Disasters implied by equity index options^{∗}

David Backus,^{†} Mikhail Chernov,^{‡} and Ian Martin^{§}

November 23, 2010

Abstract

[??] We use equity index options to quantify the distribution of consumption growth disasters. The challenge here is connecting the risk-neutral distribution of equity returns implied by options to the true distribution of consumption growth estimated from macroeconomic evidence. We attack the problem from three perspectives. First, we compare compare pricing kernels constructed from, respectively, evidence of consumption growth and power utility and true and risk-neutral distributions of equity returns derived from the time series of equity returns and option prices. Second, we tie equity returns to consumption growth and ... use true distribution of consumption (based on international macro evidence) and macro-based pricing kernel to characterize the risk-neutral distribution of returns. Third, we use the risk-neutral distribution of returns implicit in options and the macro-based pricing kernel to infer the true distribution of consumption. All three approaches suggest that options imply smaller probabilities of extreme outcomes than have been estimated from macro data. The third approach yields a viable alternative of the true distribution of consumption because it matches not only the equity premium, but also option prices and basic moments of consumption.

JEL Classification Codes: E44, G12.

Keywords: equity premium, pricing kernel, entropy, cumulants, risk-neutral probabilities, implied volatility.

∗We thank the many people who have given us advice, including David Bates, Michael Brandt, Rodrigo Guimaraes, Sydney Ludvigson, Monika Piazzesi, Thomas Sargent, Romeo Tedongap, Michael Woodford, and Liuren Wu, as well as participants in seminars at, and conferences sponsored by, the AEA, the Bank of England, Bocconi, the CEPR, the Federal Reserve, Glasgow, Harvard, IDEI/SCOR (conference on extremal events), LBS, Leicester, LSE, Minnesota, MIT, Moscow HSE, NBER (summer institute), NYU, Penn, Princeton, RCEA Rimini (workshop on money and finance), SoFiE (conference on extreme events), SIFR, Toulouse (conference on financial econometrics), and UC Davis. We also thank Campbell Harvey, an associate editor, and a referee for helpful comments on an earlier version.

† Stern School of Business, New York University, and NBER; dbackus@stern.nyu.edu.

‡ London Business School, London School of Economics, and CEPR; M.Chernov@lse.ac.uk.

§ Graduate School of Business, Stanford University, and NBER; ian.martin@gsb.stanford.edu.

1 Introduction

The field of macro-finance offers an attractive opportunity to explore links between asset returns and the real economy. Many of the most influential papers in the field do just that, most commonly by using macroeconomic data as an input to models designed to account for asset returns. There is also promise in doing the reverse: in using properties of asset returns to characterize macroeconomic risk. The growing literature on disaster risk is a clear example in which macroeconomic research might benefit from greater input from finance.

Barro (2006), Longstaff and Piazzesi (2004), and Rietz (1988) show that macroeconomic disasters — large declines in aggregate consumption growth — produce dramatic improvement in the ability of representative agent models to reproduce prominent features of US asset returns, including the equity premium. They disagree, however, about their distribution. Rietz (1988) simply chooses an arbitrary distribution and illustrates its impact. Longstaff and Piazzesi (2004) argue that a distribution based on US experience can explain only about one-half of the equity premium. Barro (2006), Barro and Ursua (2008), and Barro, Nakamura, Steinsson, and Ursua (2009) study broader collections of countries, which in principle can tell us about alternative histories the US might have experienced. In their panel of international macroeconomic data, the frequency and magnitude of disasters are significantly larger than we have seen in US history. Their estimated distribution of consumption disasters, which we refer to as the international macroeconomic evidence, has become the industry standard in both macroeconomic and finance research.

The issue, then, is the distribution of extreme negative outcomes: the shape of the left tail of the probability distribution of consumption growth. In virtually all of this research, the distribution is modelled by combining a normal component with a jump component. A jump component, in this context, is simply a mathematical device that produces nonnormal distributions. The debate in the literature concerns the parameter values of this component.

The range of opinion about jump parameters reflects the nature of the problem: they are not easily estimated from the relatively short history of US macroeconomic data. We follow a different but complementary path, estimating them from prices of equity index options. Equity index options are a useful source of additional information here, because their prices tell us how market participants value extreme events — whether they happen in our sample or not. By looking at options with different strike prices, we learn more about the distribution of equity index returns than we do from returns alone. The cyclical behavior of equity returns and the range of strike prices make this class of assets a natural choice. Other cyclical assets, including corporate and government bonds, might be studied in future work.

The idea is straightforward, but the approaches taken in the macro-finance and optionpricing literatures are different enough that it takes some work to put them on a comparable basis. The most salient differences for our purposes are these: (i) The macro-finance literature is concerned with properties of consumption growth, while the option-pricing literature is concerned with equity returns. (ii) Macro-finance is concerned with the true or objective probability distribution, while option prices characterize the risk-neutral distribution. (iii) The relation between true and risk-neutral probability distributions is different in the two literatures. In the macro-finance literature, the risk-neutral distribution is derived from the true distribution of consumption growth and the preferences of a representative agent (power utility, for example). In the option-pricing literature, the two distributions are specified separately, which leads to a different structure for the pricing kernel. These differences pose nontrivial challenges to anyone attempting to use evidence from equity options to quantify features of consumption growth.

Our objective is to compare models based on consumption and option data, respectively, but the differences in approach lead us to compare the models from several different perspectives. Each comparison uses a different analytical structure and focuses on a different aspect of the model. In our first comparison, we focus on pricing kernels, which exist in any arbitrage-free model. In macro-finance models, the properties of pricing kernels are closely related to those of consumption growth. A natural first step then is to compare the pricing kernel generated by a representative agent model calibrated to international macroeconomic data to one implied by an estimated option-pricing model. A second comparison focuses on option prices. We follow the macro-finance route, using macroeconomic data and the preferences of a representative agent to value options as well as equity, and compare these option prices to those of an estimated option-pricing model. Here and elsewhere, we report option prices as implied volatility smiles, which represent departures from normality in an intuitive way.

A third comparison delivers on our goal of estimating the distribution of consumption growth disasters from options: we compare the distribution estimated from international macroeconomic data with one derived from option prices. Here we reverse the procedure of the previous comparison, using our estimated implied volatility smile and the preferences of a representative agent to compute the consumption distribution from option prices. By construction, the distribution is consistent with both option prices and the equity premium. This approach differs from the substantial body of work that infers risk aversion from option prices, where preferences are defined directly over wealth. Consumption is not studied. See, for example, Ait-Sahalia and Lo (2000), Jackwerth (2000), Rosenberg and Engle (2002), and Ziegler (2007). In that literature, preferences are defined directly over wealth, which is proxied by equity, so the distribution of consumption is not studied.

In our comparison of pricing kernels, we find the concept of entropy (a measure of dispersion) useful. Alvarez and Jermann (2005) and Bansal and Lehmann (1997) show that the entropy of the pricing kernel of a given model is the maximum risk premium, defined as the mean excess log return. Entropy can be represented as an infinite sum of the cumulants (close relatives of moments) of the log of the pricing kernel. If the log of the pricing kernel is normal, entropy is proportional to its variance. Departures from lognormality, such as those generated by jumps, can increase cumulants, and hence entropy, thereby improving a model’s ability to account for observed excess returns. Odd cumulants, in particular, reflect the asymmetry inherent in disaster research. Despite the difference in functional forms, entropy can be computed for both the macro-finance and option-pricing models. We find that while both consumption- and option-based models generate substantial contributions to entropy from odd cumulants, the relative contribution is much smaller in the model based on option prices. We also find that the contribution of the variance is much higher in the option model, which results in greater entropy overall.

In our comparison of option prices, we follow the traditional macro-finance path: we use a consumption distribution based on macroeconomic assessments of jumps, link dividends to consumption, price assets with a representative agent with power utility, select risk aversion to match the equity premium, and use the model to value options on equity. We then compare implied volatility smiles derived this way from consumption data to those from an option model estimated by Broadie, Chernov, and Johannes (2007). We find that the volatility smile is much steeper in the consumption-based model.

In our comparison of consumption growth distributions, we do the reverse. We take estimates of the risk-neutral jump parameters from Broadie, Chernov, and Johannes’s (2007) study of equity index options. This insures that the model matches the implied volatility smile. As before, we use a representative agent with power utility and choose risk aversion to match the equity premium. This allows us to infer the true distribution of consumption growth implied by option prices. The consumption growth distribution implied by option prices agrees with consumption-based estimates on the probability of modest disasters: consumption growth more than 3 standard deviations to the left of the mean. In this respect, the distribution is similar to US data. (The Great Depression includes one year in which consumption declined by slightly more than 3 standard deviations). However, the distribution differs significantly for more extreme events: the probability of consumption growth more than 5 standard deviations to the left of its mean is much smaller than estimates based on international macroeconomic evidence. The parameter values derived this way match not only the equity premium as in the Barro calibration, but also option prices and moments of consumption growth. We think these parameter values provide a more realistic distribution of disasters in US data than the international macroeconomic evidence used in the literature.

The second and third comparisons are based on a model in which consumption growth and equity returns are perfectly correlated. There is a long tradition of similar models in the macro-finance literature, but the correlation in US data is only 0.57. We therefore consider a bivariate model in which the correlation can be set to match the evidence. If we choose parameters based on international macroeconomic evidence, the volatility smile for option prices remains much steeper than we see in the model based on option prices. Conversely, if we choose parameters to match option prices, the probability of large negative realizations of consumption growth (more than 5 standard deviations to the left of the mean) remains much smaller than suggested by the macroeconomic evidence.

We conclude that option prices imply smaller probabilities of macroeconomic disasters and their role in pricing risky assets than suggested by Barro and his coauthors. Nevertheless, we would not say we reject “the Barro model.” We would say instead that the model makes a useful point about the role of asymmetries and other departures from normality in asset pricing. Certainly there is evidence of both in prices of equity index options.

2 Preliminaries

We start with an overview of the evidence we are trying to understand and the tools we use later on to shed light on it. The tools allow us to characterize departures from lognormality, including disasters, in a convenient way.

2.1 Evidence

We provide a quick overview of US evidence on consumption growth, asset returns, and option prices.

In Table 1 we report evidence on annual consumption growth and equity returns (the S&P 500 index) for both a long sample (1889-2009) and a shorter one (1986-2009) that corresponds approximately to the option data used by Broadie, Chernov, and Johannes (2007). Similar evidence is summarized by Alvarez and Jermann (2005, Tables I-III), Barro (2006, Table IV), and Mehra and Prescott (1985, Table 1). In both samples, consumption growth and equity returns exhibit the negative skewness we would expect from occasional disasters. Our estimates of the equity premium (0.0407 in the long sample, 0.0434 in the short sample) are somewhat smaller than those reported elsewhere. One reason is that we measure returns in logs; in levels, the mean excess return on equity is 0.0571 in the long sample and 0.0613 in the short sample. Another reason is that the 2008 return (−0.38 in levels) has a significant impact on the estimated mean, particularly in the short sample.

The next issue is option prices. Options are available on the S&P 500 index and on its futures contracts. Prices are commonly quoted as implied volatilities: the value of the volatility parameter that equates the price with the Black-Scholes-Merton formula. These volatilities have two well-documented features that we examine more closely in Section 5. Similar evidence has been reviewed recently by Bates (2008, Section 1), Drechsler and Yaron (2008, Section 2), and Wu (2006, Section II). The first feature is that implied volatilities are greater than sample standard deviations of returns. Since prices are increasing in volatility, this implies that options are expensive relative to the lognormal benchmark that underlies Black-Scholes-Merton. As a result, selling options generates high average returns. The second feature is that implied volatilities are higher for lower strike prices: the well-known volatility skew. This feature is intriguing from a disaster perspective, because it suggests market participants value adverse events more than would be implied by a lognormal model. The question for us whether the extra value assigned to bad outcomes corresponds to the disasters documented in macroeconomic research.

In the following sections we use round-number versions of the estimates in Table 1 to illustrate the quantitative importance of disasters. We report the properties of numerical examples in which log consumption growth has a mean of 0.0200 (2%) and a standard deviation of 0.0350 (3.5%). Similarly, the log excess return on equity has a mean of 0.0400 and a standard deviation of 0.1800 and the log return on the one-period bond is 0.0200. Most of these numbers are similar across the long and short samples. The exception is the standard deviation of log consumption growth. We use an estimate based on the long sample because it includes the Great Depression, the one clear disaster in this sample. None of these numbers are definitive, but they are close to the values in the table and give us a starting point for considering the quantitative implications of disasters.

2.2 Jumps and disasters

We use a jump model as a way to generate departures from normality and, in particular, disasters. We follow Barro (2006) in using a two-component structure for consumption growth g_{t} = c_{t}/c_{t−1},

log g_{t+1} = w_{t+1} + z_{t+1},  (1)

with components (w_{t}, z_{t}) that are independent of each other and over time. The first component is normal: w ∼ N (μ, σ^{2}).

The second, jump, component is modelled via a Poisson mixture of normals. The central ingredient of the jump component is a random variable j (the number of jumps) that takes on nonnegative integer values with probabilities e^{−ω}ω^{j}/j!. The parameter ω (“jump intensity”) is the mean of j. Conditional on j, the jump component is normal:

z_{t}|j ∼ N (jθ, jδ^{2}) for j = 0, 1, 2. . . . .  (2)

This specification differs from the Barro model in two respects: there is a positive probability of more than one jump per year and the jump size has a parametric distribution rather than an empirical one (a histogram). If ω is small, as is the case for Barro’s calibration, the first difference is insignificant because probability of jump can be approximated by (1 − ω) when j = 0, ω when j = 1, and 0 when j > 1. If ω is large, as it is in the option model of Section 5, there can be a significant probability of multiple jumps. The difference between parametric and empirical distribution of the jump size is largely in convenience of deriving various theoretical results in the paper.

The added complexity has a number of benefits. One is that it gives us a better approximation to the empirical distribution of disasters. Another is that it is easily scaled to the different time intervals observed in option markets (it is “infinitely divisible”). For this reason and others, this specification is commonly used in work on option pricing, where it is referred to as the Merton (1976) model. In the macro-finance literature, it has been applied by Bates (1988), Martin (2007), and Naik and Lee (1990).

Cumulants is a useful concept that captures departures from normality generated by the jump model. Recall that the moment generating function (if it exists) for a random variable x is defined by

h(s; x) = E (e^{sx}) ,

a function of the real variable s. With enough regularity, the cumulant-generating function, k(s) = log h(s), has the power series expansion

k(s; x) = log E (e^{sx}) = ∑_{j=1}^{∞} κ_{j}(x)s^{j}/j!  (3)

for some suitable range of s. This is a Taylor (Maclaurin) series representation of k(s) around s = 0 in which the “cumulant” κ_{j} is the jth derivative of k at s = 0. Cumulants are closely related to moments: κ_{1} is the mean, κ_{2} is the variance, and so on. Skewness γ_{1} and excess kurtosis γ_{2} are scaled versions of the third and fourth cumulants:

γ_{1} = κ_{3}/κ_{2}^{3/2}, γ_{2} = κ_{4}/κ_{2}^{2}.  (4)

The normal distribution w has a quadratic cumulant-generating function μs+σ^{2}s^{2}/2, which implies zero cumulants after the first two. Nonzero high-order cumulants (κ_{j} for j ≥ 3) therefore summarize departures from normality. Appendix A.1 derives the cumulantgenerating function for z. Note for future reference that k(s; ax) = k(as; x) [replace s with as in (3)]. Therefore if x has cumulants κ_{j}, ax has cumulants a^{j}κ_{j}. This use of the cumulant-generating function was suggested by Martin (2009) and recurs throughout the paper.

Since the components are independent, the cumulant-generating function of log g is the sum of those for w and z. We find cumulants of log g by taking derivatives of k. The first four are

κ_{1} = μ + ωθ  (5)
κ_{2} = σ^{2} + ω(θ^{2} + δ^{2})  (6)
κ_{3} = ωθ(θ^{2} + 3δ^{2})  (7)
κ_{4} = ω(θ^{4} + 6θ^{2}δ^{2} + 3δ^{4}).  (8)

The magnitude of cumulant is not determined by the distribution of jumps sizes alone. The interaction between jump sizes and jump probabilities is critical. This point is obvious for the mean. For example ω = 0.01 and θ = −0.3 produce the same κ_{1} as ω = 1.5 and θ = −0.002.

More generally, the last observation is important for making a distinction between jumps and disasters. Disasters are negative realizations of consumption growth that are below a certain threshold −b, for large b > 0 : D_{b} = {log g ≤ −b}. Consider the probability of event D_{b} as implied by the jump model (1):

P (D_{b}) = ∑_{j=1}^{∞} P (log g ≤ −b|j) · P (j) = ∑_{j=1}^{∞} Φ (\frac{−b − μ − jθ}{\sqrt{σ^{2} + jδ^{2}}} ) · \frac{e^{−ω}ω^{j}}{j!} ,  (9)

where Φ(·) is a cumulative distribution function of a standard normal variable. It is clear that small values of ω and large jump sizes as described by the distribution N (θ, δ^{2}) will generate small probabilities for large values of b consistent with the Barro story. However, jump sizes need not be large, yet they can still generate rare extreme realizations if jumps occur frequently enough.

2.3 Pricing kernels, entropy, and cumulants

Here we describe a concept of entropy (a measure of dispersion) that we think clarifies how the risks of departures from lognormality are valued in any asset pricing model.

In any arbitrage-free environment, there is a positive random variable m (the pricing kernel) that satisfies the pricing relation,

E_{t} (m_{t+1}r_{t+1}^{j}) = 1,  (10)

for (gross) returns r^{j} on all traded assets j. Here E_{t} denotes the expectation conditional on information available at date t. In stationary ergodic settings, the same relation holds unconditionally as well; that is, with an expectation E based on the ergodic distribution. In finance, the pricing kernel is often a statistical construct designed to account for returns on assets of interest. In macroeconomics, the pricing kernel is tied to macroeconomic quantities such as consumption growth. In this respect, the pricing kernel is a link between macroeconomics and finance.

Asset returns alone tell us some of the properties of the pricing kernel, hence indirectly about macroeconomic fundamentals. A notable example is the Hansen-Jagannathan (1991) bound. We use a similar “entropy bound” derived by Alvarez and Jermann (2005) and Bansal and Lehman (1997). Both bounds relate measures of pricing kernel dispersion to expected differences in returns. With this purpose in mind, we define the entropy of a positive random variable x as

L(x) = log Ex − E log x.  (11)

We account for this use of the term shortly. Entropy has a number of properties that we use repeatedly. First, entropy is nonnegative and equal to zero only if x is constant (Jensen’s inequality). In the familiar lognormal case, where log x ∼ N (κ_{1}, κ_{2}), entropy is L(x) = κ_{2}/2 (one-half the variance of log x). We’ll see shortly that L(x) also depends on features of the distribution beyond the first two moments. Second, L(ax) = L(x) for any positive constant a. Third, if x and y are independent, then L(xy) = L(x) + L(y).

The entropy bound relates the entropy of the pricing kernel to expected differences in log returns:

L(m) ≥ E (log r^{j} − log r^{1})  (12)

for any asset j with positive returns. See Appendix A.2. Here r^{1} is the (gross) return on a one-period risk-free bond, so the right-hand side is the mean excess return or premium on asset j over the short rate. Inequality (12) therefore transforms estimates of return premiums into estimates of the lower bound of the entropy of the pricing kernel.

The beauty of entropy as a dispersion concept for the study of disasters is that it includes a role for the departures from normality they tend to generate. We can express the entropy of the pricing kernel in terms of the cumulant-generating function and cumulants of log m:

L(m) = log E (e^{log m}) − E log m
= k(1; log m) − κ_{1}(log m) = ∑_{j=2}^{∞} κ_{j}(log m)/j!.  (13)

If log m is normal, entropy is one-half the variance (κ_{2}/2), but in general there will be contributions from skewness (κ_{3}/3!), excess kurtosis (κ_{4}/4!), and so on.

Zin (2002, Section 2) points out that we can use high-order cumulants to account for properties of returns that are difficult to explain in lognormal settings. We implement his insight with a three-way decomposition of entropy: one-half the variance (the lognormal term) and contributions from odd and even high-order cumulants. Although disasters typically show up in both odd and even high-order cumulants, odd cumulants reflect their inherent asymmetry. More generally, the contribution of odd high-order cumulants represents an adaptation and extension of work on skewness preference by Harvey and Siddique (2000) and Kraus and Litzenberger (1976): adaptation because it refers to properties of the log of the pricing kernel rather than its level, and extension because it involves all odd high-order cumulants, not just skewness. We compute odd and even cumulants from the odd and even components of the cumulant-generating function:

kodd(s) = [k(s) − k(−s)]/2 = ∑_{j=1,3,...} κ_{j}(x)s^{j}/j!
keven(s) = [k(s) + k(−s)]/2 = ∑_{j=2,4,...} κ_{j}(x)s^{j}/j!.

Odd and even high-order cumulants follow from subtracting the first and second cumulants, respectively.

2.4 Risk-neutral probabilities

In option pricing models, there is rarely any mention of a pricing kernel, although theory tells us one must exist. Option pricers speak instead of true and risk-neutral probabilities. We use a finite-state iid (independent and identically distributed) setting to show how pricing kernels and risk-neutral probabilities are related.

Consider an iid environment with a finite number of states x that occur with (true) probabilities p(x), positive numbers that represent the frequencies with which different states occur (the data generating process, in other words). With this notation, the pricing relation (10) becomes

E (mr^{j}) = ∑_{x} p(x)m(x)r^{j}(x) = 1

for (gross) returns r^{j} on all assets j. A particularly simple example is a one-period bond, whose price is q^{1} = Em = ∑_{x} p(x)m(x) = 1/r^{1}. Risk-neutral (or better, risk-adjusted) probabilities are

p^{∗}(x) = p(x)m(x)/Em = p(x)m(x)/q^{1}.  (14)

The p^{∗}s are probabilities in the sense that they are positive and sum to one, but they are not the data generating process. The role of q^{1} is to make sure they sum to one. They lead to another version of the pricing relation,

q^{1} ∑_{x} p^{∗}(x)r^{j}(x) = q^{1}E^{∗}r^{j} = 1,  (15)

where E^{∗} denotes the expectation computed from risk-neutral probabilities. In (10), the pricing kernel performs two roles: discounting and risk adjustment. In (15) those roles are divided between q^{1} and p^{∗}, respectively.

Option pricing is a natural application of this approach. Consider a put option: the option to sell an arbitrary asset with future price q(x) at strike price b. Puts are bets on bad events — the purchaser sells prices below the strike, the seller buys them — so their prices are an indication of how they are valued by the market. If the option’s price is q^{p} (p for put), its return is r^{p}(x) = [b − q(x)]^{+}/q^{p} where (b − q)^{+} ≡ max{0, b − q}. Equation (15) gives us its price in terms of risk-neutral probabilities:

q^{p} = q^{1}E^{∗}(b − q)^{+}.

This highlights the role of risk-neutral probabilities in option pricing: As we vary b, we trace out the risk-neutral distribution of prices q(x) (Breeden and Litzenberger, 1978).

But what about the pricing kernel and its entropy? Equation (14) gives us the pricing kernel:

m(x) = q^{1}p^{∗}(x)/p(x).  (16)

Since q^{1} is constant in our iid world, the entropy of the pricing kernel is

L(m) = L(p^{∗}/p) = log E(p^{∗}/p) − E log(p^{∗}/p) = −E log(p^{∗}/p).  (17)

The first equality follows because q^{1} is constant [recall L(ax) = L(x)]. The second follows from the definition of entropy [equation (11)]. The last one follows from

E(p^{∗}/p) = ∑_{x}[p^{∗}(x)/p(x)]p(x) = ∑_{x} p^{∗}(x) = 1.

The expression on the right of (17) is sometimes referred to as the entropy of p^{∗} relative to p, which accounts for our earlier use of the term.

As before, entropy can be expressed in terms of cumulants. The cumulants in this case are those of log(p^{∗}/p), whose cumulant-generating function is

k[s; log(p^{∗}/p)] = log E (e^{s log(p^{∗}/p)}) = ∑_{j=1}^{∞} κ_{j}[log(p^{∗}/p)]s^{j}/j!.  (18)

The definition of entropy (11) contributes the analog to (13) in which entropy is related to cumulants:

L(p^{∗}/p) = k[1; log(p^{∗}/p)] − κ_{1}[log(p^{∗}/p)]
= ∑_{j=2}^{∞} κ_{j}[log(p^{∗}/p)]/j! = −κ_{1}[log(p^{∗}/p)].  (19)

The second line follows from k[1; log(p^{∗}/p)] = log E(p^{∗}/p) = 0 (see above). Here we can compute entropy from the first cumulant, but it is matched by an expansion in terms of cumulants 2 and above, just as it was in the analogous expression for log m. All of these cumulants are readily computed from derivatives of the cumulant-generating function (18).

To summarize: we can price assets using either a pricing kernel (m) and true probabilities (p) or the price of a one-period bond (q^{1}) and risk-neutral probabilities (p^{∗}). The three objects (m, p^{∗}, p) are interconnected: once we know two (and the price of a one-period bond), equation (14) gives us the other. That leaves us with three kinds of cumulants corresponding, respectively, to the true distribution of the random variable x, the riskneutral distribution, and the true distribution of the log of the pricing kernel. We report all three.

3 Disasters in macroeconomic models and data

Representative-agent exchange economies generate larger risk premiums when we include infrequent large declines in consumption growth. We describe the mechanism with two numerical examples that highlight the role of high-order cumulants. Here and in our study of options we restrict our attention to iid environments. There are many features of the world that are not iid, but this simplification allows us to focus without distraction on the distribution of returns, particularly the possibility of extreme negative outcomes (reported in Table 1). We think it is a reasonably good approximation for this purpose, but return to the issue briefly in Section 6.

The economic environment consists of preferences for a representative agent and a stochastic process for consumption growth. Preferences are governed by an additive power utility function,

E_{0} ∑_{t=0}^{∞} β^{t}u(c_{t}),

with u(c) = c^{1−α}/(1 − α) and α ≥ 0. We refer to α as risk aversion. The pricing kernel is

log m_{t+1} = log β − α log g_{t+1}.  (20)

With power utility, the second derivative is negative (risk aversion), the third positive (skewness preference), and the fourth negative (kurtosis aversion). The properties of the pricing kernel follow from those of consumption growth.

Entropy is

L(m) = L(e^{−α log g}) = L(e^{−αw}) + L(e^{−αz})  (21)

The entropy of the two components follows from its definition (11):

L(e^{−αw}) = (−ασ)^{2}/2  (22)
L(e^{−αz}) = ω[e^{−αθ+(αδ)^{2}/2} − 1] + αωθ.  (23)

The cumulants of log m are related to those of log g by

κ_{j}(log m) = κ_{j}(log g)(−α)^{j}/j! = (−α)^{j}κ_{j}(w)/j! + (−α)^{j}κ_{j}(z)/j!, j ≥ 2. (24)

See Section 2.3. If log consumption growth is normal, then so is the log of the pricing kernel. Entropy is then one-half the variance of consumption growth times the risk aversion parameter squared. The impact of high-order cumulants depends on (−α)^{j}/j!. The minus sign tells us that negative odd cumulants of log consumption growth generate positive odd cumulants in the log pricing kernel. Negative skewness in consumption growth, for example, generates positive skewness in the pricing kernel and thus increases the entropy of the pricing kernel. The contributions of high-order cumulants are controlled by the coefficient α^{j}/j!. Eventually the denominator grows faster than the numerator, but for moderate values of j, risk aversion can magnify the contributions of high-order cumulants (those with j ≥ 3) relative to the variance.

We can see the quantitative significance of the jump component with numerical examples based on international macroeconomic evidence. Its role is evident in Table 3 in the difference between column (1), the lognormal case, and column (2), which incorporates a Poisson jump component. In both cases, the mean and variance of log consumption growth are κ_{1}(log g) = 0.020 and κ_{2}(log g) = 0.035^{2}. In column (1), we set μ = κ_{1}(log g) and σ^{2} = κ_{2}(log g). In column (2), we set ω = 0.01, θ = −0.3 and δ = 0.15: a one percent chance of a 30 percent fall (on average) in consumption growth relative to its mean. Given these values, we adjust the parameters of the normal component to maintain the mean and variance whose theoretical values are given in (5) and (6).

The parameters of the jump component are derived from studies of international macroeconomic data by Barro (2006), Barro and Ursua (2007), and Barro, Nakamura, Steinsson, and Ursua (2009). Each of these studies looks at aggregate output or consumption over the last century or more for 20-plus countries. Martin (2009) uses the empirical distribution reported by Barro (2006) to set ω = 0.017, θ = −0.38 and δ = 0.25. Wachter (2009) uses exactly the same specification of jump sizes as Barro (2006). Barro, Nakamura, Steinsson, and Ursua (2009, Section 6.2) estimate a dynamic model, but argue that its asset pricing implications are the same as an iid model with ω = 0.0138 (corresponding to their p) and θ = −0.357 [corresponding to their log(1 − b)]. We use more modest values to avoid overstating the role of jumps and to keep the variance of the normal component positive. These numbers nevertheless suggest what may seem to be an excessively large probability of an extremely bad outcome given US history, but that’s what the international evidence implies. We return to this issue when we look at the distribution implied by options.

With these numbers, we can explore the ability of the model to satisfy the entropy bound. The observed equity premium implies that the entropy of the pricing kernel is at least 0.0400. In the lognormal case, the entropy bound implies α^{2}κ_{2}(log g)/2 = α^{2}0.0350^{2}/2 ≥ 0.0400 or α ≥ 8.08. We can satisfy the entropy bound for the equity premium, but only with a risk aversion parameter greater than 8. There’s a range of opinion about this, but some argue that risk aversion this large implies implausible behavior along other dimensions; see, for example, the discussion in Campanale, Castro, Clementi (2010, Section 4.3) and the references cited there.

When we add the jump component, a smaller risk aversion parameter suffices. Since the mean and variance of log consumption growth are the same, the experiment has a partial derivative flavor: it measures the impact of high-order cumulants, holding constant the mean and variance. The jump component z introduces negative skewness and positive excess kurtosis into log consumption growth. Both are evident in the first panel of Figure 1, where we plot cumulants 2 to 8 for log consumption growth. Each cumulant κ_{j}(log g) makes a contribution κ_{j}(log g)(−α)^{j}/j! to the entropy of the pricing kernel. The next two panels of the figure show how the contributions depend on risk aversion. With α = 2, negative skewness in consumption growth translates into a positive contribution to entropy, but the contribution of high-order cumulants overall is small relative to the contribution of the variance. That changes dramatically when we increase α. Even small high-order cumulants make significant contributions to entropy if α is large enough (see also Table 2).

Figure 2 gives us another perspective on the same issue: the impact of high-order cumulants on the entropy of the pricing kernel as a function of the risk aversion parameter α. The horizontal line is the lower bound, our estimate of the equity premium in US data. The line labelled “lognormal” is entropy without the jump component. We see, as we noted earlier, that the entropy of the pricing kernel for the lognormal case is below the lower bound until α is above 8. The line labelled “disasters” incorporates the jump component. The difference between the two lines shows that the overall contribution of high-order cumulants is positive and increases sharply with risk aversion. Table 2 implies that when α = 2 the extra terms increase entropy by 32%, but when α = 10 the increase is 850%.

It’s essential that the jumps be bad outcomes. If we reverse the sign of θ, the result is the line labelled “booms” in Figure 2. We see that for every value of α, entropy is below even the lognormal case. Table 2 shows us exactly how this works. With jumps (and α = 10), the entropy of the pricing kernel (0.5837) comes from the variance (0.0613), odd high-order cumulants (0.2786), and even high-order cumulants (0.2439). When we switch to booms, the odd cumulants change sign — see equation (24) — reducing total entropy.

The jump model increases the probability of extreme values negative consumption growth D_{b} relative to the lognormal benchmark. We see in Table 3 that the probability of log consumption growth more than three standard deviations to the left of its mean (b = −κ_{1} + 3κ_{2}^{1/2} in (9)) is 0.13% in the lognormal case [column (1)] but 0.9% in the Poisson cases [columns (2)]. This corresponds to a drop in consumption of more than 8.5%, something seen only once in US history: in 1931, when consumption fell by 9.9%. Thus a 1% event has occurred once in slightly more than a century of US history. In this respect, the examples correspond roughly to US experience.

In other respects, the example is more extreme than US history, implying larger departures from lognormality that we’ve observed. The model implies, for example, skewness of log consumption growth of −11.02 [the entry labelled γ_{1}(true) in Table 3]. In US data, our estimate is a much more modest −0.34 (the entry labelled skewness in Table 1). Excess kurtosis (γ_{2}) is similar. This is, of course, Barro’s (2006) argument: that what we’ve seen in US data may not accurately reflect the distribution of what might have happened. That leads us to study options, which in principle reflect the distribution used by market participants.

4 Risk-neutral probabilities in representative-agent models

As a warmup for our study of options, we derive the risk-neutral probabilities implied by the examples of the previous section and use them to compute the risk-neutral parameters reported in Table 3. The state spaces have continuous components, but the logic of Section 2.4 follows with integrals replacing sums where appropriate. In representative-agent models, risk aversion generates risk-neutral distributions that are shifted left (more pessimistic) relative to true distributions. The form of this shift depends on the distribution. More generally, we might think of any such shift as representing something like risk aversion.

Our first example has lognormal consumption growth. Suppose log g = w with w ∼ N (μ, σ^{2}). Then

p(w) = (2πσ^{2})^{−1/2} exp[−(w − μ)^{2}/2σ^{2}].

The pricing kernel is m(w) = β exp(−αw) and the one-period bond price is q^{1} = Em = β exp[−αμ + (ασ)^{2}/2]. Equation (14) gives us the risk-neutral probabilities:

p^{∗}(w) = p(w)m(w)/q^{1} = (2πσ^{2})^{−1/2} exp[−(w − μ + ασ^{2})^{2}/2σ^{2}].

Thus the risk-neutral distribution has the same form (normal) with mean μ^{∗} = μ − ασ^{2} and standard deviation σ^{∗} = σ. The former shows us that the distribution shifts to the left by an amount proportional to risk aversion α and risk σ^{2}. The log probability ratio is

log [p^{∗}(w)/p(w)] = [(w − μ)^{2} − (w − μ^{∗})^{2}]/2σ^{2},

which implies the cumulant-generating function

k[s; log(p^{∗}/p)] = log E (e^{s log p^{∗}/p}) = \frac{(μ − μ^{∗})^{2}}{2σ^{2}} (−s + s^{2}).

The cumulants are (evidently) zero after the first two. Entropy follows from equation (19),

L(p^{∗}/p) = \frac{(μ − μ^{∗})^{2}}{2σ^{2}} = (ασ)^{2}/2,

which is what we reported in equation (22).

In our second example, consumption growth follows the Poisson-normal mixture described by equation (2). We derive the risk-neutral distribution from the cumulant-generating function (cgf). This approach works with the other examples, too, but it is particularly convenient here. With power utility, the cgf of the risk-neutral distribution is

k^{∗}(s) = k(s − α) − k(−α).

See Appendix A.3. Since k(s) = ω[exp(sθ + (sδ)^{2}/2) − 1] (Appendix A.1), we have

k^{∗}(s) = ωe^{−αθ+(αδ)^{2}/2} [e^{s(θ−αδ^{2})+(sδ)^{2}/2} − 1]

This has the same form as k(s) and describes a Poisson-normal mixture with parameters

ω^{∗} = ωe^{−αθ+(αδ)^{2}/2}, θ^{∗} = θ − αδ^{2}, δ^{∗} = δ.  (25)

Similar expressions are derived by Bates (1988), Martin (2007), and Naik and Lee (1990). Risk aversion (α > 0) places more weight on bad outcomes in two ways: they occur more frequently (ω^{∗} > ω if θ < 0) and are on average worse (θ^{∗} < θ). Entropy is the same as equation (23).

Multi-component models combine these ingredients. If log consumption growth is the sum of independent components, then entropy is the sum of the entropies of the components, as in equation (21).

5 Disasters in option models and data

In the macro-finance literature, pricing kernels are typically constructed as in Section 3: we apply a preference ordering (power utility in our case) to an estimated process for consumption growth (lognormal or otherwise). In the option-pricing literature, pricing kernels are constructed from asset prices alone: we estimate true probabilities from time series data on prices or returns, estimate risk-neutral probabilities from the cross-section of option prices, and compute the pricing kernel from the ratio. The approaches are complementary; they generate pricing kernels from different data. The question is whether they lead to similar conclusions. Do options on US equity indexes imply the same kinds of extreme events that Barro and Rietz suggested? Equity index options are a particularly informative class of assets for this purpose, because they tell us not only the market price of equity returns overall, but the prices of specific outcomes.

5.1 The Merton model

We look at option prices through the lens of the Merton (1976) model, a functional form that has been widely used in the empirical literature on option prices. The starting point is a stochastic process for asset prices or returns. Since we’re interested in the return on equity, we let

log r_{t+1}^{e} − log r^{1} = w_{t+1} + z_{t+1}.  (26)

We use the return, rather than the price, but the logic is the same either way. As before, the components (w_{t}, z_{t}) are independent of each other and over time. Market pricing of risk is built into differences between the true and risk-neutral distributions of the components. We give the distributions the same form, but allow them to have different parameters. The first component, w, has true distribution N (μ, σ^{2}) and risk-neutral distribution N (μ^{∗}, σ^{2}). By convention, σ is the same in both distributions, a byproduct of its continuous-time origins. The second component, z, is a Poisson-normal mixture. The true distribution has jump intensity ω and the jumps are N (θ, δ^{2}). The risk-neutral distribution has the same form with parameters (ω^{∗}, θ^{∗}, δ^{∗}). The structure and notation will be familiar from Section 2.2.

The Merton model has been widely used in empirical studies of asset pricing, where the parameters of the jump component provide flexibility over the form of departures from normality. It also scales easily to different time intervals, as we show in Appendix A.5. That’s helpful here because it allows us to use the model to price options for a range of maturities. The simplest way to describe this is with the cumulant-generating function, which is proportional to the time interval. Entropy and cumulants scale the same way.

Related work supports a return process with these features. Ait-Sahalia, Wang, and Yared (2001) report a discrepancy between the risk-neutral density of S&P 500 index returns implied by the cross-section of options and the time series of the underlying asset returns, but conclude that the discrepancy can be resolved by introducing a jump component. One might go on to argue that two jumps are needed: one for macroeconomic disasters and another for more frequent but less extreme financial crashes. However, Bates (2010) studies the US stock market over the period 1926-2009 and shows that a second jump component plays no role in accounting for macroeconomic events like the Depression.

Given this structure, the pricing kernel follows from equation (16). Its entropy is

L(m) = L(p^{∗}/p)
= \frac{(μ − μ^{∗})^{2}}{2σ^{2}} + (ω^{∗} − ω) + ω [log \frac{ω}{ω^{∗}} − log \frac{δ}{δ^{∗}} + \frac{(θ − θ^{∗})^{2} + (δ^{2} − δ^{∗}^{2})}{2δ^{∗}^{2}}] . (27)

This expression and the corresponding cumulant-generating function are derived in Appendix A.6.

5.2 Parameter values

We use parameter values from Broadie, Chernov, and Johannes (2007), who summarize and extend the existing literature on equity index options. Their estimates also include stochastic volatility. We make volatility constant, but we think the simplification is innocuous for our purposes. For one thing, the volatility smile of our iid model is almost the same as the smile generated by the more general model with the volatility state variable set equal to its mean. For another, the smile in the iid model is very close to the average smile in the stochastic volatility model.

The parameters of the true distribution are estimated from the time series of excess returns on equity. We use the parameters of the Poisson-normal mixture — namely (ω, θ, δ)

— reported in Broadie, Chernov, and Johannes (2007, Table I, the line labelled SVJ EJP). The estimated jump intensity ω is 1.512, which implies much more frequent jumps than we used in our consumption-based model. With this value, the probability of 0 jumps per year is 0.220, 1 jump per year 0.333, 2 jumps 0.25, 3 jumps 0.13, 4 jumps 0.05, and 7 or more jumps about 0.001. The jumps have mean θ = −0.0259 and standard deviation δ = 0.0407. Given parameters for the Poisson-normal component, the mean μ and standard deviation σ of the normal component are chosen to match the mean and variance of excess returns to their target values (0.0400 and 0.1800^{2}, respectively). In the model, the mean excess return (the equity premium) is μ + ωθ, which determines μ. The variance is σ^{2} + ω(θ^{2} + δ^{2}), which determines σ. All of these numbers are reported in column (4) of Table 3.

The risk-neutral parameters for the Poisson-normal mixture are estimated from the cross section of option prices: specifically, prices of options on the S&P 500 over the period 1987-2003. The depth of the market varies both over time and by the range of strike prices and maturities, but there are enough options to allow reasonably precise estimates of the parameters. The numbers we report in Table 3 are from Broadie, Chernov, and Johannes (2007, Table IV, line 5). In practice, option prices identify only the product ω^{∗}θ^{∗}, so they set ω^{∗} = ω and choose θ^{∗} and δ^{∗} to match the level and shape of the implied volatility smile. Given values for (ω^{∗}, θ^{∗}, δ^{∗}), we set μ^{∗} to satisfy (15), which implies μ^{∗} + σ^{2}/2 + ω^{∗}[exp(θ^{∗} + δ^{∗2}/2) − 1] = 0.

Figure 3 shows how the jump mean θ^{∗} and standard deviation δ^{∗} affect the cross section of 3-month option prices. The relevant formulas are reported in Appendices A.4 and A.5. We express prices as implied volatilities and graph them against “moneyness,” with higher strike prices to the right. We measure moneyness as the proportional deviation of the strike from the price: (strike − price)/price. A value of zero is therefore equivalent to an atthe-money option (strike = price) or an option on the return at a strike of zero. We use 3-month rather than 1-year options because departures from lognormality (flat volatility smiles) are more obvious at the shorter maturity. In the figure, the solid line represents the implied volatility smile in the model. Since the model fits extremely well, we can take this as a reasonable representation of the data. The downward slope and convex shape are both evidence of departures from lognormality. The second line illustrates the role of the jump mean θ^{∗}: when we divide it by two, the line is flatter. By making the mean jump size smaller, we reduce the value of out-of-the-money puts. The third line illustrates the role of the jump variance δ^{∗2}: when we divide it by two, the smile has less curvature. Both lines lie below the estimated one, so the estimated parameters evidently help to account for the observed premium of implied volatilities over the true standard deviation of equity returns (0.1800 in our model).

5.3 Pricing kernel implied by options

We compute the pricing kernel from the ratio of risk-neutral to true probabilities, as in equation (16). It therefore incorporates evidence on the time series of returns as well as option prices. Its properties are reported in Tables 3 and 2 and Figure 4. We compare it with consumption-based models in the next section, but for now simply note its salient features.

The most striking feature of the pricing kernel is its entropy of 0.7747, more than an order of magnitude larger than the equity premium (0.0400) [column (3) of Table 3]. This reflects, in large part, the high price of options. Prices are high in the sense that selling them generates high average returns; see, for example, the extensive literature review in Broadie, Chernov, and Johannes (2009, Appendix A). These high average returns imply high entropy via the entropy bound, even though the model’s parameters are chosen to match the equity premium exactly. Evidently a bound based on the equity premium is too loose: other investment strategies generate significantly higher average excess returns and therefore imply higher entropy.

The large entropy is derived from the large variance of the log-pricing kernel: the contribution is 0.4720, which is 61% of the entropy. High-order cumulants have a smaller, but still substantial contribution of 0.1127 (15%) and 0.1900 (25%) from, respectively, odd and even high-order cumulants (Table 2). Like the smile itself, these numbers verify that departures from the lognormal model are quantitatively important.

Figure 4 illustrates the impact of individual cumulants. The top panel shows that high-order cumulants of equity excess returns are small relative to the variance. We know, however, that the model generates nonzero skewness and excess kurtosis (Table 3). Contributions of high-order cumulants to entropy are reported in the second panel. As we noted above, the contributions are small relative to the variance but quantitatively important. When we divide the jump mean θ^{∗} and variance δ^{∗2} by two (the third panel), the contributions decline across the board, much as when we reduce risk aversion in consumption-based models.

6 Comparing consumption- and option-based models

We have seen that macroeconomic data and option prices both suggest significant departures from the lognormal model. In this section, we explore their differences. As we have seen in the earlier sections, for our purposes, the macro and option-based modelling approaches are two steps removed from each other: the latter characterises the risk-neutral distribution of equity returns rather than the true distribution of consumption growth that used by the former. Nonetheless, each approach features a specification of the true distribution p, the risk-neutral distribution p^{∗} and their link via the pricing kernel m. Any two of the three objects imply the remaining third (section 2.4). We exploit this observation to explore the implications of the approaches from three perspectives. First, we use entropy to compare the macro-based pricing kernel to the one derived from the true and risk-neutral distributions of returns implicit in option prices. Second, we use true distribution of consumption (based on international macro evidence) and macro-based pricing kernel to characterise the riskneutral distribution of returns based on the setup of Sections 3 and 4. Third, we use the risk-neutral distribution of returns implicit in options (Section 5) and the macro-based pricing kernel to infer the true distribution of consumption.

To implement the comparisons, we need one last bit of theory that connects the observed equity premuim to the risk aversion α. This will allow us to calibrate α. We define levered equity as a claim to the dividend

d_{t} = c_{t}^{λ}.  (28)

This is not, of course, either equity or levered, but it is a convenient functional form that is widely used in the macro-finance literature to connect consumption growth (the foundation for the pricing kernel) to returns on equity (the asset of interest). See Abel (1999, Section 2.2). In the iid case, the log excess return is a linear function of log consumption growth:

log r_{t+1}^{e} − log r_{t+1}^{1} = λ log g_{t+1} + constant.  (29)

See Appendix A.7.1. This tight connection between equity returns and consumption growth overstates how closely these two variables are related, but it captures in a simple way the obvious cyclical variation in the stock market. We consider alternatives to equation (29) later in this section, but for now it is a useful simplification. The leverage parameter λ allows us to control the variance of the equity return separately from the variance of consumption growth and thus to match both. We use an excess return variance of 0.1800^{2}, so λ is the ratio of the standard deviation of the excess return (0.1800) to the standard deviation of log consumption growth (0.0350), approximately 5.1. Given these assumptions, Appendix A.7.1 derives the equity premium in terms of the cumulant-generating function of consumption growth:

E (log r_{t+1}^{e} − log r_{t+1}^{1}) = λκ_{1}(log g) + k(−α; log g) − k(λ − α; log g)  (30)

6.1 Pricing kernels

Consider a direct comparison of the pricing kernels of consumption- and option-based models. We use calibrations discussed in sections 3 and 5 to compute the respective entropies [columns (2) and (3) of Table 3]. Equation (30) yields α = 5.19 for the consumption-based model. One difference is total entropy, which is much larger in the option model. Another is the relative contribution of high-order cumulants, which is significantly smaller in the option model. High-order cumulants contribute 59% of the entropy in the consumption model (the row of Table 2 labelled “Poisson consumption growth, α = 5.19”) but only 39% in the option model (the row labelled “Merton equity returns”). Most relevant to disaster research, the contribution of odd high-order cumulants is 38% in the consumption model but only 15% in the option model. These numbers indicate significant departures from lognormality in both models, but they are relatively smaller in the option model.

6.2 Option-pricing in a consumption-based model

A second comparison between the two models involves option prices implied by the consumption model: how do they compare to those implied by the option model? This comparison focuses on the risk-neutral distribution of returns. We compute that for the consumption model in two steps. First, the parameters of the risk-neutral distribution of log consumption growth follow from power utility and the transformations described in equation (25). The results are reported in column (2) of Table 3. Second, equation (29) implies that equity excess returns are a scaled version of consumption growth with scale parameter λ. This scaling leads us to replace the parameters (σ^{∗}, ω^{∗}, θ^{∗}, δ^{∗}) with (λσ^{∗}, ω^{∗}, λθ^{∗}, λδ^{∗}). See Appendix A.5. The result has the same form as the Merton model but different parameters.

Implied volatility smiles for the consumption- and option-based models are pictured in Figure 5. Similar consumption-based option prices are reported by Benzoni, Collin-Dufresne, and Goldstein (2005) and Du (2008). What’s new is the explicit comparison to an estimated option pricing benchmark. As before, we use 3-month options to highlight departures from the lognormal model. The top line (labelled “option-based model”) refers to the model based on option prices. It’s the same as the top line in Figure 3. The bottom line (labelled “consumption-based model”) refers to the model derived from consumption data as described in the previous paragraph. The two models are clearly different. The consumption-based model has a steeper smile, greater curvature, and lower at-the-money volatility. This follows, in part, from its greater risk-neutral skewness and excess kurtosis [columns (3) and (4) of Table 3]. They suggest higher risk-neutral probabilities of large disasters (the left side of the figure) and lower probabilities of less extreme outcomes (the middle and right of the figure). These differences in the underlying distributions result in significantly different option prices.

6.3 Consumption implicit in an option-based model

Now consider the reverse: the consumption growth process implied by option prices. Here the focus is on the true distribution of consumption growth. For the option model, this involves taking the risk-neutral distribution of returns implied by option prices and computing the true distribution of consumption growth. We no longer need an estimate of the true distribution from returns data. Instead, we use power utility to link true to riskneutral parameters and equation (29) to link consumption growth to returns. This imposes different structure than the option model on the connection between true and risk-neutral parameters. For example, the restriction ω = ω^{∗} used in option model no longer holds.

The calculations are analogous to the previous comparison. First, we use the scale parameter λ to transform risk-neutral parameters for equity returns into risk-neutral parameters for consumption growth. This involves replacing (ω^{∗}, θ^{∗}, δ^{∗}) in column (3) of Table 3 with (ω^{∗}, θ^{∗}/λ, δ^{∗}/λ) for reasons outlined in Appendix A.5. Second, equations (25) link the risk-neutral parameters of consumption to their counterparts (ω, θ, δ) of the true distribution. Finally, μ and σ should be selected to match our target value for the mean and standard deviation of log consumption growth using equations (5) and (6). Risk aversion α should be selected to match the equity premium (30). The two true jump parameters ω and θ depend on α, the two true normal parameters μ and σ depend on the true jump parameters, and α depends on all the true parameters. Thus, we have to solve a system of five equations (with five unknowns).

The consumption process derived this way from option prices [column (4) of Table 3] has the same Poisson-normal distribution as the consumption process estimated from international macroeconomic evidence [column (2)]. The parameters, however, are much different. In the consumption-based model, there is a small chance (governed by the jump intensity ω = 0.01) of a large jump (the mean jump θ = −0.3 is 8.6 standard deviations of consumption growth). In the option-based model, there is a larger chance (jump intensity is ω = 1.3987) of a much smaller jump (the mean jump θ = −0.0074 is 0.21 standard deviations). Both models generate disasters in the sense that the probabilities of tail events are much larger than in the lognormal case [column (1)]. The probability of a 3 standard deviation drop in consumption (b = −κ_{1} + 3κ_{2}^{1/2} in (9)), similar to the US in the Great Depression, is about 1% in each case. However, declines in consumption of more than 5 standard deviations are much more likely in the consumption-based model (probability 0.0079) than in the option-based model (0.0001). Events of this magnitude have not been observed in US history, so the models disagree on events that have never occurred. The difference in tail probabilities is reflected in their cumulants. Skewness and excess kurtosis are −11.01 and 145.06, respectively, in the consumption-based model, but only −0.28 and 0.48 in the option-based model. The latter set of numbers is more closely aligned with sample statistics reported in Table 1.

In summary, the reverse exercise has produced a new calibration of consumption in the Barro (2006) model. This calibration can (i) match the equity premium, (ii) match implied volatilities and (iii) be consistent with what we observe about consumption. Points (i) and (ii) are direct implication of our parameter choice as outlined above. The point (iii) is a property implied by the new parameters governing the jump model of consumption. The new parameters of the true distribution of consumption were derived from the risk-neutral parameters of returns that are needed for accurate option valuation.

6.4 A bivariate model of consumption and equity returns

So far, our analysis in this section has relied on the tight connection between consumption growth and equity returns in (29). We have followed a long tradition in tying dividends to consumption. The tradition is largely a matter of convenience, because it is simpler to have one random variable rather than two, but it is motivated by the evidence connecting equity returns to business cycles. Still, it is possible to explore the possibility of imperfect correlation between consumption growth and equity returns. Figure 6 shows how the two are related in US data. There is a clear association between business cycles (represented by consumption growth) and equity returns, but the association is not perfect. The sample correlation is 0.566.

Consider, then, a bivariate model of consumption growth and dividends with arbitrary correlation between the two. Does allowing imperfect correlation affect our conclusion that consumption evidence implies more extreme outcomes than option prices? The answer is no, but it is worth working through the details. Let log g′ = (log g_{1}, log g_{2}) follow

log g_{t+1} = μ^{g} + w_{t+1} + z_{t+1},

where g_{1t+1} = c_{t+1}/c_{t} and g_{2t+1} = d_{t+1}/d_{t} are the consumption and dividend growths, respectively, and both w and z are bivariate and iid over time. The constant μ^{g} has elements μ_{i}^{g}. The first random component is bivariate normal: w_{t} ∼ N (0, Σ), where Σ has elements σ_{ij}. The second component is a Poisson mixture of bivariate normals. Jumps occur with Poisson intensity ω. Each jump generates a draw from the bivariate normal distribution N (θ, Δ), where θ and Δ have elements θ_{i} and δ_{ij}. This is direct extension of the consumption-dividend relation in (28). The parameters of the dividend process are scaled versions of the ones controlling the consumption process, as in the case of perfect correlation between consumption and divdends: μ_{2}^{g} = λμ_{1}^{g}, σ_{22} = λ^{2}σ_{11}, θ_{2} = λθ_{1}, and δ_{22} = λ^{2}δ_{11}. This process is a special case of one used by Ait-Sahalia, Cacho-Diaz, and Hurd (2009) and similar to consumption-dividend processes used by Gabaix (2010) and Longstaff and Piazzesi (2004).

Appendix A.7.2 then derives the resulting process for equity returns. In particular, the equity premium is:

E (log r_{t+1}^{e} − log r_{t+1}^{1}) = λ(μ_{1}^{g} + ωθ_{1}) + k((−α, 0)′; log g) − k((−α, 1)′; log g). (31)

Also, the joint process x′ = (x_{1}, x_{2}) = (log g_{1}, log r^{e} − log r^{1}) follows a process similar to the consumption-dividend one:

x_{t+1} = μ + w_{t+1} + z_{t+1},

with w and z are the same as above and μ has elements (μ_{1}, μ_{2}) = (μ_{1}^{g}, E (log r_{t+1}^{e} − log r_{t+1}^{1})− ωλθ_{1}) (the component ωλθ_{1} of the equity premium is subtracted in μ_{2} because it shows up in x_{2} via the mean of the jump component z_{2}).

We choose parameter values that reproduce the mean and variance of log consumption growth, the mean and variance of the log excess return on equity, and the correlation between them. Consider the components in turn. We use the parameters reported in column (2) of Table 3 for the (true) consumption process. Here we label them (μ_{1}, σ_{11}, ω, θ_{1}, δ_{11}). The parameters of the return process are scaled versions of the same numbers as described above. The jump intensity ω is, of course, the same. We choose risk aversion α to match equity premium (31). The correlation depends on the mean jumps and the correlations ρ_{w} = σ_{12}/(σ_{11}σ_{22})^{1/2} and ρ_{z} = δ_{12}/(δ_{11}δ_{22})^{1/2}. We assume that ρ_{w} = ρ_{z} = ρ. With all these assumptions the expression for correlation between consumption and returns in Appendix A.8 specializes to

Corr(x_{1}, x_{2}) = \frac{ρσ_{11} + ω(θ_{1}^{2} + ρδ_{11})}{σ_{11} + ω(θ_{1}^{2} + δ_{11})}.  (32)

We select ρ to match the correlation in the data.

Option pricing follows directly from the risk-neutral distribution over equity returns. With power utility,

ω^{∗} = ωe^{−αθ_{1}+α^{2}δ_{11}/2}, θ_{2}^{∗} = θ_{2} − αδ_{12}, δ_{22}^{∗} = δ_{22}.

See Appendix A.8. With appropriate redefinition of parameters, option pricing has the same structure as Section 5 and Appendix A.4.

If consumption growth and equity returns are perfectly correlated, this procedure reproduces the calculation of consumption-based option prices in Figure 5. For the correlation observed in the data, the result is the middle line in the same figure. We see that the two consumption-based lines are similar to each other and notably different from the smile estimated from option prices. It appears, then, that the initially assumed perfect correlation between consumption growth and the return on equity is not the source of the sharp difference between volatility smiles based on option prices and consumption data.

Now consider the consumption growth process implied by option prices as was done in the perfect consumption-dividend correlation case. First, we use the scale parameter λ to transform risk-neutral parameters for equity returns into risk-neutral parameters for consumption growth. This involves setting (ω^{∗}, θ_{1}^{∗}, δ_{11}^{∗2}) = (ω^{∗}, θ^{∗}/λ, δ^{∗2}/λ^{2}) for reasons outlined in Appendix A.5, where (ω^{∗}, θ^{∗}, δ^{∗}) are equity paramters estimated from option prices (column (3) of Table 3). Second, equations

ω^{∗} = ωe^{−αθ_{1}+α^{2}δ_{11}/2}, θ_{1}^{∗} = θ_{1} − αδ_{11}, δ_{11}^{∗} = δ_{11}

from Appendix A.8 link the risk-neutral parameters of consumption to their counterparts (ω, θ_{1}, δ_{11}) of the true distribution. The normal component of consumption growth, μ_{1} and σ_{11} should be selected to match our target value for the mean and standard deviation of log consumption growth using equations

0.02 = κ_{1}(log g_{1}) = μ_{1} + ωθ_{1}
0.035^{2} = κ_{2}(log g_{1}) = σ_{11} + ω(θ_{1}^{2} + δ_{11}).

Finally, the correlation ρ should be selected to match sample correlation between consumption and returns using (32). Risk aversion α should be selected to match the equity premium (31). The two true jump parameters ω and θ_{1} depend on α, the two true normal parameters μ_{1} and σ_{11} depend on the true jump parameters, ρ depends on true jump parameters and σ_{11}, and α depends on all the true parameters. Thus, we have to solve a system of six equations (with six unknowns).

As in the univariate case, the consumption process derived this way from option prices [column (5) of Table 3] has the same Poisson-normal distribution as the consumption process estimated from international macroeconomic evidence [column (2)]. The parameters are, however, different. With the exception of risk aversion, the difference between columns (2) and (5) is similar to the difference between columns (2) and (4), the latter being the univariate case. The need for higher risk aversion is understandable: less than perfect exposure of equities to consumption risk would generate a lower equity premium otherwise. Again, it seems that, at a qualitative level, perfect correlation between consumption growth and the return on equity that was assumed initially is not the source of the large differences between the true distributions of consumption based on option prices and consumption data.

6.5 Discussion

One might ask, in the end, whether options reliably identify extreme events (disasters) in consumption. We think the answer is yes, but let’s run through the argument. There are two degrees of separation between equity options and consumption: options identify the risk-neutral distribution of returns and we’re interested in the true distribution of consumption growth. The first link is between true and risk-neutral returns. We’ve seen that a direct estimate of the true distribution of returns shares with the estimated risk-neutral distribution modest values of skewness and excess kurtosis [column (3) of Table 3]. The second link is between returns and consumption growth. When the two are perfectly correlated, as they are in equation (29), their skewness, excess kurtosis, and tail probabilities (measured in standard deviation units) are the same. The contrast along these dimensions between distributions based on international consumption data [column (2)] and option prices [columns (3) – (5)] is striking. So, too, is the difference between implied volatilities based on option prices and consumption data (Figure 5). Finally, the evidence on the relation between returns and consumption growth (Figure 6) and the limited impact of imperfect correlation on option prices (Figure 5) and on option-implied consumption growth (columns (4) and (5) of Table 3) suggest that option prices are a reasonably good indicator of the likelihood of disasters in consumption growth. It is possible that future work using other methods will detect a significant difference between the tail behavior of consumption growth and equity returns. In the meantime, we think the evidence indicates that equity index options imply smaller probabilities of consumption disasters than the international macroeconomic evidence.

We have described, in a relatively simple theoretical setting, how option prices can be used to infer probabilities of extreme outcomes, including the infrequent sharp declines in consumption growth documented in international macroeconomic data by Barro and others. We find that the distribution of outcomes implied by option prices is less extreme than the macroeconomic evidence suggests. Nonetheless, this option-implied distribution is capable of matching the equity premium, option prices and consumption moments in a simple setting of iid consumption growth and power utility over aggregate consumption.

Can we improve upon the findings by relaxing these assumptions? In general, the answer should be yes as the most recent generation of dynamic asset pricing models is capable of capturing important features of the data by using some combination of exotic preferences and non - iid consumption dynamics. However, this work focuses on intertemporal properties of the asset prices, such as return predictability, volatility persistence and yield curve modelling. Here, we are intentionally interested in understanding the drivers of the basic unconditional moments of consumption and asset prices that are listed in Table 1. We feel that, though obviously not the final word, the models we consider are well suited for the task given their transparency and the aforementioned success in understanding these basic moments.

Consider the iid assumption. Our objective is to characterize the unconditional distribution of consumption growth, particularly the distribution of large adverse outcomes. The question is whether the kinds of time-dependence we see in asset prices are quantitatively important in assessing the role of extreme events. It’s hard to make a definitive statement without knowing the precise form of time-dependence, but there’s good reason to think its impact could be small. The leading example in this context is stochastic volatility, a central feature of the option pricing model estimated by Broadie, Chernov, and Johannes (2007). However, average implied volatility smiles from this model are very close to those from an iid model in which the variance is set equal to its mean. Furthermore, stochastic volatility has little impact on the probabilities of tail events, which is our interest here.

Power utility is the workhorse of macroeconomics and finance, but our option model suggests much higher entropy than implied by macro models with power preferences, even the ones that were calibrated to match option prices. One possibility to remedy this is to explore alternative preferences, including skewness preference (Harvey and Siddique, 2000), recursive preferences (Garcia, Luger, and Renault, 2002, and Wachter, 2009), statedependent preferences (Chabi-Yo, Garcia, and Renault, 2008), ambiguity (Drechsler, 2008, and Liu, Pan, and Wang, 2005), learning (Shaliastovich, 2008), and habits (Bekaert and Engstrom, 2010, and Du, 2008). Another promising avenue is heterogeneity across agents. Certainly there is strong evidence of imperfect risk-sharing across individuals and good reason to suspect that this affects asset prices. Bates (2008), Chan and Kogan (2002), Guvenen (2009), Longstaff and Wang (2008), and Lustig and Van Nieuwerburgh (2005) are notable examples. The question for future work is whether these extensions provide a persuasive explanation for prices of equity index options.

7 Final remarks

We have used prices of equity index options to infer probabilities of large negative realizations of consumption growth that we can compare to the macroeconomic evidence summarized by Barro and coauthors. The exercise faces two issues from the start: equity index options refer to equity, not consumption, and they reflect risk-neutral rather than true probabilities. We address these issues in a number of ways. All of them indicate that the probabilities of extreme adverse events implied by option prices are smaller than we see in international macroeconomic data. On the broader question of the role of disasters in asset pricing, we find that a pricing kernel constructed from option prices includes substantial contributions from high-order cumulants. In this sense, the departures from lognormality suggested by the disaster literature remain quantitatively important in the option pricing model. Furthermore, the contribution of odd high-order cumulants is suggestive of the market pricing of return asymmetries noted in research on skewness preference.

This exercise had two useful byproducts. One is the reminder that matching the equity premium may not be enough. There’s growing evidence that other trading strategies can generate average returns that are substantially higher, so that models designed to account for the equity premium may not be able to account for higher returns on other assets. The other is the value of transform methods, particularly cumulant-generating functions. These are not new to finance, but they are nevertheless extremely helpful. We find them not only a source of intuition and compact notation, but a convenient approach to a number of practical problems.

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Table 1 Properties of consumption growth and asset returns

Variable  Mean Std Dev Skewness Kurtosis

(a) Consumption growth and returns, annual, 1889-2009

Consumption growth  0.0198  0.0350  −0.34  1.11
Return on equity  0.0587  0.1795  −0.61  0.43
Excess return on equity 0.0407  0.1812  −0.72  0.91

(b) Consumption growth and returns, annual, 1986-2009

Consumption growth  0.0178  0.0150  −0.87  0.66
Return on equity  0.0641  0.1845  −1.24  1.56
Excess return on equity 0.0434  0.1808  −1.39  2.01

Notes. Entries are statistics computed from annual observations for the US economy. Mean is the sample mean, Std Dev is the standard deviation, Skewness is the standard measure of skewness, Kurtosis is the standard measure of excess kurtosis. Consumption growth is log(c_{t}/c_{t−1}) where c is real per capita consumption. Returns are logarithms of gross real returns and the excess return is the difference between the log-returns on equity and the one-year bond. The one-year bond is the Treasury security of maturity closest to one year. Equity is the S&P 500. Consumption and return data are from Shiller (2009).

Table 2 Components of entropy for model economies

High-Order Cumulants

Model  Entropy  Variance/2  Odd  Even

Normal consumption growth

α = 2  0.0025  0.0025  0  0
α = 5  0.0153  0.0153  0  0
α = 10  0.0613  0.0613  0  0
α = 8.92^{∗}  0.0487  0.0487  0  0

Poisson consumption growth

α = 2  0.0033  0.0025  0.0007  0.0002
α = 5  0.0356  0.0153  0.0132  0.0071
α = 10  0.5837  0.0613  0.2786  0.2439
α = 10, θ = +0.3 (boom)  0.0265  0.0613  –0.2786  0.2439
α = 5.19^{∗}  0.0400  0.0165  0.0151  0.0084

Models fit to option prices

Merton equity returns  0.7747  0.4720  0.1127  0.1900
Implied consumption growth  0.0478  0.0464  0.0013  0.0002

Notes. Entries include entropy of the pricing kernel and its components for a variety of models. Entropy is the sum of contributions from the variance and from odd and even high-order cumulants (those of order j ≥ 3). An asterisk denotes a value of α that matches the observed equity premium.

Table 3 Parameter values and properties of model economies

Bivariate

Lognormal Poisson Merton Implied  Implied
Cons Gr Cons Gr Returns Cons Gr Cons Gr
Parameter  (1)  (2)  (3)  (4)  (5)

Preferences

α 8.92 5.19 — 8.70 14.94

True distribution

μ  0.0200  0.0230  0.0792  0.0303  0.0269
σ  0.0350  0.0100  0.1699  0.0253  0.0265
ω  —  0.0100  1.5120  1.3987  1.3452
θ  — −0.3000 −0.0259 −0.0074  -0.0051
δ  —  0.1500  0.0407  0.0191  0.0191

Risk-neutral distribution

μ^{∗}  0.0091  0.0225  0.0584  0.0247  0.0233
ω^{∗}  —  0.0642  1.5120  1.5120  1.5120
θ^{∗}  — −0.4168 −0.0542 −0.0105  -0.0105
δ^{∗}  —  0.1500  0.0981  0.0191  0.0191

Skewness, excess kurtosis, and tail probabilities

γ_{1} (true skewness)  0  −11.02  −0.04  −0.28  -0.19
γ_{2} (true kurtosis)  0  145.06  0.02  0.48  0.41
γ_{1}^{∗} (risk-neutral skewness)  0  −4.51  −0.25  −0.38  -0.36
γ_{2}^{∗} (risk-neutral kurtosis)  0  21.96  0.30  0.53  0.49
γ_{1} (log m skewness)  0  11.02  −0.12  0.28  0.18
γ_{2} (log m kurtosis)  0  145.06  2.21  0.48  0.41
P (D_{b}) (tail prob ≤ −3 st dev)  0.0013  0.0090  0.0032  0.0081  0.0056
P (D_{b}) (tail prob ≤ −5 st dev)  0.0000  0.0079  0.0000  0.0001  5 · 10^{−5}

Entropy

L(m) = L(p^{∗}/p)  0.0487  0.0400  0.7747  0.0478  0.1424

Notes. Entries are parameters and properties of examples with different specifications of disasters. The labels at the top of the columns describe the model used and the variable on which it is based. Columns (1)-(2) and (4)-(5) are based on consumption growth. In each one, log consumption growth has a mean of 0.0200 and a standard deviation of 0.0350. Risk aversion α is chosen to match the mean equity premium (0.0400). Column (3) is the Merton model parameterized to option prices and equity returns. Columns (4) and (5) take this model, scales the risk-neutral parameters to fit consumption growth, and sets the true parameters by applying the relations implied by power utility. γ_{1} and γ_{2} are the traditional measures of skewness and excess kurtosis, defined in equation (4). We report versions for the true distribution of log consumption growth or the log return on equity, the risk-neutral distribution, and the distribution of the pricing kernel. Tail probabilities refer to P (D_{b}) in (9) for b = −κ_{1} + Nκ_{2}^{1/2}, N = 3 or 5.

Figure 1 Poisson jumps: cumulants of log consumption growth and contributions to entropy

Notes. The panels graph terms in the power series expansion of entropy, equation (13), for the consumption-based asset pricing model with a Poisson-normal jump component. The top panel plots the jth cumulant of log consumption growth, κ_{j}(log g), against its order j. The next two panels plot the contribution to entropy of the jth term, κ_{j}(log m)/j! = (−α)^{j}κ_{j}(log g)/j!, against j for risk aversion α equal to 2 and 10, respectively. The model and parameter values are reported in Section 3 and column (2) of Table 3.

Figure 2 Poisson jumps: entropy of the pricing kernel

Notes. The lines represent versions of the consumption-based asset pricing model with Poisson jumps outlined in Section 3. The middle line is the lognormal model: log consumption growth is normal and there is no jump component. The top line shows how a Poisson component (an infrequent large negative realization) increases entropy. The bottom line shows how this changes when the jump is positive (an infrequent large positive realization).

Figure 3 Option model: implied volatility smiles for 3-month options

Notes. The lines represent implied volatility “smiles” for the Merton model with estimated parameters and some alternatives. Moneyness is measured as the proportional difference of the strike from the price, (strike − price)/price. For the solid line the parameters are those reported in column (4) of Table 3. For the second line, we have divided the jump mean θ^{∗} by two. For the bottom line, we have divided the jump variance δ^{∗2} by two.

Figure 4 Option model: cumulants of equity returns and contributions to entropy

Notes. The figure summarizes properties of the estimated Merton model using parameters reported in column (4) of Table 3. The top panel shows cumulants of the log excess return on equity based on its (estimated) true distribution. The second panel shows contributions to entropy of the cumulants of log m. The contribution of order j is κ_{j}(log m)/j! where κ_{j}(log m) is the jth cumulant of log m. The third panel is the same with θ^{∗} and δ^{∗2} each divided by two.

Figure 5 Option models: implied volatility smiles based on option and consumption data

Notes. The lines represent implied volatility “smiles” for the Merton model with three different sets of parameters. The top line is the one we saw in Figure 3 and uses parameters estimated from option prices. The bottom line uses parameters estimated from consumption data, extrapolated to equity returns using equation (29). The middle line is based on a bivariate model of consumption growth and equity returns with correlation chosen to match US data.

Figure 6 Equity returns and consumption growth in US data

Notes. The dots represent annual observations of the log excess return on equity and the log of per capita consumption growth over the period 1889-2009. The data come from Shiller (2009), updated as needed.

A Appendix

A.1 The cumulant-generating function of Poisson-normal mixtures

We’ll look at a Poisson-normal mixture shortly, but it is useful to start with a Poisson random variable z that equals j with probability e^{−ω}ω^{j}/j! for j = 0, 1, 2, . . .. Recall that the power series representation of the exponential function is

e^{ω} = ∑_{j=0}^{∞} ω^{j}/j!.

From this we see that the probabilities sum to one. The moment-generating function is

h(s) = ∑_{j=0}^{∞} e^{−ω}ω^{j}/j!e^{sj} = ∑_{j=0}^{∞} e^{−ω}(ωe^{s})^{j}/j! = exp[ω(e^{s} − 1)].

The cumulant-generating function is therefore

k(s) = log h(s) = ω(e^{s} − 1).

Cumulants follow by differentiating.

The Poisson-normal mixture has a similar structure. Conditional on j, z is normal with mean jθ and variance jδ^{2}. The conditional moment-generating function is exp[(sθ + s^{2}δ^{2}/2)j]. The mgf for the mixture is the probability-weighted average,

h(s) = ∑_{j=0}^{∞} e^{−ω}ω^{j}/j! exp[(sθ + s^{2}δ^{2}/2)j] = exp (ω[e^{sθ+(sδ)^{2}/2} − 1]) ,

which implies the cgf

k(s) = ω [e^{sθ+(sδ)^{2}/2} − 1] .

The same approach can be used for jumps with other distributions. If we set θ = 1 and δ = 0, we get the cgf of the original Poisson.

We find cumulants by taking derivatives of k. The first five are

κ_{1} = ωθ
κ_{2} = ω(θ^{2} + δ^{2})
κ_{3} = ωθ(θ^{2} + 3δ^{2})
κ_{4} = ω(θ^{4} + 6θ^{2}δ^{2} + 3δ^{4})
κ_{5} = ωθ(θ^{4} + 10θ^{2}δ^{2} + 15δ^{4}).

Here you can see that the sign of the odd moments is governed by the sign of θ. Negative odd cumulants evidently require θ < 0.

A.2 The entropy bound

The entropy bound (12) is derived by Alvarez and Jermann (2005) as a byproduct of their Proposition 2. Bansal and Lehmann (1997, Section 2.3) have a similar result that treats variation in the short rate differently [the term L(q^{1}) in (35) below]. We derive the bound like this:

• Bound on mean log return. Since log is a concave function, Jensen’s inequality and the unconditional version of the pricing relation (10) imply that for any positive return r,

E log m + E log r ≤ log(1) = 0,

with equality if and only if mr = 1. Therefore no asset has higher expected (log) return than the inverse of the pricing kernel:

E log r ≤ −E log m.  (33)

The asset with this return is sometimes called the “growth optimal portfolio.” We call it the “high-return asset.”

• Short rate. A one-period (risk-free) bond has price q_{t}^{1} = E_{t}m_{t+1}, so its return is r_{t+1}^{1} = 1/E_{t}m_{t+1}.

• Entropy of the one-period bond price. With the bound in mind, our next step is to express E log r^{1} in terms of unconditional moments. The entropy of the one-period bond price does the trick:

L(q^{1}) = log Eq^{1} − E log q^{1} = log Em + E log r^{1}.  (34)

• Entropy bound. (33) and (34) imply

L(m) ≥ E (log r^{j} − log r^{1}) + L(q^{1}).  (35)

Inequality (12) follows from L(q^{1}) ≥ 0 (entropy is nonnegative). In practice, L(q^{1}) is small; in the iid case, it is zero.

We find the loglinear perspective of the entropy bound convenient, but the familiar Hansen-Jagannathan bound also depends (implicitly) on high-order cumulants of log m. The bound is

Var(m)^{1/2}/Em ≥ E (r^{j} − r^{1}) /Var(r^{j} − r^{1})^{1/2} = SR,

where SR is the Sharpe ratio. If k(s) is the cumulant generating function for log m, the bound depends on

Em = E (e^{log m}) = e^{k(1)}
Var(m) = E(m^{2}) − (Em)^{2} = e^{k(2)} − e^{2k(1)},

Since k(1) and k(2) involve high-order cumulants of log m, the bound does, too. The squared Sharpe ratio is bounded below by

Var(m)/E(m)^{2} = e^{k(2)−2k(1)} − 1.

If the cumulants are small (true for a small enough time interval), this is approximately k(2) − 2k(1). Expressed in similar form, entropy is k(1) − k′(0).

A.3 Risk neutral distributions with power utility

A similar approach reveals the connection between true and risk-neutral cumulants of log consumption growth log g = w (w because it is easier to type). The cumulant generating function for the true distribution is

k(s) = log E (e^{sw}) .

The pricing kernel is m(w) = βe^{−αw}, which implies q^{1} = βk(−α). Risk-neutral probabilities are p^{∗}(w) = p(w)m(w)/q^{1} = p(w)e^{−αw}/k(−α) . The cumulant generating function is therefore

k^{∗}(s) = k(s − α) − k(−α).  (36)

This is a standard math result. We find its cumulants by differentiating:

κ_{n}^{∗} = ∑_{j=0}^{∞} κ_{n+j}(−α)^{j}/j!.

Thus risk-neutral cumulants depend on higher-order true cumulants. Positive excess kurtosis, for example, reduces risk-neutral skewness.

A.4 Risk-neutral option pricing

We review option pricing in the Merton model, starting with its primary ingredient, the Black-Scholes-Merton formula. For convenience, we define options on returns rather than prices and drop the time subscripts. All of the parameters in what follows refer to the risk-neutral distribution.

Let the risk-neutral distribution of the return on an arbitrary asset be lognormal: log r ∼ N (log r^{1} + κ_{1}, κ_{2}). The pricing relation (15) implies the restriction κ_{1} + κ_{2}/2 = 0, which we’ll hold in reserve. The BSM formula is the solution to

q^{p} = q^{1}E^{∗}(b − r)^{+}.

The implicit integral on the right includes the terms

q^{1}b Prob(r ≤ b) = q^{1}bN(d)
−q^{1}E^{∗}(r|r ≤ b) = −q^{1} ∫_{−∞}^{log b} e^{log r}(2πκ_{2})^{−1/2} exp[−(log r − log r^{1} − κ_{1})^{2}/2κ_{2}] d log r
= − exp(κ_{1} + κ_{2}/2)N(d − κ_{2}^{1/2}),

with

d = (log b − log r^{1} − κ_{1})/κ_{2}^{1/2}

and N is the standard normal cdf. We use this to define the function

q^{p}(b) = f(b; κ_{1}, κ_{2}) = q^{1}bN(d) − exp(κ_{1} + κ_{2}/2)N(d − κ_{2}^{1/2}).  (37)

In the conventional BSM formula, we set κ_{1} + κ_{2}/2 = 0 and simplify, but this version is more useful in what follows.

The Merton model with normal jumps is a Poisson-weighted average of BSM option prices. The model is described in Section 5.1 and has (risk-neutral) parameters (μ, σ, ω, θ, δ). The first two pertain to the normal component, the remainder to the Poisson-normal mixture. Option prices in this setting are

q^{p}(b) = ∑_{j=0}^{∞}(e^{−ω}ω^{j}/j!)f(b; κ_{1j}, κ_{2j})

with κ_{1j} = μ + jθ and κ_{2j} = σ^{2} + jδ^{2}.

A.5 Two scaling issues

Two scaling issues come up in the paper. The first is the relation between equity returns and consumption growth: for most of the paper, log equity returns are a linear function of log consumption growth. The second is time: option prices for intervals other than one year depend on the distribution of returns over other time intervals.

Consider the relation between the distributions of x and λx for some scale factor λ. Consumption growth and equity returns have this structure if we ignore intercepts [equation (29)]. The general result follows from this property of cgfs: k(s; λx) = k(λs; x). If x (think log consumption growth) has the Poisson-normal structure of Section 2.2, its cgf is

k(s; x) = μs + (σs)^{2}/2 + ω[e^{θs+(δs)^{2}/2} − 1].  (38)

The cgf for λx (think excess returns) is therefore

k(s; λx) = μλs + (σλs)^{2}/2 + ω[e^{θλs+(δλs)^{2}/2} − 1].

This has the same form with (μ, σ, ω, θ, δ) replaced by (λμ, λσ, ω, λθ, λδ). A similar result applies to the relation between the true distribution of x = log g and the risk-neutral distribution of λx with power utility. Given (36), their cgfs are connected by

k^{∗}(s; λx) = k^{∗}(λs; x) = k(λs − α; x) − k(−α; x).

In words, we compute the cgf of the risk-neutral distribution of λx by, first, computing the cgf of the risk-neutral distribution of x and, second, scaling by λ. It’s important the steps be done in that order.

The second issue concerns the time interval. In an iid setting, suppose the cgf (38) applies to the distribution over a unit time interval. The cgf for an arbitrary time interval τ > 0, if it exists, is the cgf for a time interval of one multiplied by τ. In the Poisson-normal case, we have

k(s; τ) = τμs + τ(σs)^{2}/2 + τω[e^{θs+(δs)^{2}/2} − 1].

The cgf has the same form as (38) with (μ, σ^{2}, ω, θ, δ^{2}) replaced by (τμ, τσ^{2}, τω, θ, δ^{2}).

A.6 Cumulant-generating functions based on true and risk-neutral probabilities

We derive the salient features of models in which the true and risk-neutral distributions are Poisson mixtures of normals with different parameters.

We start with a normal example that serves as a component of the Poisson mixture. Let the log return follow (26), where z = 0 and w has true distribution of N (μ, σ^{2}) and risk-neutral distribution N (μ^{∗}, σ^{∗2}). The density functions are

p(w) = (2πσ^{2})^{−1/2} exp[−(w − μ)^{2}/2σ^{2}]
p^{∗}(w) = (2πσ^{∗2})^{−1/2} exp[−(w − μ^{∗})^{2}/2σ^{∗2}].

This differs from the examples in Section 4 in allowing the variance to differ between the two distributions. In continuous time, σ^{∗} = σ is needed to assure absolute continuity of the true and risk-neutral probability measures with respect to each other. In discrete time, there is no such requirement; see, for example, Buhlmann, Delbaen, Elbrechts, and Shiryaev (1996). The risk-neutral pricing relation (15) implies μ^{∗} + σ^{∗2}/2 = 0.

We can derive all of the relevant properties from these inputs. The log probability ratio is

log[p^{∗}(w)/p(w)] = (1/2) log φ + [(w − μ)^{2} − φ(w − μ^{∗})^{2}]/2σ^{2},

where φ = σ^{2}/σ^{∗2} > 0. The moment-generating function of the log probability ratio is

h(s; log p^{∗}/p) = E (e^{s log p^{∗}/p})
= ∫_{−∞}^{∞} p^{∗}(w)^{s}p(w)^{1−s}dw
= (2πσ^{2})^{−1/2}φ^{s/2} ∫_{−∞}^{∞} exp{−[(1 − s)(w − μ)^{2} + sφ^{2}(w − μ^{∗})^{2}]/2σ^{2}}dw
= φ^{s/2}[1 − s(1 − φ)]^{−1/2} exp (\frac{s(s − 1)(μ^{∗} − μ)^{2}}{2σ^{∗2}[1 − s(1 − φ)]})

for 1 − s(1 − φ) > 0 (automatically satisfied if s = 0 or s = 1). The last line follows from completing the square. Thus the cumulant-generating function is

k(s; log p^{∗}/p) = (s/2) log φ − (1/2) log[1 − s(1 − φ)] + (\frac{s(s − 1)(μ^{∗} − μ)^{2}}{2σ^{∗2}[1 − s(1 − φ)]}) .

Entropy is minus the first derivative evaluated at zero:

− κ_{1}(log p^{∗}/p) = (1/2)[log φ + 1 − φ] + (μ − μ^{∗})^{2}/2σ^{∗2}.  (39)

If φ = 1 (σ^{∗} = σ), we have

k(s; log p^{∗}/p) = s(s − 1)(μ^{∗} − μ)^{2}/2σ^{2},

and the only nonzero cumulants are the first two. Otherwise, high-order cumulants are generally nonzero.

Now let’s ignore the normal component and focus on z. Both the true and risk-neutral distributions have Poisson arrivals and normal jumps, but the parameters differ. Conditional on a number of jumps j, the density functions are

p(z|j) = e^{−ω}ω^{j}/j! · (2πjδ^{2})^{−1/2} exp[−(z_{j} − jθ)^{2}/(2jδ^{2})]
p^{∗}(z|j) = e^{−ω^{∗}}ω^{∗j}/j! · (2πjδ^{∗2})^{−1/2} exp[−(z_{j} − jθ^{∗})^{2}/(2jδ^{∗2})].

The moment generating function for log p^{∗}/p is

h(s; log p^{∗}/p) = ∑_{j=0}^{∞} e^{−ω}ω^{j}/j! [e^{s(ω−ω^{∗})+js log(ω^{∗}/ω)}h(s; z)^{j}] .

Using (39) we have

h(s; z) = φ^{s/2}[1 − s(1 − φ)]^{−1/2} exp (\frac{s(s − 1)(θ^{∗} − θ)^{2}}{2δ^{∗2}[1 − s(1 − φ)]}) ,

where φ = δ^{2}/δ^{∗2}. Therefore the cumulant-generating function is

k(s; log p^{∗}/p) = s(ω − ω^{∗})
+ ω [(ω^{∗}/ω)^{s}φ^{s/2}[1 − s(1 − φ)]^{−1/2} exp (\frac{s(s − 1)(θ^{∗} − θ)^{2}}{2δ^{∗2}[1 − s(1 − φ)]}) − 1] .

Entropy is minus the first derivative evaluated at zero:

− κ_{1}(log p^{∗}/p)  (40)
= (ω^{∗} − ω) + ω[log(ω/ω^{∗}) − 1/2 · log φ + 1/2 · (φ − 1)] + ω(θ − θ^{∗})^{2}/2δ^{∗2}.

Because the normal and Poisson mixture components are independent, their cumulantgenerating functions are additive. Therefore, the entropy for the full model is the sum of the entropy of the normal case [equation (39) with φ = 1] and the entropy of the Poisson mixture of normals [equation (40)].

A.7 Equity premium

Most of our analysis is loglinear, which allows us to express asset prices and returns as functions of cumulant-generating functions of (say) the log of consumption growth. The notation is wonderfully compact. The idea and many of the results follow Martin (2009).

A.7.1 Perfectly correlated consumption and dividend growth

Let’s start with the short rate. A one-period risk-free bond sells at price q_{t}^{1} = E_{t}m_{t+1} and has return r_{t+1}^{1} = 1/q_{t}^{1} = 1/E_{t}m_{t+1}. In the iid case, the short rate is constant and equals

log r^{1} = − log E(m)
= − log β − log E (e^{−α log g}) = − log β − k(−α; log g).

The second equality is based on the definition of the pricing kernel, equation (20). The last one follows from the definition of the cumulant-generating function k, equation (3).

We now turn to equity, defined as a claim to a dividend process d_{t} = c_{t}^{λ}. If the price-dividend ratio on this claim is q^{e}, the return is

r_{t+1}^{e} = g_{t+1}^{λ}(1 + q_{t+1}^{e})/q_{t}^{e}.

In the iid case, q^{e} is constant. The pricing relation (10) and our power utility pricing kernel (20) then imply

q^{e}/(1 + q^{e}) = E (βg^{λ−α}) = βE (e^{(λ−α) log g}) .

Thus we have, in compact notation,

log [q^{e}/(1 + q^{e})] = log β + k(λ − α; log g)
log r_{t+1}^{e} = λ log g_{t+1} − log β − k(λ − α; log g)
log r_{t+1}^{1} = − log β − k(−α; log g)
log r_{t+1}^{e} − log r_{t+1}^{1} = λ log g_{t+1} + k(−α; log g) − k(λ − α; log g).

The equity premium is therefore

E (log r_{t+1}^{e} − log r_{t+1}^{1}) = λκ_{1}(log g) + k(−α; log g) − k(λ − α; log g)
= L(e^{−α log g}) − L(e^{(λ−α) log g})
= ∑_{j=2}^{∞} κ_{j}(log g)[(−α)^{j} − (λ − α)^{j}]/j!.

The second line follows because the first-order cumulants cancel. The third is the usual cumulant expansion of entropy. They tell us that the equity premium is the entropy of the pricing kernel minus a penalty (entropy must be positive). It hits its maximum when λ = α, in which case equity is the high return asset.

A.7.2 A bivariate model of consumption and dividend growth

As before, one-period risk-free bond sells at price q_{t}^{1} = E_{t}m_{t+1} and has return r_{t+1}^{1} = 1/q_{t}^{1} = 1/E_{t}m_{t+1}. In the iid case, the short rate is constant and equals

log r^{1} = − log E(m)
= − log β − log E (e^{−α log g_{1}}) = − log β − k((−α, 0)′; log g).

The last equality follows from using a cumulant-generating function of a bivariate variable log g = (log g_{1}, log g_{2})′ given by

k(s) = log E(e^{s′ log g}) = s′μ^{g} + s′Σs/2 + ω [e^{s′θ+s′Δs/2} − 1] .

We now turn to equity, defined as a claim to a dividend process d_{t} If the price-dividend ratio on this claim is q^{e}, the return is

r_{t+1}^{e} = e^{log g_{2t+1}}(1 + q_{t+1}^{e})/q_{t}^{e}.

In the iid case, q^{e} is constant. The pricing relation (10) and our power utility pricing kernel (20) then imply

q^{e}/(1 + q^{e}) = E (βe^{−α log g_{1}+log g_{2}}) .

Thus we have, in compact notation,

log [q^{e}/(1 + q^{e})] = log β + k((−α, 1)′; log g)
log r_{t+1}^{e} = log g_{2t+1} − log β − k((−α, 1)′; log g)
log r_{t+1}^{1} = − log β − k((−α, 0)′; log g)
log r_{t+1}^{e} − log r_{t+1}^{1} = log g_{2t+1} + k((−α, 0)′; log g) − k((−α, 1)′; log g).

The equity premium is therefore

E (log r_{t+1}^{e} − log r_{t+1}^{1}) = κ_{1}(log g_{2}) + k((−α, 0)′; log g) − k((−α, 1)′; log g).

A.8 Risk-neutral distribution of returns in a bivariate model

Consider a bivariate model in which equity returns are (potentially) less closely tied to consumption growth. Let x′ = (x_{1}, x_{2}) = (log g, log r^{e} − log r^{1}) have the two-component structure used throughout the paper:

x_{t+1} = w_{t+1} + z_{t+1}.

The first component is bivariate normal: w_{t} ∼ N (μ, Σ) where μ and Σ have elements μ_{i} and σ_{ij}, respectively. The second component is a Poisson mixture of bivariate normals. As in Section 2.2, jumps occur with Poisson intensity ω so that the probability of j jumps is e^{−ω}ω^{j}/j!. Each jump adds a draw from the bivariate normal distribution N (θ, Δ).

Option pricing (indeed equity pricing) requires us to deal with the bivariate distribution of equity returns and the pricing kernel. We derive the risk-neutral distribution of returns from its cumulant-generating function. The (joint) cgf is

k(s) = log E(e^{s′x}) = s′μ + s′Σs/2 + ω [e^{s′θ+s′Δs/2} − 1] .

The logic here is virtually identical to the univariate case outlined in Appendix A.1. Derivatives of this expression lead to these formulas:

Var(x_{i}) = σ_{ii} + ω(θ_{i}^{2} + δ_{ii})
Cov(x_{1}, x_{2}) = σ_{12} + ω(θ_{1}θ_{2} + δ_{12}).

The correlation is

Corr(x_{1}, x_{2}) = \frac{σ_{12} + ω(θ_{1}θ_{2} + δ_{12})}{[σ_{11} + ω(θ_{1}^{2} + δ_{11})]^{1/2}[σ_{22} + ω(θ_{2}^{2} + δ_{22})]^{1/2}}
= \frac{ρ_{w}(σ_{11}σ_{22})^{1/2} + ω[θ_{1}θ_{2} + ρ_{z}(δ_{11}δ_{22})^{1/2}]}{[σ_{11} + ω(θ_{1}^{2} + δ_{11})]^{1/2}[σ_{22} + ω(θ_{2}^{2} + δ_{22})]^{1/2}}

with ρ_{w} = σ_{12}/(σ_{11}σ_{22})^{1/2} and ρ_{z} = δ_{12}/(δ_{11}δ_{22})^{1/2}.

The remaining step is to find the risk-neutral distribution for returns. With power utility, the cgf corresponding to the risk-neutral distribution is

k^{∗}(s_{1}, s_{2}) = k(s_{1} − α, s_{2}) − k(−α, 0).

The logic is analogous to Appendix A.3. The cgfs corresponding to marginal distributions follow from setting the other elements of s equal to zero. (This follows from the definitions of the marginal distribution and cgf.) Thus the cgf for x_{2} is k(0, s_{2}). The risk-neutral cgf for the log equity excess return is therefore

k^{∗}(0, s_{2}) = s_{2}(μ_{2} − ασ_{12}) + s_{2}^{2}σ_{22}/2 + ωe^{−αθ_{1}+α^{2}δ_{11}/2} [e^{s_{2}(θ_{2}−αδ_{12})+s_{2}^{2}δ_{22}/2} − 1]
= s_{2}μ_{2}^{∗} + s_{2}^{2}σ_{22}/2 + ω^{∗} [e^{s_{2}θ_{2}^{∗}+s_{2}^{2}δ_{22}/2} − 1]

with the implicit definitions

μ_{2}^{∗} = μ_{2} − ασ_{12}, θ_{2}^{∗} = θ_{2} − αδ_{12}, δ_{22}^{∗} = δ_{22}
ω^{∗} = ωe^{−αθ_{1}+α^{2}δ_{11}/2}.

This has the same form as the Merton model with suitably defined parameters. Options can therefore be priced using the methods of Appendix A.4. Similarly, k^{∗}(s_{1}, 0) implies the risk-neutral distribution of consumption growth:

μ_{1}^{∗} = μ_{1} − ασ_{11}, θ_{1}^{∗} = θ_{1} − αδ_{11}, δ_{11}^{∗} = δ_{11}.