The Volatility Forecast That Needs No Model: Britten-Jones and Neuberger

"Implied volatility" is one of the most used phrases in finance, and for decades it rested on an awkward foundation.

The traditional definition is circular in a troubling way: implied volatility is the volatility number you have to plug into the Black-Scholes formula to make it spit out the price the market is actually charging. It is the market's price, laundered through a model.

The problem is that Black-Scholes is wrong. We know it is wrong. Its central assumption, constant volatility, is contradicted by the very existence of the volatility smile. So when you invert a formula you know to be false, what exactly is the number that comes out? It is a price quoted in strange units. It is not, in any clean sense, the market's forecast of volatility.

And it gets worse, because you get a different number for every strike price. The smile means an option struck at 90 implies one volatility and an option struck at 110 implies another. Which one is "the" market forecast? Nobody could say. People used the at-the-money one, out of habit, and hoped for the best.

Mark Britten-Jones and Anthony Neuberger found a way to get the market's volatility forecast without any pricing model at all.

The problem: you want the forecast, not the model's fingerprints

What you actually want to know is a clean, well-defined quantity: what does the market expect the total amount of squared movement to be, between now and expiry? That is a question about the market's beliefs. It should not depend on which pricing model happens to be fashionable.

The insight is that this information is genuinely in the option prices. It is just that inverting one option through one model is a clumsy way to extract it.

The key idea via analogy: read the whole book, not one page

Suppose you want to know how much a crowd expects it to rain. You could look at the price of one umbrella and try to back out the expectation using some model of umbrella demand, which requires you to trust your model of umbrella demand.

Or you could observe that the crowd is also betting on rainfall directly, across every possible level, and just read the whole book of bets.

Options are bets on where the price ends up. There are options at every strike: deep out-of-the-money puts betting on a crash, at-the-money options betting on ordinary moves, far out-of-the-money calls betting on a melt-up. Taken together, the whole strip of option prices across all strikes tells you the market's entire probability distribution for the future price.

Britten-Jones and Neuberger showed that if you take that whole strip and combine the prices in a particular weighted way, out pops the risk-neutral expected total variance between now and expiry. And it pops out without you having to assume anything at all about how volatility behaves. No constant volatility, no Heston, no jumps, no nothing. Just the requirement that the price moves continuously.

The weighting has a nice interpretation. Options far out-of-the-money get weighted heavily, in a way that is inversely related to the square of their strike price. Which means the tails matter. The market's volatility forecast is dominated by the price of far out-of-the-money options, precisely the ones the old at-the-money-implied-volatility convention was ignoring.

This quantity is exactly the fair price of a variance swap, a contract that pays you the realized variance and charges you a fixed rate. So the paper also tells you how to price and hedge one of the most important volatility products in existence, with a portfolio of ordinary options.

Why it mattered

• The VIX was rebuilt on it. This is the headline. In 2003, the CBOE abandoned its old VIX methodology, which was based on Black-Scholes implied volatilities of a handful of near-the-money options, and replaced it with a model-free calculation of the kind this paper describes, using the full strip of out-of-the-money options. The "fear gauge" that appears on every financial news channel is a direct application.
• It made variance a tradeable asset. Variance swaps, VIX futures, VIX options: an entire volatility trading complex rests on the insight that expected variance can be extracted from and replicated with a portfolio of vanilla options.
• It gave researchers a clean measure. Any study of the variance risk premium, including the work of Carr and Wu and of Bollerslev, Tauchen and Zhou, needs a model-free measure of expected variance. This paper is where it comes from.
• It resolved the "which strike" embarrassment. The answer to "which implied volatility is the market's forecast" turns out to be: none of them individually, all of them together.

The honest limitations

• You need options at every strike, and you do not have them. The theory calls for a continuum of strikes from zero to infinity. The market gives you a few dozen discrete strikes over a limited range. Everything outside that range must be extrapolated, and everything between the strikes must be interpolated. The tails of the calculation, which the weighting says are important, are exactly where the data is worst and the extrapolation most arbitrary. This is a genuine and well-documented source of error in the VIX.
• Jumps break the exactness. The clean result assumes the price moves continuously. If the price can jump, the formula picks up an error term. It is usually small, but it is there, and it is largest in exactly the turbulent conditions people care about.
• It is a risk-neutral expectation, not a real-world forecast. This is the subtlety that trips up commentators constantly. The number you extract is what the market charges for variance, not what it expects variance to be. Those differ by a risk premium, and that premium is substantial: investors will systematically overpay for protection. The VIX is therefore a biased forecast of realized volatility, running persistently above it. That bias is not a flaw, it is a price, and studying it became its own field.
• Illiquid options poison the calculation. Deep out-of-the-money options are thinly traded with wide spreads, and their prices carry heavy weight in the formula.

The one-line takeaway

Britten-Jones and Neuberger showed that the market's forecast of future variance can be read off the entire strip of option prices at once, with no pricing model whatsoever, which finally gave "implied volatility" an honest definition, priced the variance swap, and led directly to the modern VIX.