The Variance Risk Premium: Why Implied Volatility Usually Exceeds Realized Volatility

The variance risk premium is the gap between the variance that option prices imply for the future and the variance that actually occurs. For stock indexes it has been large and negative from the buyer’s point of view: on average, index options have priced in more volatility than the market then delivered, so buyers of volatility have paid a premium and sellers have collected it.

In short: implied volatility is a price, not a forecast. Because options pay off most in crashes, when investors’ wealth is lowest, buyers accept paying more than expected realized volatility, much as homeowners pay more for insurance than their expected losses. Academic work measuring this gap directly (Carr and Wu, Review of Financial Studies, 2009) and indirectly through delta-hedged options (Bakshi and Kapadia, Review of Financial Studies, 2003) finds the premium is strongly significant for the S&P 500, smaller for individual stocks, and not explained by ordinary market beta. The flip side is that sellers earn it as a small, steady stream punctuated by large, sudden losses.

This article assumes you know what implied volatility is and how it differs from historical volatility (see implied volatility explained). It describes a mechanism and the evidence for it. It is not a recommendation to buy or sell volatility.

A common misconception: “options are overpriced, so selling them is free money”

If implied volatility is above realized volatility most of the time, the tempting conclusion is that options are simply mispriced and selling them is an edge. The evidence supports a different reading. The premium is the price of a specific risk: that volatility itself jumps, usually at the same moment the stock market falls. A seller of index volatility is short exactly the scenario in which almost every other asset also loses money. Whether the premium is “too high” depends on how much that scenario costs you, which is a question about your whole portfolio, not about the average gap.

Measuring it properly: variance swaps

Comparing an at-the-money implied volatility with later realized volatility is a rough approach, because implied volatility differs across strikes. Peter Carr and Liuren Wu’s approach is cleaner. They use a variance swap: an over-the-counter contract that, at expiry, pays the difference between realized variance over the period and a fixed variance swap rate agreed at the start. Because a swap costs nothing to enter, the swap rate is the market’s risk-neutral expected value of realized variance.

The key result, which Carr and Wu show holds in general as long as prices move continuously, is that this swap rate can be synthesized from a static portfolio of out-of-the-money puts and calls across all strikes, each weighted by 1/K² (one over the strike squared), plus dynamic trading in futures. Jumps break the replication, but they show the error is third order in the jump size and small in realistic settings. This is the same formula Cboe uses for the VIX, which since 2003 approximates the 30-day variance swap rate on the S&P 500 (the mechanics are in the VIX in depth).

With a synthetic swap rate (call it SW) and realized variance (RV) over the same 30 days, the variance risk premium is simply:

RP = RV - SW (in variance units), or
LRP = ln(RV / SW), the log version, which Carr and Wu note is the continuously compounded excess return to buying variance at the swap rate.

A negative number means buyers of variance lost on average and sellers gained.

What Carr and Wu found

In the working-paper version of “Variance Risk Premia” (Carr and Wu, 2004, published in the Review of Financial Studies in 2009 as “Variance Risk Premiums”), the authors build daily 30-day synthetic variance swap rates from OptionMetrics data for five stock indexes and 35 individual stocks, from January 1996 to February 2003. Their summary statistics, quoted in volatility percentage points for readability:

Underlying	Mean realized volatility	Mean synthetic swap rate	Mean log premium ln(RV/SW)	t-statistic
S&P 500 (SPX)	18.82	24.41	-0.594	-9.48
S&P 100 (OEX)	19.88	24.61	-0.509	-7.83
Dow Jones (DJX)	19.67	24.71	-0.525	-7.03
Nasdaq-100 (NDX)	37.54	40.19	-0.207	-4.42

Four findings matter for practitioners.

The index premium is large. A mean log premium of -0.594 means that buying 30-day S&P 500 variance at the synthetic swap rate returned, on average, a continuously compounded -59% per month. In simple terms, e^(-0.594) = 0.552, so the typical position lost about 45% of its stake each month. Carr and Wu describe the excess return as “over -50 percent per month” for the S&P indexes and the Dow.
Single stocks are different. The mean premium in variance terms was insignificant for all but three of the 35 stocks, though the log premium was significantly negative for 21 of them. Their interpretation: the market prices variance risk that is linked to the overall market, not each stock’s own variance.
Beta does not explain it. Index variance has a strongly negative correlation with index returns, which gives variance a negative CAPM beta. Carr and Wu find this beta explains only a small part of the premium, and Fama-French factors do not explain it either. They conclude that either the market for variance risk is highly inefficient or variance risk is largely an independent factor that the market prices heavily.
Implied variance still forecasts. Regressing log realized variance on the log swap rate gives slopes they cannot statistically distinguish from one for any of the five indexes. Option prices contain real information about future volatility. They are biased upward by a fairly stable premium, not useless.

What Bakshi and Kapadia found

Gurdip Bakshi and Nikunj Kapadia (“Delta-Hedged Gains and the Negative Market Volatility Risk Premium”, Review of Financial Studies, 2003) approached the same question without variance swaps. Their test: buy an option, hedge its delta with the underlying, and rebalance. If volatility risk carries no premium, the hedged position should earn zero on average beyond the risk-free rate. If buyers pay a premium, it should lose.

Using S&P 500 index options from January 1988 to December 1995 (starting after the 1987 crash to avoid mixing pre-crash and post-crash pricing), they report:

Delta-hedged long calls underperformed zero by about 0.05% of the index level on average across strikes and maturities, and about 0.13% for at-the-money calls. For at-the-money options, that was about 8% of the option’s value.
The underperformance was largest at the money, smaller away from the money, and larger when volatility was high.
Measures of crash fear (the skewness of the risk-neutral distribution) explained part of the losses on short-dated options, but the volatility risk premium was the dominant explanation at all maturities. Delta-hedged losses also appeared in a pre-crash sample.

To put the 0.13% in contract terms with today’s index levels: on a hypothetical S&P 500 level of 6,000, 0.13% is 7.80 index points, or $780 per SPX contract over the life of each hedged option (the multiplier is 100). Small per trade, but systematic.

Our measurement: the VIX against what followed, 1990-2026

The simplest public version of this comparison uses the VIX itself. We took Cboe’s published daily VIX closes and S&P 500 closes and, for each day from January 2, 1990 to October 9, 2026, compared that day’s VIX with the realized volatility of the S&P 500 over the following 21 trading days (annualized from daily log returns, with no mean subtracted). This is our calculation from Cboe’s index history files as of November 9, 2026.

Period	Average VIX	Average subsequent realized volatility	Average gap (points)	Days VIX was higher
1990-1999	18.5	13.0	5.4	92%
2000-2009	22.1	18.8	3.3	81%
2010-2019	16.9	13.2	3.7	84%
2020-2026	20.7	16.8	3.8	85%
Full sample	19.4	15.3	4.1	86%

The median gap was 4.7 points. The pattern is consistent with a Cboe-published study by Oleg Bondarenko (“Historical Performance of Put-Writing Strategies”, 2019), which reports an average VIX of 19.3% against average S&P 500 realized volatility of 15.1% from 1990 to 2018.

The averages hide the shape. The largest misses in our sample all cluster in February 2020: on February 20, 2020, the VIX closed at 15.6, and realized volatility over the next 21 trading days was 87.5. A gap of minus 72 points erased many years of average plus 4-point gaps for anyone short variance at size.

When the premium is largest

The premium is not constant. Bakshi and Kapadia found the underperformance of delta-hedged calls was larger at times of higher volatility. Carr and Wu went further: they estimate the risk-neutral variance of volatility itself, as the difference between the variance swap rate and the square of the at-the-money implied volatility (which approximates a volatility swap rate), and find that the premium becomes more negative when this “volatility of volatility” is high, with positive slopes for most indexes and stocks, more than half of them statistically significant. In plain terms, buyers pay the most for variance when the market is most uncertain about where volatility is heading, which is also when sellers face the most risk. Our decade table shows the same instability in a cruder form: the average gap ranged from 3.3 to 5.4 points across decades, and the share of days on which the VIX exceeded subsequent realized volatility ranged from 81% to 92%.

Why the gap exists: crash risk compensation

Three mechanisms, all supported by the papers above, explain why buyers pay up.

Volatility rises when markets fall. Bakshi and Kapadia cite the well-documented negative relation between returns and volatility. A long volatility position pays off in the states investors fear most, so it works as insurance and is priced like insurance.
Variance risk is its own factor. Carr and Wu’s finding that beta and Fama-French factors cannot explain the premium suggests investors care about volatility shocks themselves, not only about the market falling.
Limited supply of insurance. Writers of index options must hold capital against crash losses, and that capital is expensive exactly when it is needed. A premium is what attracts it. (How margin rules set that capital is the subject of portfolio margin explained.)

Worked example: why a variance seller’s losses are convex

Variance swaps are traded in vega notional: the approximate P&L per volatility point near the strike. The variance notional is vega notional divided by twice the strike, and P&L is variance notional times (strike² - realized²). Take a seller of a 30-day variance swap with a strike of 20 and $10,000 vega notional, so variance notional is 10,000 / (2 x 20) = $250 per variance point.

Realized volatility	Strike² - realized²	Seller’s P&L
12	400 - 144 = 256	+$64,000
15	400 - 225 = 175	+$43,750
18	400 - 324 = 76	+$19,000
20	0	$0
25	400 - 625 = -225	-$56,250
40	400 - 1,600 = -1,200	-$300,000
60	400 - 3,600 = -3,200	-$800,000
80	400 - 6,400 = -6,000	-$1,500,000

The best case for the seller (realized near zero) is capped at $100,000. The loss is unbounded and grows with the square of realized volatility. A month like March 2020, with realized volatility in the 80s, costs the seller 15 times the maximum possible gain. A volatility swap with the same $10,000 vega would lose $600,000 at 80; the variance swap loses $1.5 million, which is the convexity that variance buyers pay for and variance sellers are paid to hold.

How the premium shows up in listed products: the Cboe PUT index

Most investors encounter the premium through option selling programs. Cboe publishes benchmark indexes for the two simplest ones:

The Cboe S&P 500 PutWrite Index (PUT) tracks a hypothetical strategy that sells one-month at-the-money SPX puts, typically on the third Friday of each month, and holds Treasury bills sized to cover the maximum possible loss on the puts (full cash collateral). It was launched in 2007 with history back to June 30, 1986. The retail version of the same idea is in cash-secured puts explained.
The Cboe S&P 500 BuyWrite Index (BXM) holds the S&P 500 and sells one-month calls with the strike closest to and at or above the index level shortly before 11:00 a.m. ET on the roll date. Per Cboe’s BuyWrite Indices methodology (as of November 2026), the new call is deemed sold at the volume-weighted average price between 11:30 a.m. and 1:30 p.m. ET, a window extended from 30 minutes on November 19, 2010.

Bondarenko’s 2019 study reports that from June 1986 through 2018, PUT had an annual compound return of 9.54% against 9.80% for the S&P 500, with a standard deviation of 9.95% against 14.93%, giving a Sharpe ratio of 0.65 against 0.49. He attributes the risk-adjusted outperformance to the volatility risk premium, and estimates that PUT’s negative exposure to volatility risk accounts for 0.29% of its average monthly excess return.

Worked example: one month of put writing

With a hypothetical S&P 500 at 6,000, a 30-day at-the-money put priced at 18% implied volatility (Black-Scholes, 4% interest) costs 113.71 points, or $11,371 per contract, about 1.9% of the strike. Fully collateralized, the seller holds $600,000 in T-bills per contract.

S&P 500 at expiry	Put payoff (x 100)	Seller’s P&L
6,300 (+5%)	$0	+$11,371
6,000 (flat)	$0	+$11,371
5,700 (-5%)	$30,000	-$18,629
5,400 (-10%)	$60,000	-$48,629
4,800 (-20%)	$120,000	-$108,629

The seller wins the same amount in every flat or rising month and gives back many months of premium in one bad one.

What the drawdowns looked like

From Cboe’s daily index data (our calculation, using the S&P 500 price index without dividends as the comparison):

Episode	PUT	BXM	S&P 500 price index
Calendar 2008	-26.8%	-28.7%	-38.5%
Peak-to-trough 2007-2009	-37.1% (May 19, 2008 to Mar 9, 2009)	-40.1% (Dec 26, 2007 to Mar 9, 2009)	-56.8% (Oct 9, 2007 to Mar 9, 2009)
Feb 19 to Mar 23, 2020	-28.9%	-30.2%	-33.9%
Calendar 2022	-7.7%	-11.4%	-19.4%
Calendar 2020	+2.1%	-2.8%	+16.3%

Two lessons. Put writing is not a hedge: in 2008 and 2020 it lost most of what the index lost. And it lags in sharp recoveries: in 2020, when stocks rebounded quickly from the March low, PUT ended the year up 2.1% while the price index was up 16.3%, because after the crash the strategy kept selling puts and capped its own upside at the premium.

What can go wrong when you try to harvest it

The losses arrive together. Short volatility loses in the same weeks that equity, credit and liquidity all deteriorate. Margin requirements rise at the worst moment, which can force positions closed at the bottom.
The average gap is smaller than it looks after costs. Index options have bid-ask spreads, and the hedged versions of the trade (as in Bakshi and Kapadia’s delta-hedged portfolios) also pay to rebalance. The academic premium is measured at mid-market or model prices.
Single-stock options behave differently. Carr and Wu found the premium is weak for individual stocks in variance terms. Selling single-stock volatility mostly earns compensation for idiosyncratic jumps (earnings, takeovers, fraud), which is a different risk. See earnings and the implied move.
Out-of-sample risk. The premium is an average over decades that include several crashes. A seller’s realized result depends heavily on whether a crash happens in their particular holding period, and on surviving it.
Buying volatility is not the safe mirror image. The premium means long-volatility positions lose money on average, which is one of the reasons studied in why most options buyers lose.

The short version

The variance risk premium is the persistent excess of option-implied variance over realized variance. Carr and Wu measured it directly with synthetic variance swaps and found it strongly negative for the S&P 500 (a mean log return to variance buyers of -0.59 per month from 1996 to 2003) and much weaker for individual stocks. Bakshi and Kapadia found the same sign in delta-hedged S&P 500 options from 1988 to 1995. From 1990 to 2026 the VIX exceeded the S&P 500’s subsequent 21-day realized volatility on 86% of days, by about 4 points on average. The premium exists because volatility spikes when markets crash, and it is collected as many small gains and a few very large losses, as the PUT index drawdowns of 2008 and 2020 show.