Part X · Edges, one by one · Chapter 10C

The variance risk premium

Why options are priced above the moves they insure

Where we are

Chapter 10B took the first foundation premium, the equity risk premium. This chapter takes the second, the variance risk premium. Earlier chapters met it from several sides: as insurance (Chapter 4C), as the gap between implied and realized volatility (Chapter 5E) and as the hedged seller's profit (Chapter 7A). Here it is judged as an edge, with its evidence, the one trade this book means by harvesting it, how that trade fails and the rule for sizing it.

What you need first

Implied and realized volatility, the rule of 16, and the premium's size in volatility points (Chapter 5E), with its units (Chapter 4C).
The delta-hedged profit and loss (P&L) as a bet on realized against implied volatility (Chapter 7A).
Why the Sharpe ratio flatters short volatility, the Jurek-Stafford finding and the stress scenarios S1 to S3 (Chapter 1D).
Path dependency and margin (Chapter 1E), and tail correlation (Chapter 1F).
The VIX and the VIX3M (Chapter 9A), and XSP with its Section 1256 treatment (Chapters 2A and 2E).

In plain terms

Option-implied volatility consistently exceeds the volatility that follows, because option buyers pay a premium for insurance. The buyers are asset managers hedging portfolios, issuers of structured products hedging what they sold, and retail traders buying calls. The sellers earn the premium for accepting crash risk: most months it pays them a little, and in a crash it costs them a lot.

1 · The gap, from 1990 to 2018

From 1990 to 2018 the VIX averaged 19.3%, while the realized volatility (RV) of SPX averaged 15.1%. That gap of 4.2 volatility points compounded through systematic put-writing into the record of Bondarenko's PUT index.

Worked example: What 4.2 volatility points are worth

Step 1: the gap in variance. The premium is defined on variances, the squares of volatilities. The gap is $0.193^2 - 0.151^2 = 0.0372 - 0.0228 = 0.0144$ , or 1.44 percentage points of variance, as Chapter 4C explained.

Step 2: the gap on one option. An at-the-money option's price is close to $0.4\,S\,\sigma\sqrt{T}$ , since $\varphi(0) = 0.399$ in the formulas of Chapter 5C. For one month, $\sqrt{T} = \sqrt{1/12} = 0.289$ , so at 19.3% the option costs $0.4 \times 0.193 \times 0.289 = 2.23\%$ of the index, and at 15.1% it would cost 1.75%.

Step 3: the seller's share. The difference, $0.4 \times 0.042 \times 0.289 = 0.49\%$ of the index a month, is roughly what a delta-hedged seller keeps when realized volatility comes in at 15.1% (Chapter 7A). On one XSP put with the index at a hypothetical 6,000, a notional of $60,000, that is about $290 a month, before costs and before the months in which realized volatility exceeds implied.

Over the same years, the PUT index and SPX compare as the table shows. The last row, the Calmar ratio of Chapter 1D, is computed from the rows above it.

1990 to 2018	PUT index	SPX
Compound annual growth rate (CAGR)	9.54%	9.80%
Volatility	9.95%	14.93%
Sharpe ratio	about 0.65	about 0.49
Maximum drawdown	−33%, in the GFC	−51%
Calmar ratio	9.54/33 = 0.29	9.80/51 = 0.19

The GFC is the global financial crisis of 2008. Selling puts earned almost the index's return with two-thirds of its volatility and two-thirds of its worst loss.

The figure prices the same option at every expiry out to six months. Drag the realized volatility up toward the implied and watch the band, the seller's edge, close.

implied vol 19.3%

realized vol 15.1%

days to expiry 21

premium at implied = 2.23% ($1,337)

worth at realized = 1.74% ($1,046)

seller's edge = 0.485% ($291)

variance gap = 0.0144

√T = √(21/252) = 0.289

What the gap between implied and realized volatility is worth. The price of an at-the-money option on the index, 0.4 S σ √T, against trading days to expiry: once at the implied volatility sold and once at the realized volatility that follows. The band between the two is roughly what a delta-hedged seller keeps, and the readouts give it in dollars on one XSP contract with the index at a hypothetical 6,000, a notional of $60,000. Widen the gap and the band widens with it. Lengthen the expiry and it grows only as the square root of time.

2 · The formal object

The variance risk premium (VRP) at time $t$ is implied variance minus the variance investors expect to realize over the option's life, from $t$ to its expiry $T$ :

\text{VRP}_t = \sigma_{I,t}^2 - \mathbb{E}_t\big[\sigma_{R,t,T}^2\big]

(10C.1)

Here $\sigma_{I,t}$ is the implied volatility at time $t$ , $\sigma_{R,t,T}$ the realized volatility between $t$ and $T$ , and $\mathbb{E}_t$ the expectation given what is known at $t$ . With the long-run averages above, the premium is the 0.0144 of Step 1. For SPX it is positive in about 80% of months. Across its full distribution it has a mean of about +4 volatility points, the unit the Edge Book quotes it in. But its payoff to a seller has negative skew: −2.09 for the PUT index, against −0.81 for SPX (Chapter 1D).

The formula also says why sellers earn it. Chapter 7A showed that a delta-hedged short option, a short volatility position, gains over each short interval its gamma times the gap between implied variance and the squared move that actually occurs. Quiet days pay the seller a little; a day with a large move costs far more than a quiet day pays. The VRP is the average of that stream, and the skew is its shape.

3 · The evidence

Carr and Wu (2009), in the Review of Financial Studies (RFS), give the canonical academic decomposition of the VRP across asset classes.
Bollerslev, Tauchen and Zhou (2009, RFS) show the VRP predicting future returns.
Drechsler and Yaron (2011, RFS) explain the VRP as exposure to a time-varying risk of disaster.
Bondarenko (2014, 2019) analyzes the PUT index, the source of the table above.
Jurek and Stafford (2015, Journal of Finance, JOF) is required reading. After skew is priced, hedge fund returns are indistinguishable from those of mechanical put-writing (Chapter 1D).
Bakshi, Crosby and Gao (2024, Mathematical Finance) build a formal framework for an investor averse to ambiguity. After risk is priced, a naive harvest of the VRP is unattractive.

Ambiguity aversion is a dislike of not knowing the probabilities themselves, beyond the risk that known probabilities describe. You meet it whenever a trial's confidence interval is wide: the same point estimate is less attractive when the uncertainty around it is larger.

4 · Why it exists

The VRP is a risk premium, Source 1 of Chapter 1G: option buyers want insurance against bad outcomes, and option sellers demand a premium to take the other side. The premium is compensation for crash risk. The Jurek-Stafford result sharpens the point: the apparent "alpha" of premium selling is largely compensation for accepting negative skew. Chapter 1G's decomposition adds a smaller institutional part, from asset managers' hedging demand, and a small behavioral part, from retail call buying in bull markets.

5 · "Put-writing" is not one thing

Selling puts comes in variants that behave very differently. The delta of a put, read loosely as its chance of finishing in-the-money (Chapter 5C), sets how far below the index its strike sits:

Variant	What it is	How it behaves
ATM monthly, cash-secured	Sell a one-month at-the-money put, holding its strike in cash	Closest to the academic VRP
OTM 30-delta short put	Sell an out-of-the-money (OTM) put with a delta of −0.30	Better risk-adjusted, smaller premium
Far-OTM 5-delta tail put	Sell a put far below the index	Structurally dangerous: a small premium with catastrophic tails
Weekly or 0DTE selling	Sell options with days or hours to expiry	A different vega and gamma profile, and much higher operational risk
Put spread or iron condor	Sell a put, buy a lower one, and perhaps do the same with calls	Defined risk but inferior expected return: CNDR underperforms massively
Short variance swap	Pay realized variance and receive a fixed level	The cleanest theoretical exposure, out of retail reach

When this book says VRP harvest, it means one thing: selling at-the-money, monthly, cash-secured puts on XSP. The other variants are different strategies, with their own evidence and their own risks.

The iron condor row marks a disagreement between this book's two traditions. The Dealer Mechanics Lab teaches the condor for regimes in which realized volatility is expected to come in below implied (Chapter 12C). It counts condor writing among the short-premium strategies with positive long-run returns (Chapter 5E). The Edge Book rejects running condors systematically for retail accounts (Chapter 10I). Chapter 12C itself reports the CNDR index's underperformance.

6 · In practice

In practice

Preferred vehicle: XSP cash-secured, monthly, at-the-money puts. XSP is the mini-SPX, one tenth of the index, and its options are taxed as Section 1256 contracts.
Passive vehicle: the PUTW ETF, with one caution: it changed its strategy in 2023 to weekly options, a different variant from the table's first row.
The haircut: a Sharpe ratio of 0.65 academically, 0.30 to 0.45 for PUTW live, and 0.20 to 0.40 for a self-managed XSP program in a user account (Chapters 1D and 13A).
Contract size: each XSP put needs its strike in cash. With the index at a hypothetical 6,000, one contract secures $60,000, so a $150,000 sleeve holds two contracts and $30,000 of spare cash.

The two traditions of this book also weigh the premium itself differently. The Dealer Mechanics Lab calls the VRP the most important source of edge available to retail traders (Chapters 4C and 5E). The Edge Book is more guarded about what a retail account can keep. It reads the apparent alpha of premium selling as payment for negative skew, and finds a naive harvest unattractive once crash aversion is priced. It also expects a self-managed account to turn the academic Sharpe ratio of 0.65 into 0.20 to 0.40. It still holds the VRP as a foundation premium (Chapter 10A), but caps the sleeve at 10% to 15% of the portfolio, as the rule below and Chapter 10E's allocation table do.

7 · When it fails

When it fails

Crashes: the PUT index lost 33% in the GFC, against 51% for SPX, and 28% in March 2020.
Negatively skewed events: gaps like 1987's can force liquidation under portfolio margin (PM), when the loss and the margin requirement jump together (Chapter 1E).
Path dependency: even cash-secured puts mark to market (MTM) badly in stress, and they can require holding through periods of 3 months under water.

8 · A decision rule

Decision rule

Allocate 10% to 15% of the portfolio to at-the-money, monthly, cash-secured XSP put-writing, with three kill criteria:

Halt new sales when the VIX divided by the VIX3M exceeds 1.05 for 3 days.
Cut the size by 50% when 21-day realized volatility exceeds 30%.
Stop and reassess when the sleeve's drawdown, peak to trough (PTP), exceeds 20%.

Worked example: The three kill criteria with numbers

Step 1: the term structure. Suppose the VIX is 24 and the VIX3M is 22. The ratio is $24/22 = 1.09$ , above 1.05, the backwardation that marks stress (Chapter 5E). If it stays above 1.05 for 3 trading days, you sell no new puts.

Step 2: realized volatility. By the rule of 16 of Chapter 5E, 30% annualized is about $30/16 = 1.9\%$ a day. If SPX has been moving about 1.9% a day for the past 21 trading days, halve the sleeve.

Step 3: the drawdown. A $150,000 sleeve that peaked at $160,000 and now stands at $120,000 is in a drawdown of $1 - 120/160 = 25\%$ , more than 20%, so stop selling and reassess before doing anything else.

Familiar ground

The kill criteria work like a trial's prespecified stopping rules for harm. A data monitoring committee does not wait for the final analysis when a safety boundary is crossed, and it does not redraw the boundary after seeing the data. Writing the three triggers down before the first sale does the same job, because the moment they fire is the moment your judgment is least reliable.

Common mistake: "the premium is +4 volatility points, so time is on my side"

The belief is that because the VRP averages +4 volatility points, selling puts is sure to make money over time. The VRP is a long-run average premium with heavy negative skew. In any 12-month window you can lose 30% or more of the position, even while the average premium stays positive. Any promise holds only over very long horizons, 5 years or more, and only if you size the position to survive the path.

What to carry forward

The VRP is implied variance minus expected realized variance. From 1990 to 2018 the VIX averaged 19.3% against 15.1% realized, a 4.2-point gap, positive in about 80% of months.
It pays sellers for crash risk, and the cost shows in the tail. The PUT index nearly matched SPX's compound return, 9.54% against 9.80%, with two-thirds of its volatility, but its returns have a skew of −2.09. Jurek and Stafford read most premium-selling "alpha" as this payment.
"VRP harvest" in this book means at-the-money, monthly, cash-secured XSP puts. Other variants, from 5-delta tails to condors and 0DTE, are different strategies.
Allocate 10% to 15% of the portfolio, halt new sales when VIX/VIX3M exceeds 1.05 for 3 days, halve at 30% realized volatility, and stop at a 20% sleeve drawdown.

9 · Check yourself

Question 1

State the VRP. Why is it positive?

Answer

It is implied variance minus the realized variance expected over the option's life, $\sigma_{I,t}^2 - \mathbb{E}_t[\sigma_{R,t,T}^2]$ . It is positive because option buyers, such as asset managers hedging portfolios, structured-product issuers and retail call buyers, pay for insurance. Sellers demand a premium for bearing the crash risk they take on.

Question 2

What does Jurek and Stafford (2015) show about VRP "alpha"?

Answer

After skewed payoffs are priced correctly, broad hedge fund returns are indistinguishable from those of mechanical put-writing on the S&P 500. The apparent alpha of premium selling is largely compensation for accepting negative skew, not skill. For you, it means a simple, mechanical XSP program is the benchmark, and cleverness in timing or strikes is unlikely to beat it.

Question 3

Why do variants of put-writing produce very different profit and loss profiles?

Answer

Strike and tenor change what you sell. A far-OTM 5-delta put collects a small premium and loses catastrophically in a crash, while a 30-delta put collects less than an at-the-money put but has better risk-adjusted returns. Weekly and 0DTE options carry a different vega and gamma profile and more operational risk. Spreads and condors cap the loss but give up expected return, as the CNDR record shows.

Question 4

A sleeve of XSP put-writing sees the VIX at 26 and the VIX3M at 24 for 4 days, while SPX has moved about 2% a day for 21 trading days. What do the kill criteria say?

Answer

The ratio is $26/24 = 1.08$ , above 1.05 for more than 3 days, so halt new sales. A 2% daily move is about $2\% \times 16 = 32\%$ annualized by the rule of 16, above 30%, so cut the sleeve's size by 50%. The third criterion depends on the drawdown: if the sleeve is more than 20% below its peak, stop and reassess.