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The pricing kernel puzzle in forward looking data

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Abstract

The pricing kernel puzzle concerns the locally increasing empirical pricing kernel, which is inconsistent with a risk-averse representative investor in a single period, single state variable setting. Some recent papers worry that the puzzle is caused simply by the mismatch of backward looking subjective and forward looking risk-neutral distributions of index returns. By using a novel test and forward looking information only, we generally confirm the existence of a u-shaped pricing kernel puzzle in the S&P 500 options data. The evidence is weaker for tests against an alternative with a risk-neutral investor and for longer horizons.

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Notes

  1. See e.g. (Cochrane 2000, p. 50).

  2. Dybvig (1988) and Beare (2011) provide another interesting view on the puzzle under the assumption of a complete market. Buying the market portfolio is the cheapest way to obtain the payoff of the market portfolio if and only if the pricing kernel is monotonically decreasing. Or stated differently, if the pricing kernel is locally increasing, there exists a portfolio of Arrow-Debreu securities that stochastically dominates the market return. See also Beare and Schmidt (2014a) for an empirical study on this issue.

  3. For additional models of the pricing kernel puzzle, see Cuesdeanu and Jackwerth (2017).

  4. B-splines are scaled truncated power functions, which can be used instead of polynomial splines (de Boor 1978, p. 96). Their theoretical characterization makes them favorable for the estimation procedure in Linn et al. (2017).

  5. The methodology applied in Bliss and Panigirtzoglou (2004), Linn et al. (2017), as well as in our paper is forward looking in the sense that no historical return time series is used to explicitly compute the subjective density. Nevertheless, monthly realized returns are needed for calibrating the pricing kernel, see Sect. 2.3. Recovering the pricing kernel without using past returns but relying on more strict assumptions on the data generating process was pioneered by Ross (2015); see also the follow-up studies by Jackwerth and Menner (2017) and Jensen et al. (2017).

  6. Using a different number of equally spaced values on the return axis does not change the results, see Sect. 5.2.

  7. Consider also Figs. 1, 2, and 3 where such estimated pricing kernels are depicted. After scaling the pricing kernel, it can take on any value and is therefore not restricted by the initial value of 5.

  8. Our method is robust to extending the pricing kernel below 0.8 and above 1.2 at the slopes of the last interior segments instead of horizontally. Moreover, the results do not change when fixing different basis points on the return axis, see Sect. 5.2.

  9. Our formulation is equivalent to expressing the subjective density in terms of state prices \(\pi _t\) as the risk-free discount factor \(e^{-r_{f,t} \Delta t}\) cancels out when normalizing the subjective densities:

    \( p_t=\frac{\pi _t}{m} \Big / \int _{0}^{\infty } \frac{\pi _t}{m} dR_t = \frac{e^{-r_{f,t} \Delta t} q_t}{m} \Big / \int _{0}^{\infty } \frac{ e^{-r_{f,t} \Delta t} q_t}{m} dR_t = \frac{q_t}{m} \Big / \int _{0}^{\infty } \frac{q_t}{m} dR_t \).

  10. See Gneiting and Raftery (2007) for a comprehensive discussion of score functions.

  11. A similar approach is used in Shackleton et al. (2010) for mixing forecasts from past returns and forward looking option prices.

  12. We thank an anonymous referee for stressing this point.

  13. See Beare and Moon (2015) for a detailed discussion. Intuitively, when simulating with a flat pricing kernel, the differences between the simulated restricted and unrestricted log scores are larger than the differences when simulating with a decreasing pricing kernel. Hence, the difference we obtain from the data has to be much larger in order to reject the null hypothesis, making the risk-neutral version of the test the most conservative way to test pricing kernel monotonicity.

  14. See Cramer (1928).

  15. Theoretically, one could also consider overlapping returns, e.g. monthly returns shifted by on day at a time. While the estimation of the unrestricted and restricted pricing kernel would be no issue in such setting, the simulation of the p values of the \(\Delta \) statistic is no longer straightforward. One would have to draw overlapping returns stemming from daily non-parametric densities with a monthly horizon. It is not clear at all how to do this without making restrictive assumptions on the data generating process.

  16. The pricing kernel derived from an exponential utility function is given by \(m^{exp.}=\exp (-a \cdot R_t)\). Bliss and Panigirtzoglou (2004) identified \(a=6.33\) as the optimal value when looking at a monthly horizon and so we repeated our base procedure for \(a=2,4,6,8,\text { and }10\). Again, all p values decrease in a. All our tests, except the Berkowitz (2001) test, reject pricing kernel monotonicity at least at the \(10\%\) level.

  17. In particular, Cuesdeanu (2016) argues that the tilde shaped pricing kernels in the canonical papers of Ait-Sahalia and Lo (2000), Jackwerth (2000), and Rosenberg and Engle (2002) can be explained by the sample period (characterized by a relatively low variance-risk-premium) in combination with the non-observability of out-of-the-money calls.

  18. A model that could potentially explain w-shaped pricing kernels in returns is Bakshi et al. (2015). The authors describe an economy where investors are long and short volatility, and aggregation leads to a u-shaped pricing kernel in the volatility dimension. Since high (low) absolute returns are typically associated with high (low) volatility, a u-shaped pricing kernel in the volatility dimension could mechanically imply a w-shape in the return dimension.

  19. The Berkowitz (2001) test might gain in power as the higher moments of the returns realized in this subsample are less extreme. Compared to the full sample, sample kurtosis drops from 7.18 to 3.74 and sample skewness is less negative: −1.24 in the full sample versus −0.66 in the low volatility subsample.

  20. In case of linear interpolation, a pricing kernel is non-decreasing if and only if its values at the spline knots are non-increasing.

  21. This GMM approach is equivalent to setting \(\widetilde{\Omega }_L\) to the identity matrix in Knüppel (2015).

  22. Note that Linn et al. (2017) discuss the results for all numbers of moment conditions up to \(L=50\) in the working paper version only. In the published version of their paper they only show the results for \(L \in \{ 4, 5, 6, 7, 8, 9 \}\) and claim that the best pricing kernel obtains when the number of pricing kernel parameters equals to the number of moment conditions. They find that for the S&P 500, choosing \(L=7\) delivers the highest p values for the Berkowitz, Cramér van Mises, and Kolmogorov-Smirnov test statistics. Note that the p values in their Table 2 are upwards biased since fitting the moments of a distribution helps in fitting the distribution itself. Only the p values corresponding to the rows labelled ’No pricing kernel’ are unbiased since no pricing kernel optimization happened there.

  23. The maximal realized return in our sample is at about 1.11.

References

  • Ait-Sahalia, Y., & Lo, A. W. (2000). Nonparametric risk management and implied risk aversion. Journal of Econometrics, 94, 9–51.

    Article  Google Scholar 

  • Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–858.

    Article  Google Scholar 

  • Babaoglu, K., Christoffersen, P., Heston, S., & Jacobs, K. (2017). Option valuation with volatility components, fat tails, and nonlinear pricing kernels. Review of Asset Pricing Studies, forthcoming.

  • Bakshi, G., Madan, D. B., & Panayotov, G. (2010). Returns of claims on the upside and the viability of U-shaped pricing kernels. Journal of Financial Economics, 97, 130–157.

    Article  Google Scholar 

  • Bakshi, G., Madan, D. B., & Panayotov, G. (2015). Heterogeneity in beliefs and volatility tail behavior. Journal of Financial and Quantitative Analysis, 50, 1389–1414.

    Article  Google Scholar 

  • Barone-Adesi, G., Dall’O, H., & Vovchak, V. (2012). Is the pricing kernel monotone? Journal of Finance and Risk Perspectives, 1, 43–69.

    Google Scholar 

  • Barone-Adesi, G., Engle, R. F., & Mancini, L. (2008). A GARCH option pricing model with filtered historical simulation. Review of Financial Studies, 21, 1223–1258.

    Article  Google Scholar 

  • Beare, B., & Moon, J.-M. (2015). Nonparametric tests of density ratio ordering. Econometric Theory, 31, 471–492.

    Article  Google Scholar 

  • Beare, B., & Schmidt, L. (2014a). Empirical implications of the pricing kernel puzzle for the return on contingent claims. In Working paper, University of California San Diego.

  • Beare, B., & Schmidt, L. (2014b). An empirical test of pricing kernel monotonicity. Journal of Applied Econometrics, 31, 338–356.

    Article  Google Scholar 

  • Beare, B. K. (2011). Measure preserving derivatives and the pricing kernel puzzle. Journal of Mathematical Economics, 47, 689–697.

    Article  Google Scholar 

  • Benzoni, L., Dufresne, C. P., & Goldstein, R. S. (2011). Explaining asset pricing puzzles associated with the 1987 market crash. Journal of Financial Economics, 101, 552–573.

    Article  Google Scholar 

  • Berkowitz, J. (2001). Testing density forecasts with applications to risk management. Journal of Business and Economic Statistics, 19, 465–474.

    Article  Google Scholar 

  • Bliss, R. R., & Panigirtzoglou, N. (2004). Option-implied risk aversion estimates. Journal of Finance, 59, 407–446.

    Article  Google Scholar 

  • Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of Business, 51, 621–651.

    Article  Google Scholar 

  • Brown, D. P., & Jackwerth, J. C. (2012). The pricing kernel puzzle: reconciling index option data and economic theory. Contemporary Studies in Economic and Financial Analysis, 94, 155–183.

    Article  Google Scholar 

  • Chabi-Yo, F. (2012). Pricing kernels with stochastic skewness and volatility risk. Management Science, 58, 624–640.

    Article  Google Scholar 

  • Chabi-Yo, F., Garcia, R., & Renault, E. (2009). State dependence can explain the risk aversion puzzle. Review of Financial Studies, 21, 973–1011.

    Article  Google Scholar 

  • Chaudhuri, R., & Schroder, M. D. (2015). Monotonicity of the stochastic discount factor and expected option returns. Review of Financial Studies, 28, 1462–1505.

    Article  Google Scholar 

  • Christoffersen, P., Heston, S., & Jacobs, K. (2013). Capturing option anomalies with a variance-dependent pricing kernel. Review of Financial Studies, 26, 1962–2006.

    Article  Google Scholar 

  • Christoffersen, P., & Jacobs, K. (2004). The importance of the loss function in option valuation. Journal of Financial Economics, 72, 291–318.

    Article  Google Scholar 

  • Cochrane, J. H. (2000). Asset pricing (1st ed.). Princeton, USA: Princeton University Press.

    Google Scholar 

  • Cramer, H. (1928). On the composition of elementary errors. Scandinavian Actuarial Journal, 59, 13–74.

    Article  Google Scholar 

  • Cuesdeanu, H. (2016). Empirical pricing kernels: A tale of two tails and volatility? In Working paper, University of Konstanz.

  • Cuesdeanu, H., & Jackwerth, J. C. (2017). The pricing kernel puzzle: Survey and outlook. Annals of Finance, forthcoming.

  • de Boor, C. (1978). A practical guide to splines (1st ed.). New York: Springer.

    Book  Google Scholar 

  • Dybvig, P. H. (1988). Distributional analysis of portfolio choice. Journal of Business, 63, 369–393.

    Article  Google Scholar 

  • Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. In Global derivatives and risk management, Madrid.

  • Gneiting, T., & Raftery, A. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102, 359–378.

    Article  Google Scholar 

  • Golubev, Y., Haerdle, W. K., & Timofeev, R. (2014). Testing monotonicity of pricing kernels. Advances in Statistical Analysis, 98, 305–326.

    Article  Google Scholar 

  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343.

    Article  Google Scholar 

  • Jackwerth, J., & Menner, M. (2017). Does the Ross recovery theorem work empirically? In Working paper, University of Konstanz.

  • Jackwerth, J. C. (2000). Recovering risk aversion from option prices and realized returns. Review of Financial Studies, 13, 433–451.

    Article  Google Scholar 

  • Jackwerth, J. C. (2004). Option-implied risk-neutral distributions and risk-aversion. Charlotteville, USA: Research Foundation of AIMR.

    Google Scholar 

  • Jensen, C. S., Lando, D., & Pedersen, L. H. (2017). Generalized recovery. Journal of Financial Economics, forthcoming.

  • Knüppel, M. (2015). Evaluating the Calibration of multi-step-ahead density forecasts using raw moments. Journal of Business & Economic Statistics, 33, 270–281.

    Article  Google Scholar 

  • Linn, M., Shive, S., & Shumway, T. (2017). Pricing kernel monotonicity and conditional information. Review of Financial Studies, forthcoming.

  • Mitchell, J., & Hall, G. S. (2005). Evaluating, comparing and combining density forecasts using the KLIC with an application to the Bank of England and NIESR fan charts of inflation. Oxford Bulletin of Economics and Statistics, 67, 995–1033.

    Article  Google Scholar 

  • Polkovnichenko, V., & Zhao, F. (2013). Probability weighting functions implied in options prices. Journal of Financial Economics, 107, 580–609.

    Article  Google Scholar 

  • Rosenberg, J. V., & Engle, R. F. (2002). Empirical pricing kernels. Journal of Financial Economics, 64, 341–372.

    Article  Google Scholar 

  • Ross, S. (2015). The recovery theorem. Journal of Finance, 70, 615–648.

    Article  Google Scholar 

  • Shackleton, M. B., Taylor, S. J., & Yu, P. (2010). A multi-horizon comparison of density forecasts for the S&P 500 using index returns and option prices. Journal of Banking and Finance, 34, 2678–2693.

    Article  Google Scholar 

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Authors and Affiliations

  1. University of Konstanz, PO Box 134, 78457, Konstanz, Germany

    Horatio Cuesdeanu & Jens Carsten Jackwerth

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  1. Horatio Cuesdeanu
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  2. Jens Carsten Jackwerth
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Correspondence to Horatio Cuesdeanu.

Additional information

We received helpful comments and suggestions from Brendan Beare, Peter Christoffersen, Malte Knüppel, Dietmar Leisen, Christian Skov Jensen, Fabian Krüger, Thierry Post, Thorsten Hens, and Axel Kind. We thank seminar participants at the University of Konstanz, the University of Lancaster, the University of Toronto, the University of Sydney, and the University of Queensland.

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Cuesdeanu, H., Jackwerth, J.C. The pricing kernel puzzle in forward looking data. Rev Deriv Res 21, 253–276 (2018). https://doi.org/10.1007/s11147-017-9140-8

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