Why Insurance Is Always Overpriced: Carr and Wu on Variance Risk Premiums

There is a persistent, uncomfortable, extremely profitable fact about options markets: the market charges more for volatility than volatility actually turns out to cost.

Buy insurance against big market moves, and on average, you lose. Sell it, and on average, you win. This is not a market inefficiency in any simple sense. It is a risk premium, the same in principle as the premium you earn for holding stocks instead of cash. But it is a strange one, because it runs the wrong way round: the person who sells insurance gets paid, and what they are being paid for is bearing the risk of catastrophe.

Peter Carr and Liuren Wu set out to measure this premium properly, cleanly and at scale. Their paper is the definitive empirical account of it.

The problem: how do you even measure the price of variance?

To measure a premium you need two numbers:

1. What the market charges for future variance.
2. What variance actually turned out to be.

The second is not hard. Realized volatility, computed from high-frequency data, gives you a good measurement of what actually happened.

The first is the tricky one, and it had traditionally been done badly. The old approach was to take the Black-Scholes implied volatility of an at-the-money option and call that "the market's price of volatility." But that number is the output of a model everyone agrees is wrong, and it ignores all the information sitting in the out-of-the-money strikes, which is precisely where the interesting stuff lives.

The key idea via analogy: build the insurance contract synthetically

Carr and Wu's contribution starts with a construction. Imagine a contract, a variance swap, that simply pays you whatever the realized variance of the stock turns out to be over the next month, in exchange for a fixed rate agreed today. That fixed rate is the market's price of variance. No model, no inversion, just a price.

The problem is that variance swaps are traded over the counter, opaquely, and only for a handful of underlyings. You cannot get a long, clean history of them across dozens of stocks.

So Carr and Wu build them synthetically. Drawing on the model-free logic that Britten-Jones and Neuberger had established, they show that the fair rate on a variance swap can be replicated by a specific weighted portfolio of ordinary options across all strikes. Options are liquid, transparent, and have long histories. So: assemble the option portfolio, read off its value, and you have manufactured the variance swap rate for any asset with a listed option chain.

Now you have both numbers. The variance risk premium is simply the difference: what you would have been paid on the swap (realized variance) minus what you had to pay for it (the swap rate). Do this every month, for five stock indices and thirty-five individual stocks, over a large options dataset, and look at what you find.

What they found

The premium is large, it is negative for the buyer, and it is remarkably systematic.

For stock indices in particular, the variance swap rate sits persistently above the variance that subsequently gets realized. Someone buying variance protection on an index pays consistently more than the protection ends up being worth. The seller collects. The size of this gap is not a rounding error, it is economically substantial.

The pattern is much stronger for indices than for individual stocks, and that asymmetry is the most interesting economic result in the paper. Why would that be?

The natural explanation is that what investors are truly terrified of is not any single company blowing up. That risk you can diversify away. What they cannot diversify away is the whole market falling at once. Index variance is a bet on exactly that scenario, so protection against it is what everybody wants and nobody wants to sell. Hence the enormous premium. Individual stock variance is a mixture of market-wide risk and idiosyncratic company risk, and the idiosyncratic part carries little or no premium because it is diversifiable.

Carr and Wu also investigated whether this premium can be explained away as compensation for standard market risk, the ordinary CAPM story. It cannot. The premium on variance is not simply the equity risk premium in disguise. Variance appears to be a separate, independently priced risk factor, which is a significant claim about the structure of asset markets.

Why it mattered

• It is the reference measurement. If you want to know what the variance risk premium is and how it behaves, this is the paper. It is the empirical anchor for the entire field.
• It legitimised volatility selling as a strategy. Selling variance swaps, selling straddles, selling index puts, running short-volatility ETFs: all of these harvest the premium documented here. The paper explains what the return actually is and why it exists.
• And implicitly, it explains why volatility selling blows up. A premium paid for bearing catastrophe risk is a premium you collect steadily and then give back violently. The strategy has a long history of doing exactly that, most spectacularly in February 2018.
• It established variance as its own risk factor. The finding that variance risk is priced separately from market risk shaped subsequent asset pricing research.
• The replication result is practical. Manufacturing a variance swap from vanilla options is not just an academic device. It is how these products are actually hedged.

The honest limitations

• Replication is imperfect in practice. The theory wants options at every strike from zero to infinity. Reality gives you a truncated, discrete grid, so the synthetic variance swap rate carries approximation error, and the error is worst in the tails, which carry the most weight.
• Jumps introduce a wedge. The clean replication argument assumes continuous price paths. Real prices jump, which puts an error term into the construction, largest during exactly the crisis periods of most interest.
• A risk premium is not free money. The uncomfortable question that the paper cannot fully answer is why the premium is so large. Is it fair compensation for a genuinely terrifying risk, or is it partly an anomaly created by structural demand for portfolio insurance? If it is a risk premium, then earning it means occasionally being obliterated, and the historical record is not reassuring.
• The sample is a bull market. The core sample period is heavily weighted toward relatively benign conditions. Short-volatility strategies look wonderful in such samples, which is precisely how they lure people in.
• It is a US equity study. The findings, and especially the index-versus-single-name contrast, are documented in one market.

The one-line takeaway

Carr and Wu manufactured variance swaps out of ordinary options to measure what the market charges for volatility, compared it to what volatility actually delivered, and found a large, persistent gap that pays the seller of protection, concentrated overwhelmingly in index options, because the one risk nobody can diversify away is the whole market falling at once.