A dynamic equilibrium model for U-shaped pricing kernels

Abstract

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of the risk-neutral return distribution relative to the physical distribution, negative expected returns on deep OTM call options and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows a time-changed Lévy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on aggregate consumption as well as the investor’s risk aversion.

Keywords:

• Stock index
• U-shaped pricing kernel
• Stochastic activity rate
• Aggregate consumption
• Physical distribution
• Risk-neutral distribution
• Realized variance

Acknowledgements

The author wishes to thank an anonymous referee and Daisuke Yoshikawa at Hokkai-Gakuen University for helpful comments and suggestions, which have made this a much improved paper.

Notes

No potential conflict of interest was reported by the author.

1 See Pennacchi (2008), Cochrane (2009), Duffie (2010), and Back (2010) as excellent monographs for the standard theory of asset pricing.

2 See Schoutens (2003) for example.

3 As more realistic modelling, we can formulate

where is an independent noise process such that for all . However, in this paper we omit such a noise process for simplicity. Even if the noise process is introduced to the model, we can easily develop the formulas as a simple extension of the result in this paper.

Additional information

Funding

The author gratefully acknowledges the financial support of JSPS KAKENHI [grant number 26380402]. His work is also supported by the Research Institute for Innovation Management at Hosei University.