The Pricing Kernel Puzzle: A Real Phenomenon or a Statistical Artifact?

Date01 June 2020
AuthorSajid Anwar,Hammad Siddiqi
Published date01 June 2020
DOIhttp://doi.org/10.1111/irfi.12207
The Pricing Kernel Puzzle: A Real
Phenomenon or a Statistical
Artifact?*
HAMMAD SIDDIQI
AND SAJID ANWAR
,
School of Business, University of the Sunshine Coast, Maroochydore DC,
Queensland, Australia and
School of International Economics and Trade, Shanghai Lixin University of
Accounting and Finance, Shanghai, China
ABSTRACT
A large literature nds evidence that the pricing kernels estimated from
option prices and historical returns are not monotonically decreasing in mar-
ket returns. We show that the inevitable coalescence of contingencies, espe-
cially in the left-tail, associated with estimating a distribution function may
give rise to a nonmonotonic empirical pricing kernel even if the actual pric-
ing kernel is monotonic. Hence, the observed nonmonotonicity of pricing
kernels may be a statistical artifact rather than a real phenomenon. We argue
that empirical work should explicitly correct for this effect by widening the
option pricing bounds associated with monotonic pricing kernels.
JEL Codes: G13; G12
Accepted: 29 May 2018
I. INTRODUCTION
Most economics and nance models require the pricing kernel/stochastic dis-
count factor/intertemporal marginal rate of substitution to be monotonically
decreasing in the aggregate payoff. Starting from the work of Jackwerth (2000)
and Ait-Sahalia and Lo (2000), researchers have been using a combination of
options and historical returns data to estimate pricing kernels. Typically, this
literature nds a nonmonotonic pricing kernel, which is either U-shaped or
positively sloped in the middle. This nding is known as the pricing kernel puz-
zle. Several explanations for this puzzle have been put forward in the literature
(e.g., Hens and Reichlin 2012 and Siddiqi and Quiggin 2017).
Pricing kernel is typically estimated by dividing the risk-neutral density by
the physical density. Linn et al. (2018) argue that the empirical pricing kernel
may falsely appear nonmonotonic because it essentially involves dividing a
* The authors are grateful to an anonymous reviewer for helpful comments and suggestions. How-
ever, all remaining errors and/or imperfections are their own responsibility.
© 2018 International Review of Finance Ltd. 2018
International Review of Finance, 20:2, 2020: pp. 485491
DOI: 10.1111/ir.12207

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