How the Conditional Kernel Reweights the Future
This chapter examines the mechanism that turns the VIX, the term spread, and other observable conditions into a changing map of state prices. The key question is how to estimate a pricing kernel when option prices look forward but the physical probability of future returns cannot be observed directly. Understanding that bridge lets the reader see what “conditional” adds: the price of a given future return can change when the information environment changes, even if the return itself has not changed.
The problem is harder than estimating a volatility index. Options provide valuable information about how claims are priced, but they do not hand the researcher a complete, directly observed probability distribution of future outcomes. A useful method must connect the forward-looking option market to a physical distribution, allow the connection to vary with conditions, and impose enough restrictions that the resulting distribution remains coherent.
A forward-looking density needs a bridge
The Federal Reserve paper begins with the projected pricing kernel for a one-month S&P 500 return. The risk-neutral density is inferred from option prices. In the standard option-pricing relationship, the curvature of call prices across strikes contains information about the density that prices those claims. This is a forward-looking object: it reflects what is embedded in prices at the end of the current month for the next month’s return.
The physical density is different. It describes the probabilities assigned to realized returns rather than the prices of claims on those returns. If a researcher estimates it from a long history of past returns, the information timing does not line up. The historical sample looks backward, while the risk-neutral density looks forward. That mismatch can make the resulting pricing kernel partly a comparison between different information sets rather than a clean measure of how the market prices the same future states.
A second common route is to specify a return model—such as a volatility model or a jump model—and use it to generate a conditional physical distribution. This solves the timing problem more directly, but it introduces a different dependence: the shape of the estimated kernel may reflect the chosen return dynamics, distributional assumptions, and estimation specification. The Federal Reserve paper illustrates the stakes by comparing several existing approaches. In specific normal and crisis months, including September 2008 and March 2020, the resulting conditional-kernel estimates differ widely across historical, GARCH, stochastic-volatility, and stochastic-volatility-with-jumps specifications.
The modified conditional density integration method takes another route. It begins with the option-implied risk-neutral density and represents a scaled pricing kernel as a flexible function of the future market return. In the paper’s main specification, that function is a second-order polynomial. Its coefficients are not fixed once and for all. They vary with conditioning variables observed at the current date. In schematic form, the kernel can be read as a return function whose coefficients respond to the current information set:
[ m_t(r) = b_{0,t} + b_{1,t}r + b_{2,t}r^2, ]
with each coefficient changing as the conditioning variables change. The point is not that a quadratic is the only possible shape. It is that a relatively parsimonious function can still allow the relationship between a future return and its price to move through time.
The horizon and data discipline are part of the meaning of the estimate. The paper works with monthly observations from January 1996 through December 2020 and focuses on a 30-day return horizon. It filters the option data, interpolates and extrapolates implied volatilities where necessary, and constructs a fine strike grid before recovering the risk-neutral density. These steps are not incidental housekeeping. A density estimated from sparse or uneven strikes can create artificial curvature, and a kernel estimated for one month should not be described as if it were a long-run price of risk.
The conditional coefficients also should not be understood as a switch between two permanent regimes. The main specification lets observable information enter the coefficients continuously. A small change in a conditioning variable can therefore change the estimated level, slope, or curvature of the kernel rather than simply moving an observation from a “low-risk” box to a “high-risk” box. This is what makes the method useful for the title question: it can represent two information environments that share a broad volatility label while assigning different prices across the return axis.
Once a candidate kernel is specified, the conditional physical density is recovered from the risk-neutral density divided by the scaled kernel. This is the bridge between prices and probabilities. The method uses a statistical property of a cumulative distribution function: when a correctly specified physical cumulative distribution is evaluated at its own realized draw, it has a uniform distribution. The estimation uses moment conditions from that property to find parameters that make the inferred distribution behave like a valid physical distribution over time.
That construction has a crucial boundary. The inferred physical density is not observed independently of the kernel. It is produced jointly by the model’s restrictions and the option data. The result is informative precisely because the method uses forward-looking information consistently, but it remains an estimate whose behavior depends on its approximation, instruments, and constraints.
The paper adds two checks that make the bridge more economically meaningful. First, the implied conditional physical density should integrate to one. A density that does not integrate to one cannot be a coherent probability distribution, even if it happens to match some of the moment conditions. The original density-integration approach could leave this property imperfectly controlled. The modified method adds a P-density restriction and penalizes deviations from the required total mass.
Second, the estimation includes Euler equations for the market return and the risk-free asset. These equations require the pricing kernel to price the market and the risk-free payoff consistently. This does not make the kernel a complete model of every asset. In fact, the paper explicitly notes that the empirical kernel is projected on the market return and does not capture a component orthogonal to that return. The Euler restrictions instead give the estimated object a direct connection to the assets used in its construction.
The sequence matters. Option prices do not directly become physical probabilities. They first provide a risk-neutral pricing distribution. Conditioning information changes the kernel that relates the two distributions. Probability and Euler restrictions then test whether the resulting map is coherent and relevant to the market and risk-free asset. The final state-price interpretation comes after those links, not before them.
This design is why the paper calls the approach semi-nonparametric or almost model-free. It does not impose a specific stochastic return dynamic such as a particular GARCH or jump process. It still makes choices about the polynomial order, the support of returns, option filtering, interpolation, positivity, instruments, and GMM moments. “Almost model-free” means that the return dynamics are not specified in the usual parametric way; it does not mean that the estimate is free of modeling decisions.
Conditioning turns a static ratio into a changing map
The next question is which information should be allowed to move the kernel. The Federal Reserve paper begins with 12 candidates because a variable can be economically plausible without adding reliable statistical information. The candidates cover market risk, higher moments, left-tail risk, sentiment, economic activity, credit conditions, the Treasury term spread, and recent market performance. The author estimates a conditional kernel with one variable at a time, then examines combinations.
The individual tests support six variables under the paper’s criteria: the VIX, sentiment, the term spread, industrial-production growth, credit spread, and the market return. The VIX is the most informative individual variable. Its coefficient changes both the overall level and the slope of the estimated kernel. In the reported specification, a higher VIX is associated with a lower and less steep kernel in negative-return regions. That is not simply a statement that high-volatility months have lower stock returns. It says that the price assigned to a future negative return changes with the current volatility environment.
The selection stage is not just a search for variables that correlate with the VIX or with realized returns. The paper examines the significance of the time-varying coefficients and uses Hansen’s over-identification test to ask whether the moment restrictions remain jointly plausible. A variable can look economically sensible yet fail the restrictions once it is used to condition the kernel. Conversely, a variable that adds a statistically supported change is useful because it alters the state-price mapping, not merely because it forecasts a market headline.
This distinction also limits the causal reading. The estimation shows that a variable contains information relevant to the fitted conditional kernel under the paper’s moments. It does not show that changing that variable would mechanically cause investors to reprice every future state. The term spread may summarize several expectations about growth, inflation, policy, and bond-market conditions; the VIX may summarize several forms of option demand and uncertainty. The model uses them as observables that organize the pricing relationship, not as isolated levers with one identified effect.
The paper then compares five bivariate specifications, each combining the VIX with one additional variable: sentiment, term spread, industrial-production growth, credit spread, or market return. Conditioning improves on the unconditional kernel in all five comparisons. But the tests do not treat every improvement as equally persuasive. The VIX and term spread are selected as the preferred pair because the added term spread is informative, the over-identification restriction is not rejected at the 5% level, and the relevant Wald tests support the additional conditioning information.
The term spread matters for a reason that the VIX cannot duplicate. It is the difference between the 10-year and 2-year Treasury yields and is used in the paper as a leading economic indicator. A higher or lower spread is not simply another reading of current option volatility. It carries information about the market’s forward-looking economic environment. When it is added to the VIX, the model can distinguish two situations that would otherwise be compressed into the same risk-state label: one with a given level of implied volatility but a more favorable outlook, and another with that volatility level but less favorable forward-looking conditions.
This is the conditional answer to the title. The VIX supplies an important coordinate, while the term spread supplies a different coordinate. The pair need not be interpreted as a complete description of the economy. It is enough that the second coordinate changes the estimated relationship between future returns and state prices after the first coordinate has been included.
The evidence also explains why adding variables is not automatically an improvement. When the paper adds a third conditioning variable to the VIX and term spread, the additional coefficients are not statistically significant as a group, the over-identification tests reject the specification, and the trivariate estimate is not statistically distinguishable from the bivariate one under the reported Wald comparison. A larger information set can create more flexibility without creating more reliable information. The preferred model is therefore not the model with the most inputs; it is the model whose added inputs improve the relationship without making the restrictions implausible.
The resulting kernel varies most visibly in the left tail of the return distribution. When the VIX is low or the term spread is relatively high, the estimated kernel in negative-return states is higher and steeper. When the VIX is high or the term spread is low, it is lower and less steep. The paper offers an economic interpretation: when a severe negative state is less expected, investors may value resources delivered in that state more highly or become more willing to save against it. When the outlook is already adverse, the estimated state-price pattern is different. The interpretation is economically plausible, but it is not a separately identified causal estimate of patience or fear.
That qualification matters because the shape is not uniformly settled across all returns. The average conditional kernel is generally downward sloping through the central range, but monotonicity is not rejected only up to about 8% of the monthly market return and is rejected in deep right tails. The result is therefore not “the kernel always slopes downward” or “the VIX determines the curve.” The supported conclusion is that conditioning variables change the estimated shape, especially where negative-return states matter most.
Nor does the preferred pair become a universal sufficient statistic. It is selected from this paper’s candidate variables, sample, one-month horizon, option data, approximation, and testing procedure. Another sample or a different horizon might make other information more useful. The important transferable lesson is methodological: when a summary statistic is used to explain state prices, test whether other observable conditions change the mapping rather than assuming that the summary is complete.
The next step is to follow the consequences of that changing map. If conditional information mainly reshapes prices in negative-return states, it should alter how the equity premium is allocated across the distribution and how well the kernel prices tail-sensitive options during crises. That is where the difference between an unconditional average and a conditional estimate becomes economically visible.