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The pricing kernel puzzle: survey and outlook

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Abstract

It has been a while since the literature on the pricing kernel puzzle was summarized in Jackwerth (Option-implied risk-neutral distributions and risk-aversion, The Research Foundation of AIMR, Charlotteville, 2004). That older survey also covered the topic of risk-neutral distributions, which was itself already surveyed in Jackwerth (J Deriv 2:66–82, 1999). Much has happened in those years and estimation of risk-neutral distributions has moved from new and exciting in the last half of the 1990s to becoming a well-understood technology. Thus, the present survey will focus on the pricing kernel puzzle, which was first discussed around 2000. We document the pricing kernel puzzle in several markets and present the latest evidence concerning its (non-)existence. Econometric studies are detailed which test for the pricing kernel puzzle. The present work adds much breadth in terms of economic explanations of the puzzle. New challenges for the field are described in the process.

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Notes

  1. For the fit of a continuous time Levy-processes see Carr et al. (2002).

  2. Note that Friedman and Savage (1948) introduced their utility function for individuals and not for the representative investor. In particular, their concern was with small stakes gambling such as buying a lottery ticket. Chetty and Szeidl (2007) provide a microeconomic motivation for Friedman–Savage utility via consumption commitments (e.g. housing), for which the spending cannot easily be adjusted. Again, this is a model of individual investors, and it is not obvious that the convexities would survive aggregation to a representative investor. See Ingersoll (2014) for related results on another partially convex utility function, namely cumulative prospect theory.

  3. Zdorovenin and Pezier (2011) use a close variant, too, and are subject to the same critique as Kostakis et al. (2011).

  4. See also Lioui and Malka (2004) for reported differences due to using either only call or only put options.

  5. Compare Linn et al. (2014), who can only establish the pricing kernel puzzle for the FTSE 100 but not for the S&P 500. Cuesdeanu and Jackwerth (2017) attribute this result to (i) a lack of scaling so that the physical distributions of Linn et al. (2014) are not integrating to one and (ii) a mismatch in their optimization (based on moments of the uniform distribution via GMM) and their measurement of fit (based on the Cramer van Mises statistic).

  6. In particular, they find that the empirical volatility pricing kernel is u-shaped; a fact that is not captured by any option pricing model so far. A related observation by Boes et al. (2007) is that that the risk-neutral distribution, conditional on a low spot volatility, does not exhibit negative skewness.

  7. For a study on forecasting option returns, see Israelov and Kelly (2017).

  8. Branger et al. (2011) do not confirm their result in more recent data, thus documenting the presence of the pricing kernel puzzle in the data. They further argue that stochastic volatility, stochastic jump option pricing models, which also have jumps in the volatility process, can explain those call option returns.

  9. The empirical evidence is consistent with Branger et al. (2011); compare for the theoretical results also Chaudhuri and Schroder (2015).

  10. Carr and Yu (2012) replace the assumptions on the utility function of a representative investor by assuming that the dynamics of the numeraire portfolio under the physical measure are being driven by a bounded diffusion. Walden (2017) extends Ross (2015) recovery to unbounded diffusion processes and Huang and Shaliastovich (2014) to the state dependent, recursive preferences of Epstein and Zin (1989). Schneider and Trojani (2015) suggest recovery based on assumptions on the signs of risk premia on different moments of market returns.

  11. This point is also made in Borovicka et al. (2015) who attribute these problems to “misspecified recovery,” which happens when the pricing kernel has non-trivial martingale components.

  12. In such setting, Shefrin (2008a, b) coins the term sentiment for the ratio of the mixture of the different subjective distributions and the physical distribution. His ideas become clearer when one assumes that the shapes of the subjective distributions and the physical distributions remain the same but the mean is low for the pessimists, high for the optimists, and in between for the physical distribution, see Shefrin (2008b, Fig. 1).

  13. Although the model captures stochastic volatility and jumps, the risk-aversion functions turn negative for high return states, implying a u-shaped pricing kernel. Such behavior contradicts the standard assumption of a risk-averse representative investor. This leads to the question, if stochastic volatility, stochastic jump models are typically incapable of fitting the historical risk-neutral and physical distribution simultaneously, or if the assumptions on the functional form of the risk-premium parameters are mis-specified in such models.

  14. A number of papers show that such additional state variables seem to be empirically needed in order to explain option prices, see for example Buraschi and Jackwerth (2001), Coval and Shumway (2001), and Constantinides et al. (2013).

  15. Benzoni et al. (2011) offer a similar model but do not show the model pricing kernel in the return dimension, and one cannot easily determine if it exhibits the pricing kernel puzzle; the pricing kernel in the dimension of consumption is monotonically decreasing by assumption.

  16. For a survey of ambiguity aversion and its relevance for asset pricing, see Epstein and Schneider (2010). For an alternative formulation of ambiguity aversion through Choquet expected utility, see Bassett et al. (2004).

  17. Unfortunately, we cannot easily analyze the derivative of the pricing kernel with respect to returns. The resulting expressions are intractable and cannot be nicely segregated into, say, an income and a substitution effect.

  18. Note that alternatively, one could also use \(U(x)=\frac{x^{1-\gamma }-1}{1-\gamma }\) with \(\gamma \in (0,1)\) but the above formulation allows for a great range of risk aversion coefficients.

  19. Marroquin-Martinez and Moreno (2013) extend Cochrane and Saa-Requejo (2000) and Bernardo and Ledoit (2000) to settings with stochastic volatility and find the resulting bounds to be tighter than in the original papers.

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    Horatio Cuesdeanu & Jens Carsten Jackwerth

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  1. Horatio Cuesdeanu
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We would like to thank George Constantinides, Maria Grith, and Alexandros Kostakis for helpful comments.

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Cuesdeanu, H., Jackwerth, J.C. The pricing kernel puzzle: survey and outlook. Ann Finance 14, 289–329 (2018). https://doi.org/10.1007/s10436-017-0317-9

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