Semiparametric estimation of asset pricing kernel

Abstract

This article empirically studies the pricing kernel implicit in option prices. Based on the cross-sectional fits alone, no significant difference can be detected between models with different factor dynamics. A cubic pricing kernel provides almost perfect fits in the sample. Nonlinearity in the pricing kernel is crucial for in-sample performance. Both excess kurtosis and skewness are very important. The claim-based market line sharply distinguishes various estimates of the pricing kernel and tracks the market sentiment. However, a well-specified factor dynamics model improves the out-of-sample pricing performance. With a well-specified factor dynamics model, the linear pricing kernel beats the other competitors at a 2-week horizon.

Acknowledgements

This article is based on the author's Ph.D. thesis at the School of Business, Queen's University (Canada). The author is grateful to Dr. Edwin Neave for his superb guidance and thoughtful inspiration. The author acknowledges invaluable comments and suggestions from Peter Christoffersen, Lew Johnson, Frank Milne, Wulin Suo, two anonymous reviewers, and participants at French Finance Association 2003 conference (Lyon) and Northern Finance Association 2003 conference (Quebec city).

Notes

¹ Examples include Aït-Sahalia and Lo (1998, 2000), Bliss and Panigirtzoglou (2004), Chernov and Ghysels (2000), Jackwerth (2000) and Rosenberg and Engle (2002). See Section ‘Literature on option-implied SPD and pricing kernel’ for more details.

² Gallant and Tauchen (1992).

³ In this article, market sentiment refers to market participants’ perception of the market condition or their attitude towards risks. Formally, it is measured by the excess return of the pricing kernel in the CBML.

⁴ See Cochrane (2001) for a complete discussion on asset pricing kernel.

⁵ In a discrete contingent claim world, the payoff of any financial asset is a bundle of contingent claims. Suppose an investor constructs a portfolio M consisting of q_k /p_k (1 + r) claims on each state k (q is the risk neutral probability and p is the objective probability), then this is the reference portfolio used in the CBML in Equation 2. While p and q are either observable or estimable, this is a tradable portfolio (at least from a theoretical perspective).

⁶ In the extreme case of risk neutrality when the risk neutral probabilities are the same as the objective probabilities, the pricing kernel is a constant. Thus, is zero and the expected excess return is zero.

⁷ Although the variance of the pricing kernel obviously depends on its specification (linear, quadratic or cubic), the magnitude of the excess return of the pricing kernel is an empirical issue. The excess returns in Equation 3 will be estimated in Section V.

⁸ We work with the log pricing kernel m(z_t , θ _t ) to ensure positivity of the pricing kernel M(z_t , θ _t ), which in turn guarantees that the pricing kernel does not admit arbitrage.

⁹ Rosenberg and Engle (2002) estimate a GJR-GARCH model of the S&P 500 index returns and then conduct simulations to get the transition densities. The SNP approach is more general than and actually nests their model.

¹⁰ See Gallant and Tauchen (2001) for details of model specifications, estimation method, model selection process and some working examples. The application of SNP in this article closely follows Gallant and Tauchen (2001).

¹¹ Research concerning the effect of skewness on asset valuation include Ingersoll (1975) and Rubinstein (1973). Barone-Adesi (1985), Lin and Wang (2003) and Harris et al. (2004) provide empirical evidence.

¹² Previous research (e.g. Fama, 1965; Aggarwal and Rao, 1990) has documented that US common stock returns are distributed with more returns in extreme tails; Watana (2000) provides evidence in the Japanese market.

¹³ While univariate specifications of the pricing kernel can be thought as projections of the true pricing kernel onto the space spanned by the return of the underlying asset, they work perfectly like the true pricing kernel to price or hedge other securities whose returns lie in the same space.

¹⁴ See Gallant and Tauchen (1992) for a description of the simulation algorithm for SNP densities.

¹⁵ The close price is adjusted for all applicable splits, spinoffs and dividend distributions.

¹⁶ For example, Jackwerth and Rubinstein (1996) show that since the crash of 1987 S&P 500 index options exhibit pronounced volatility smiles.

¹⁷ Chernov and Ghysels (2000) also estimate the SNP model with S&P 500 daily log returns and their fitted model is a semiparametric ARCH model. At the time of their work, the SNP version of GARCH model had not been introduced.

¹⁸ To check the robustness of the results above, we conduct a similar analysis with a 4-week-to-maturity sample from 1996 to 1999. The results (not shown) display very similar patterns to those in Table 2.

¹⁹ Brown and Jackwerth (2004) call this phenomenon the pricing kernel puzzle and investigate several possible explanations. They rule out hypothesis such as data imperfections and methodological problems. They provide a representative agent model where volatility is a function of a second momentum state variable. That model is capable of generating the empirical patterns of the pricing kernel. Benzoni et al. (2005) demonstrate that the puzzle can be rationalized if the agent is endowed with Epstein–Zin preference and if the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. Shefrin (2005) offers behavioural explanations to this observation.