Volatility Risk Premium: Formula, Calculation and Evidence

Overlapping measurement windows, not independent trades. The pooled mean's 95% calendar-block bootstrap interval is -1.48 to +1.49 volatility points; the mean is imprecise.

Four definitions of the volatility risk premium

"VRP" is used for four different numbers. They differ in units (volatility or variance) and in timing (known today, or known only after the options expire). Most disagreements about VRP are two people using different cells of this grid.

The expected variance premium in asset-pricing research compares risk-neutral and physical expectations of the same future variance: \(K_{var,t}(T) - E_t^P[RV^2_{t,t+T}]\). The option-strip variance-swap rate \(K_{var}\) differs from squared ATM implied vol because it incorporates prices across strikes [1–3]. Our ATM-based differences are observable proxies; they do not directly estimate both expectations or the return on an ATM straddle.

All differences here use the seller's sign convention: implied minus realized. A measured spread, a forecast, a tested trading rule, and an implemented return are separate results. The last requires a specified contract, execution, hedging, costs and capital.

The formula, step by step

1. Specify the implied measure and horizon. This guide uses constant-maturity 30-calendar-day ATM IV. It measures one part of the surface, whereas a variance-swap rate uses prices across strikes. A one-week ATM IV can differ by several points (Table 2 shows 10.0% against 13.0% on the same day).
2. State the realized-vol window and timing. The study uses the last 20 sessions for a trailing reading and the next 21 for a forward outcome. 21 sessions approximates 30 calendar days; holidays and weekends make the match inexact. A forecast of future RV is a third, separately estimated input.
3. Compute daily log returns from closes (or a better estimator, see Yang-Zhang vs close-to-close).
4. Annualize with 252 trading days. Using 365 raises realized vol by \(\sqrt{365/252} - 1 \approx 20\%\).
5. Subtract in vol points, or square both first for the variance premium.

Worked example: SPY at the Oct 1, 2025 close

On Oct 1, 2025, SPY had been quiet for a month. 30-day ATM implied vol was 13.03% and the trailing 20-session realized vol was 6.31%, so the trailing premium was a wide +6.72 points. On Oct 10, SPY fell −2.7%.

Step by step, with every input in the CSV and spreadsheet:

1. Implied vol. 30-day ATM implied vol at the Oct 1 close (constant maturity, from FlashAlpha's daily SVI fit): \(\sigma_{IV}\) = 13.03%.
2. Trailing realized vol. 20 daily log returns of the 16:00 closing midpoints, Sep 4 to Oct 1: mean +0.1896%, sample standard deviation 0.003977, so RV = 0.003977 × √252 × 100 = 6.31%.
3. Trailing premium. 13.03 − 6.31 = +6.72 vol points; in variance, 169.9 − 39.9 = 130.0 vol points squared.
4. Forward realized vol. The mean daily variance over the next 21 sessions (Oct 2 to Oct 30), each day the sum of squared 5-minute returns plus the squared overnight return, is 0.00006398; annualized, 12.70%.
5. Forward premium. 13.03 − 12.70 = +0.34 vol points; in variance, 169.9 − 161.2 = 8.6 vol points squared.

Calculations use unrounded inputs. Subtracting the displayed two-decimal values can differ by 0.01 point; the public calculation summary preserves full precision.

A wide reading can reflect a quiet past. The trailing difference was +6.72 points; the forward difference was +0.34, using the same implied vol. The large initial spread mainly described calm September trading. It did not preserve that margin over the volatility that arrived in October. This explains why a historical comparison is different from a forecast of the next month's gap.

Reproducing the FlashAlpha API

The historical API replays the VRP dashboard for any past minute. For 15:55 ET on Oct 1, 2025, its realized-vol and premium fields reproduce exactly from the archive:

Table 2. The FlashAlpha historical VRP response for SPY at 15:55 ET on Oct 1, 2025, reproduced from the archive.

Three conventions matter: atm_iv uses the first expiry at least a week out; replay RV stops at the previous close; and the term_vrp 30-DTE bucket subtracts rv_30d, which uses 30 trading returns, from roughly 30-calendar-day IV (13.25 − 8.16 = 5.09). That bucket aligns the IV tenor to 30 DTE but does not make the realized lookback the same duration. For a specific comparison, recompute RV over the intended sessions and preserve the timestamp.

Trailing vs forward: what does the observed spread predict?

The trailing premium subtracts the last month's realized vol from implied vol. The forward premium subtracts the next month's. They share the implied vol, so the question is whether trailing realized vol is a good stand-in for the realized vol to come.

The original spread-only regression did not improve on the historical-mean benchmark: out-of-sample R² ranged from −0.09 to −0.00. IV also had higher marginal correlation with forward RV than trailing RV did on all 11 symbols. That correlation ordering does not establish that historical RV has no incremental information. An unconstrained combination can use the two inputs differently from their simple difference.

The simple spread forecast did not improve the prediction. A negative out-of-sample R² means this model made larger forecast errors than a forecast based on the past average. In this specification, knowing the trailing spread did not help forecast the gap that followed. That is a result about a tested model, rather than evidence that every possible VRP model must fail.

Does trailing RV add information beyond IV alone?

We fit IV-only, RV-only, IV+RV and a constrained IV−RV model on identical forecast origins within each symbol. The target is next-21-session RV; a second regression targets the forward IV-minus-RV difference. IV+RV lowered observed MSE on 3 of 11 symbols, with gains from -6.58% to +3.86% versus IV-only.

The pointwise 95% improvement interval lay entirely above zero for 0 of 11 symbols with 25-session HAC lags, and 0 with 63-session lags. This is a bounded historical comparison, not proof that RV can never help. Joint variance forecasting in the literature [6] addresses a related question in different units.

Subtracting two inputs does not automatically extract a forecast. IV and historical RV can each carry information, but taking their difference fixes how they are combined. The joint comparisons did not establish a consistent improvement from adding historical RV. They support testing the combination, rather than assuming that a larger dashboard spread signals a better future outcome.

Study: 11 symbols, 2017 to 2026

Volatility premium vs variance premium

The volatility difference was positive on most days and slightly positive on average. The variance difference against ATM implied variance was negative on average for 10 of 11 symbols. Squaring weights large moves more: 20% implied against 40% realized gives \(20^2 - 40^2 = -1{,}200\) volatility-points squared; 20% implied against 12% realized gives \(20^2 - 12^2 = 256\). These are measurement differences, not returns on a specified investment.

The tails explain why the averages disagree. Squaring gives large volatility increases much more influence: a few severe windows can outweigh many small positive gaps. The episode check also shows that the negative ATM-variance average is sensitive to 2020. Its sign describes this sample and measure; it is not a fixed property of every period or the return on an option strategy.

A variance swap uses an option-strip-implied strike [3]; an option strategy also depends on strikes, skew, path, hedging, fills and costs. Summing overlapping variance differences is a descriptive tail diagnostic, not an equity curve.

Does a high VRP reading help?

We test one question: is the ATM-based forward variance difference higher when the trailing spread is in the top third of its own prior-year history than when it is in the bottom third?

Top-third readings had higher mean differences on 8 of 11 symbols, with 1 significant before correction and 0 after Holm correction. This does not establish that every z-score rule fails or that a profitable strategy is impossible. It also does not validate entry timing or position sizing. Any such rule needs its own point-in-time execution and cost test.

The rule's direction was more encouraging than the strength of its evidence. High-reading groups had higher mean forward variance differences for most symbols, but the estimated advantage was too uncertain to survive the cross-symbol test correction. This rule therefore did not establish a dependable way to select better future outcomes. A larger trailing reading can come from higher IV, lower historical RV, or both; those situations need not imply the same future volatility.

Choices that change the number

Table 6. The trailing premium (30-day IV minus trailing RV) by realized-vol choice, pooled over 11 symbols: mean, share of days positive, median distance from the 20-session close-to-close version, and correlation with the forward premium.

• Realized-vol window. 5-session realized vol made the trailing premium +1.84 points on average against +0.33 for 20 sessions, with a median distance of 4.57 points on any given day.
• Estimator. Yang-Zhang instead of close-to-close moved it by a median 1.72 points. The pooled correlations were small (Table 6); the common-sample sensitivity below separates estimator and calendar choices.
• Implied-vol tenor. One-week against 30-day ATM vol: 10.03% against 13.03% on Oct 1, 2025.
• Whether today's close is in the window. 6.31% through Oct 1 against 6.39% through Sep 30 in the worked example.
• Forward realized vol from closes or from intraday data. They differed by a median 1.65 points over 21 sessions.
• Annualization. 365 instead of 252 raises realized vol by about 20%.

Different platform readings can both be calculated correctly. The estimator, window and timestamp are part of the metric itself. Reconcile those choices before comparing the displayed numbers. A convention that produces a larger spread has not, by that fact alone, produced a better forecast.

Do the estimator and calendar approximation change the conclusion?

The four measurement specifications use the same 21,635 eligible symbol-days and conservative common training endpoints. Actual 30-calendar-day windows contain 17–22 sessions in this sample. For SPY, replacing 21 sessions with those actual dates changes RV by a median 0.06 volatility points; using daily instead of intraday forward RV changes it by 1.13 points.

The spread-only premium regression has positive out-of-sample R² on 1 symbol under the original common-sample specification and 1 under calendar matching, versus 0 with matched daily estimators and 0 with matched intraday estimators. The small exceptions matter: the broad result is limited predictive performance in these tests, not universal failure under every reasonable measurement choice.

Spread-target out-of-sample R2 on common sensitivity origins. Forecast counts are smaller than eligible-origin counts because each fit needs initial training; all four specifications use the same forecast origins within each symbol. Each column uses its own expanding-mean target benchmark.

Public evidence, Python and downloads

import numpy as np

def realized_vol(closes, window=20):
    """Annualized close-to-close realized vol (%) over the last `window` returns."""
    r = np.diff(np.log(np.asarray(closes, float)))[-window:]
    return 100 * np.std(r, ddof=1) * np.sqrt(252)

def vrp(iv_30d_pct, rv_pct):
    """Volatility premium (vol points) and variance premium (vol points squared)."""
    return iv_30d_pct - rv_pct, iv_30d_pct ** 2 - rv_pct ** 2

# SPY, Oct 1, 2025 (closes from the CSV download, window = "trailing" rows plus the seed row)
# realized_vol(trailing_closes) -> 6.31; vrp(13.03, 6.31) -> (+6.72, 130.0)

The files are free with a FlashAlpha account. Sign in or create one, and the download starts. Everything on this page stays open to read.

The volatility risk premium from the FlashAlpha API

/v1/vrp/{symbol} returns the premium in one call on the Alpha plan. What each field is, so you can compare it with your own numbers:

• vrp.atm_iv: ATM implied vol of the first expiry at least 7 days out, the mean of the four nearest out-of-the-money contracts.
• vrp.rv_5d to rv_30d: close-to-close realized vol, sample standard deviation, × √252. The historical replay uses the archive's 16:00 closing midpoints through the previous session.
• vrp.vrp_5d to vrp_30d: atm_iv minus each window. variance_risk_premium: atm_iv² − rv_20d².
• term_vrp: expiry buckets of 7, 14, 30 and 60 DTE. The 30-DTE bucket uses approximately 30-calendar-day IV and rv_30d from 30 trading returns. State both durations rather than calling them identical.
• vrp.z_score and percentile: today's premium against its own recent history, a description rather than a forecast.

The historical API returns the same response for any past timestamp (?at=), which is how Table 2 was produced and how you can test a VRP rule without lookahead.

import httpx

H = {"X-Api-Key": "YOUR_API_KEY"}
live = httpx.get("https://lab.flashalpha.com/v1/vrp/SPY", headers=H, timeout=30).json()
past = httpx.get("https://historical.flashalpha.com/v1/vrp/SPY",
                 params={"at": "2025-10-01T15:55:00"}, headers=H, timeout=60).json()
for d in (live, past):
    v = d["vrp"]
    t30 = next(t for t in d["term_vrp"] if t["dte"] == 30)
    print(d["as_of"], "vrp_20d", v["vrp_20d"], "| 30-day bucket", t30["iv"], "-", t30["rv"], "=", t30["vrp"])

curl -H "X-Api-Key: YOUR_API_KEY" "https://lab.flashalpha.com/v1/vrp/SPY"
curl -H "X-Api-Key: YOUR_API_KEY" "https://historical.flashalpha.com/v1/vrp/SPY?at=2025-10-01T15:55:00"

Limitations

• 11 large, liquid symbols; the premium on smaller stocks, and around earnings, can behave differently.
• The variance premium is measured against ATM implied variance, not a variance swap rate, and before trading costs.
• 21 sessions approximates the 30 calendar days of the implied vol; the match is not exact.
• The 5-minute benchmark is itself an estimate of realized variance.
• The signal test covers one simple rule. Other rules need their own out-of-sample test.
• The historical input files are the archived extraction used here. Session ordering checks and outcome embargoes do not prove that every vendor value was available without later revisions at its original timestamp.
• Daily forward windows overlap and symbols share market shocks. Symbol-days are observations, not independent experiments; reported intervals are approximate and pointwise.

Related reading

• Yang-Zhang vs close-to-close: which realized volatility to use
• Variance risk premium vs volatility risk premium
• Realized vs implied volatility
• Do volatility forecasts help option sellers?

Frequently asked questions

References

1. Carr, P., & Wu, L. (2009). Variance risk premiums. The Review of Financial Studies, 22(3), 1311-1341. https://doi.org/10.1093/rfs/hhn038
2. Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. The Review of Financial Studies, 22(11), 4463-4492. https://doi.org/10.1093/rfs/hhp008
3. Demeterfi, K., Derman, E., Kamal, M., & Zou, J. (1999). A guide to volatility and variance swaps. The Journal of Derivatives, 6(4), 9-32. Related author-hosted research note on replication and skew.
4. Bakshi, G., & Kapadia, N. (2003). Delta-hedged gains and the negative market volatility risk premium. The Review of Financial Studies, 16(2), 527-566. https://doi.org/10.1093/rfs/hhg002
5. Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1), 246-256. https://doi.org/10.1016/j.jeconom.2010.03.034
6. Bali, T. G., & Zhou, H. (2011). Risk, uncertainty, and expected returns. Federal Reserve FEDS 2011-45, section 2.1. Joint variance forecasting and the expected-premium definition. This is context, not the model replicated by our volatility-unit comparison.
7. Clark, T. E., & West, K. D. (2007). Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics, 138(1), 291-311. https://doi.org/10.1016/j.jeconom.2006.05.023.

Bakshi and Kapadia [4] examine delta-hedged option gains; that trading outcome differs from the ATM spread measured here. Related results should be compared with their contract, sign and payoff definitions intact.

How to cite this page

@misc{flashalpha_vrp_2026,
  author    = {Dobrowolski, Tomasz},
  title     = {Volatility Risk Premium: Formula, Calculation and Evidence},
  year      = {2026},
  version   = {1.2},
  publisher = {FlashAlpha},
  url       = {https://flashalpha.com/articles/volatility-risk-premium-formula-how-to-calculate-vrp},
  note      = {Accessed: YYYY-MM-DD}
}

Changelog. Version 1.2 (2026-11-10): adds the calculator, public evidence record, joint forecasts in volatility units and publication integrity checks; preserves the v1.1 variance forecasts, bootstrap and episode checks; and corrects additional inference and timing details. Version 1.1 added partial-history signal coverage and robustness checks. Version 1.0 introduced the historical SPY reconciliation and 11-symbol study. First published May 10, 2026.

Review and interests. This release has automated calculation, input-alignment, publication and interface checks. It does not claim a named external methodological review or independent reproduction from separately obtained data. FlashAlpha sells the analytics and historical-data service described in this guide.