What a Volatility Number Leaves Out
This chapter isolates the measurement problem behind the conditional pricing kernel: what does the VIX summarize, and what information about future return states remains outside it? The answer supports a practical judgment. Two markets can carry the same broad volatility reading while assigning different prices to downside protection, because variance, distributional shape, physical probability, and compensation for risk are related but distinct quantities.
The distinction is not a technical footnote. If an investor, policymaker, or analyst reads a volatility index as a complete account of market fear, they may miss a change in the price of a specific state—especially a severe negative return. To see why, it helps to move from the single number back to the option prices from which it is built.
The VIX summarizes variance, not the whole distribution
The VIX is commonly described as a fear gauge, but the more exact description used by the BIS study is more useful: it is a risk-neutral expectation of S&P 500 return volatility. “Risk-neutral” means that the measure is extracted from prices that make option payoffs tradable and internally consistent. It does not mean that the VIX is a survey of investors’ objective forecast. “Expectation of volatility” means that the index summarizes dispersion in returns. It does not mean that it records the price or probability of every possible outcome separately.
A convenient way to make the limitation visible is to imagine two distributions with the same variance. Variance is built from squared deviations from the mean, so a moderate move upward and a moderate move downward contribute symmetrically to that summary. But the same total can be produced by very different shapes. One distribution can be more negatively skewed, with a relatively thin upside and a pronounced left tail. Another can put more of its dispersion into ordinary two-sided movements. Their VIX readings may be alike even though protection against a crash is more valuable in the first market.
The VIX therefore answers a question like, “How much total movement is being priced?” It is less suited to the question, “Which part of the movement is most expensive?” That second question requires information about the shape of the distribution and the relative prices of claims tied to different regions of it.
The Federal Reserve study makes this limitation part of its research design. Before selecting a final conditional specification, it considers 12 candidate conditioning variables. They include the VIX, risk-neutral skewness and kurtosis, a left-tail-risk index, investor sentiment, the Treasury term spread, several economic-activity measures, credit spread, and the lagged market return. The point is not that an analyst should always use a long list. It is that the pricing kernel is being asked to respond to more than one dimension of market information. The VIX is important—it is the most informative individual variable in the reported tests—but importance is not completeness.
A volatility index also differs from realized volatility. The VIX is forward-looking and inferred from option prices; realized volatility is calculated from past returns. The two can move differently around news. For the state-price question, however, the more important distinction is between a symmetric volatility summary and the relative price of downside outcomes.
The loss of information occurs when a rich surface of option prices is compressed into one headline number. Options differ by strike, maturity, and sensitivity to the underlying index. A volatility index is designed to summarize a broad slice of that surface in a standardized way. That standardization makes the index useful for comparing conditions over time, but it also means that many different surfaces can map to similar headline readings. The index is a good coordinate system for total implied movement; it is not a complete description of the surface from which it was calculated.
This is why the location of an option on that surface matters. An out-of-the-money put close to the current index level protects against a less extreme decline than a deep out-of-the-money put. The Federal Reserve paper’s candidate variables include higher moments and a left-tail-risk index for precisely this reason: large losses are not fully characterized by the average size of all movements. The BIS study uses 25-delta options and deeper 10-delta options to examine two different degrees of moneyness. Moving farther into the tail changes the question from general dispersion toward the market price of a more remote loss state.
A simple thought experiment makes the compression problem concrete. Suppose two option markets both imply the same overall variance for the next month. In the first, the prices of calls and puts are relatively balanced around their comparable deltas. In the second, investors bid up deep out-of-the-money puts while call prices change little. A variance index can remain similar because the extra insurance cost is concentrated in a narrow portion of the surface. The second market is not saying exactly the same thing as the first: it is placing a different price on a particular class of negative outcomes. That difference is the raw material from which a conditional state-price interpretation begins.
OTM options reveal the price of the downside
The BIS study uses risk reversals to observe that relative price. For options with the same maturity and delta, it compares the implied volatility of an out-of-the-money call with that of an out-of-the-money put. In compact notation, the risk reversal at delta (\delta) is
[ RR_\delta = IV_{Call,\delta} - IV_{Put,\delta}. ]
In equity markets, the put is often more expensive than the comparable call. Its implied volatility is higher because a claim that pays after a large fall is valuable insurance. The risk reversal is consequently negative on average under the paper’s sign convention; the study works with absolute values when presenting its positive crash-risk gauges. The magnitude tells us how much more expensive the downside protection is relative to the upside option at a comparable distance from the current price.
That measure is not simply “more volatility.” It is information about asymmetry. A market can have a stable VIX while the put wing becomes more expensive relative to the call wing. The total variance summary may not move much because the change is concentrated in a particular region of strikes. Conversely, the VIX can rise because the market anticipates broad two-sided turbulence even if the relative price of downside insurance changes less.
The distinction is economically meaningful because a put and a call transfer risk in opposite directions. The buyer of a deep out-of-the-money put pays a premium today for compensation after a sharp decline. The seller receives that premium but bears the loss in the state where the protection is needed. A risk reversal therefore contains information about who is paying to transfer crash exposure and how costly that transfer has become. It is not a neutral count of how likely the crash is.
The BIS evidence shows why the proxy is useful but should not be overinterpreted. The paper reports that monthly 25-delta and 10-delta risk reversals are highly correlated with a more direct tail-risk measure, with correlation coefficients of 0.8745 and 0.8863 respectively. That is strong evidence that the simple option measure tracks a broader downside-risk concept in the reported sample. It does not turn the risk reversal into a direct probability of a crash. A high premium can reflect a higher perceived likelihood of a loss, a higher price of bearing that loss, or both.
There is also a practical trade-off. To recover a full option-implied distribution or a direct comparison between risk-neutral and statistical tail probabilities, researchers need more granular option and return information. The BIS authors use risk reversals partly because they are available at a frequency useful for event studies. A simpler measure can be more timely and easier to compare, but it gives up some detail about the entire tail. The right conclusion is not that one measure is correct and the other is wrong. They answer different questions at different costs.
The same distinction applies to the relationship between a risk reversal and the VIX. The VIX is a compact measure of total implied variance. The risk reversal is a compact measure of skew and the cost of downside insurance. They can move together in a crisis, but they need not move by the same amount or at the same time. Treating them as interchangeable erases precisely the information needed to understand why equal VIX readings can imply different state prices.
From option prices to a state-price ratio
A second distinction is needed before the pricing-kernel mechanism can be understood: the difference between risk-neutral and physical probability. The risk-neutral density is the probability-like distribution that reproduces option prices after risk compensation is embedded in those prices. The physical density is the distribution used to describe the likelihood of realized returns. It is tempting to call the first the market’s forecast, but that would confuse a pricing measure with a belief measure.
The pricing kernel links them. For a projected one-month market return, the Federal Reserve paper writes the kernel as the risk-neutral density divided by the physical density, with risk-free discounting:
[ M_t(r) = \frac{1}{1+r_{f,t}\tau}\frac{f_t^*(r)}{f_t(r)}. ]
The notation is less important than the logic. If a future return state is expensive in option prices relative to its physical likelihood, the ratio is high. If the state is relatively cheap after accounting for its likelihood, the ratio is lower. The kernel is therefore a map of state-dependent compensation: it tells us how the market values payoffs delivered in different parts of the return distribution.
This also explains how two equal VIX readings can lead to different state prices through more than one channel.
The first channel is the risk-neutral distribution itself. The VIX summarizes its variance, but two risk-neutral distributions can share that variance while differing in skewness, kurtosis, or the weight of the far-left tail. The VIX can remain the same while the risk reversal changes. Since the kernel uses the full density rather than just its second moment, the price assigned to a negative-return state can change.
The second channel is the physical distribution. Even if the option-implied variance and much of the risk-neutral shape were similar, investors could assign different physical probabilities to future returns because the economic outlook had changed. A negative return that is surprising in a favorable environment may be valued differently from the same return when a downturn is already expected. The term spread is relevant here because it supplies forward-looking economic information distinct from the VIX.
The third channel is compensation. A state price is not just the probability of receiving a payoff. It also reflects how valuable that payoff is when it arrives and how much investors require to bear the risk of the opposite outcome. A dollar delivered during a market collapse can be especially valuable to a portfolio that is otherwise losing money. Its price can therefore be high even if the physical probability of the collapse is not high. Conversely, when an adverse state is already expected and portfolios are positioned for it, the marginal compensation attached to that state can differ.
This is also why a traded option should not be confused with a single state price. A put with one strike pays across a range of outcomes below that strike, while a call pays across outcomes above its strike. The option price is a bundle of state-contingent payoffs. Researchers can use prices across strikes to infer a risk-neutral density, but the density is still a pricing object. To learn what the same range of returns means under physical probability, they need an additional relation between prices and beliefs. The pricing kernel supplies that relation, but estimating it requires assumptions or restrictions that deserve attention.
The difference between a probability and a price becomes especially important in a crisis. If a state is both likely and damaging, its insurance may be expensive for two reasons: the event is no longer remote, and investors value protection against it more urgently. If policy communication reduces the price of that insurance, the change could reflect a lower perceived likelihood, a lower compensation for bearing the risk, an improvement in the ability of intermediaries to absorb losses, or several of these at once. The option quote records the market price; it does not, by itself, identify which underlying belief or preference moved.
For the same reason, an analyst who sees a high VIX should not jump directly to a conclusion about the physical probability of a crash. The VIX may rise because the market expects more two-sided uncertainty, because risk-bearing capacity has fallen, because hedging demand has intensified, or because the whole option surface has shifted in a way that a single index cannot describe. The pricing-kernel framework is valuable precisely because it asks how these pieces fit together rather than treating them as synonyms.
The diagram is a compact warning against category errors. Risk-neutral variance is not downside skew. Downside skew is not a physical crash probability. A physical probability is not a state price. The pricing kernel is the relationship that helps connect them, and it can vary with information available at the time.
These distinctions also explain why the Federal Reserve paper’s evidence is more informative than a simple correlation between the VIX and returns. The study estimates the projected pricing kernel and allows its coefficients to vary with conditioning variables. It is asking how the relationship between future return states and prices changes when the information environment changes. That is a more demanding question than whether high VIX months tend to have low stock returns.
A careful reader should consequently reject several shortcuts. The VIX is not a direct probability that the market will fall by a given percentage. A high risk reversal is not proof that the fall will occur. A low risk reversal is not proof that the downside has disappeared. Nor does a risk-neutral density, taken alone, reveal how likely a state is under the physical measure. Each observation becomes useful only when its pricing role is kept distinct.
The next question is methodological: if options provide forward-looking prices but physical probabilities are not directly observable, how can the two distributions be estimated together without simply imposing a model that decides the answer in advance? That is the problem the conditional pricing-kernel approach is designed to address.