.

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## Related topics

- [Implied Volatility](/options/implied-volatility/)
- [Implied and Realized Volatility](/options/iv-and-rv/)
- [Short Straddle](/options/short-straddle/)
- [Option Stress Testing](/options/option-stress-testing/)

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## Authoritative sources

- [Variance Risk Premiums](https://doi.org/10.1093/rfs/hhn038) — Peter Carr and Liuren Wu
- [Expected Stock Returns and Variance Risk Premia](https://doi.org/10.1093/rfs/hhp008) — Tim Bollerslev, George Tauchen, and Hao Zhou
- [Cboe Volatility Index Mathematics Methodology](https://cdn.cboe.com/resources/indices/Cboe_Volatility_Index_Mathematics_Methodology.pdf) — Cboe Global Indices
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — Options Clearing Corporation