(Almost) Model‐Free Recovery

• Author: FABIO TROJANI,PAUL SCHNEIDER
• Published date: 01 February 2019
• DOI: http://doi.org/10.1111/jofi.12737
• Date: 01 February 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 1 •FEBRUARY 2019

(Almost) Model-Free Recovery

PAUL SCHNEIDER and FABIO TROJANI∗

ABSTRACT

Under mild assumptions, we recover the model-free conditional minimum variance

projection of the pricing kernel on various tradeable realized moments of market

returns. Recovered conditional moments predict future realizations and give insight

into the cyclicality of equity premia, variance risk premia, and the highest attainable

Sharpe ratios under the minimum variance probability.The pricing kernel projections

are often U-shaped and give rise to optimal conditional portfolio strategies with

plausible market timing properties, moderate countercyclical exposures to higher

realized moments, and favorable out-of-sample Sharpe ratios.

ASEMINAL FINDING IN BREEDEN and Litzenberger (1978) shows that the condi-

tional forward-neutral distribution of an asset return in arbitrage-free option

markets is uniquely identified by the second derivative of the European call

price function with respect to the option’s strike. No corresponding model-free

result is available for the conditional distribution of returns under the physical

belief.

Recent model-based recovery theorems study conditions on physical tran-

sition dynamics and pricing kernels under which the physical belief can be

determined from Arrow-Debreu prices alone. Broadly speaking, this literature

shows that recovery is possible whenever physical probabilities are induced

by a family of path-independent pricing kernels and satisfy further stability

conditions.1Path-independence is essential for these theorems, as, in general,

only a particular probability different from the physical probability can be

∗Paul Schneider is at USI Lugano and the Swiss Finance Institute. Fabio Trojani is at the Uni-

versity of Geneva and Swiss Finance Institute. We are indebted to Editor Ken Singleton and two

anonymous referees for key comments that strongly improved our paper. Our thanks for helpful

discussions and comments also go to Jaroslav Boroviˇ

cka, Peter Carr, Gianluca Cassese, Jerome

Detemple, Damir Filipovic, Patrick Gagliardini, Stefano Giglio, Peter Gruber,Leonid Kogan, Abra-

ham Lioui, Eric Renault, Steve Ross, Mirela Sandulescu, Olivier Scaillet, Raman Uppal, Christian

Wagner,Liuren Wu, and workshop participants at Boston University, Brown University, EDHEC

Business School, EPFL, IFSID Montreal, MIT, Morgan Stanley, University of Zurich, SFI Annual

Research Days, and USI. We are indebted to Dacheng Xiu for providing us with parameters and

state variables. Financial support from the Swiss Finance Institute (Project “Term structures and

cross sections of asset risk premia,”) and the Swiss National Science Foundation (Projects “Trad-

ing Asset Pricing Models,” “Model-Free Asset Pricing,” and “Higher Order Robust Resampling

and Multiple Testing Methods”) is gratefully acknowledged. Wehave read the Journal of Finance

disclosure policy and have no conflicts of interest to disclose.

1These conditions are naturally nested by the requirement of recurrent physical state dynamics.

Recurrence is the requirement that the expected number of times a stochastic process visits every

DOI: 10.1111/jofi.12737

323

324 The Journal of Finance R

recovered. Under further stability assumptions, this probability can be iden-

tified with the long forward probability in Hansen and Scheinkman (2009).2

Hence, the physical belief remains unidentifiable without further information.

Acknowledging the impossibility of identifying physical probabilities without

additional restrictions, we step back from the recovery approaches in the

literature and ask a different question: Which family of pricing kernels is

consistent with mild sign restrictions on (i) particular covariances of market

returns with the pricing kernel and (ii) the risk premiums for variance and

higher order moment risks, given various pricing constraints for the tradeable

second and higher realized moments of returns?3We first characterize the

kernel projections that exactly price tradeable second and higher realized

return risks under the given sign restrictions. We next determine the minimum

variance kernel projection (MVP) consistent with the most conservative lower

bound on the maximal Sharpe ratio in the economy. Finally, we show that mild,

economically motivated and empirically supported restrictions are sufficient to

obtain an informative minimum variance probability, which corresponds to an

MVP that is tradeable with delta-hedged option portfolios. The MVP naturally

defines a lower bound on the maximal Sharpe ratio that is attainable in any

economy satisfying the initial economic constraints. Our recovery approach

therefore corresponds to an (almost) model-free kernel selection that recovers

the MVP under a conservative max-min Sharpe ratio criterion.

We determine the MVP from mild hypotheses on the risk premia of certain

tradeable realized moments of returns. These hypotheses can include, for

instance, a negative risk premium for particular variance payoffs, a positive

risk premium for payoffs that are increasing in market returns, or ranking

relations between the Sharpe ratios of various higher moments. Together

with the maintained no-arbitrage assumption that equivalent physical and

forward conditional moments are defined on a common information set,

these hypotheses imply a binding lower bound for the variance of kernel

projections, which is elicited from Arrow-Debreu prices alone. In this sense,

our methodology is (almost) model-free, as it depends exclusively on reasonable

and testable assumptions about particular risk premia, which allow us to

avoid hypotheses on the structure of the economy that might be difficult to

test in practice. For instance, we make no ex-ante assumption on the state

subset of its state space is infinite. See Carr and Yu (2012), Ross (2015), Walden (2017), and Qin

and Linetsky (2017), among others, for various recovery theorems based on recurrent physical

dynamics.

2See Boroviˇ

cka, Hansen, and Scheinkman (2016) for a thorough discussion. This probability

also yields a unique long-run factorization of the pricing kernel into transitory and permanent

(martingale) components.

3While, theoretically, the market equity premium is often assumed positive in equilibrium,

further natural assumptions can be motivated for the risk premia on higher moments. For instance,

theoretical and empirical literature on market variance risk typically concludes that the risk

premium for even realized moment risk is negative; see Carr and Wu (2009), Martin (2013), and

Schneider and Trojani (2014), among others. Similarly, the risk premium for odd realized moment

risk, such as skewness risk, tends to be positive; see Kozhan, Neuberger, and Schneider (2013)and

Schneider and Trojani (2014).

(Almost) Model-Free Recovery 325

space of returns, Markovianity or stationarity of the relevant state processes

under the forward or physical probabilities, semimartingale properties, or

path-independence of the pricing kernel. Clearly, the cost of this generality

arises from our assumptions on the risk premia of particular payoffs, which

are nonetheless easier to interpret theoretically and to validate empirically.

Our empirical analysis covers 25 years of monthly Standard & Poor’s (S&P)

500 option data from January 1990 to August 2016. We find that our MVPs

produce useful insights into the conditional structure of pricing kernels and

their dependence on the investment horizon. MVPs for horizons above one

quarter are usually U-shaped. In contrast, monthly short-horizon projections

can be downward-sloping in states of moderate market uncertainty. More

informative economic assumptions may also strengthen the U-shape of the

MVP. For instance, the requirement that variance Sharpe ratios be at least as

large as market Sharpe ratios (labeled as constraint “SR” below) already leads

to consistently U-shaped projections at monthly horizons.

MVPs naturally embed potentially useful information about the time-varying

structure of any probability belief that supports these projections. We find that

the recovered moments of returns are highly time-varying and countercyclical.

Annualized monthly and annual equity premia are less than 1% in periods of

low and transitory volatility,but can be as large as about 22% and 10%, respec-

tively, in turbulent markets, such as during the financial crisis of 2008. Recov-

ered first and second moments have predictive power for future returns and sec-

ond realized moments of returns. They also provide insights into the cyclicality

of equity premia, variance risk premia, and highest attainable Sharpe ratios.

Tradeable minimum variance projections (MVPs) allow us to study the

properties of optimal risk-return trade-offs under the minimum variance

probability. Optimal delta-hedged option portfolios that short the MVP imply

a countercyclical Sharpe ratio that is decreasing (increasing) with the invest-

ment horizon in periods of large (moderate) uncertainty. In phases of greater

uncertainty,the optimal investment strategy is tilted to short out-of-the-money

call option payoffs, in order to replicate the countercyclical U-shapedness of

MVPs. We find that, overall, these optimal option portfolios have plausible

market timing properties, moderate exposures to higher realized moments,

and favorable out-of-sample Sharpe ratios when SR constraints are considered.

Our paper is related to different fields in the literature. The empirical

properties of pricing kernels in index option markets have been considered

by a large number of authors. A¨

ıt-Sahalia and Lo (1998), Jackwerth (2000),

Rosenberg and Engle (2002), A¨

ıt-Sahalia and Duarte (2003), Chabi-Yo, Garcia,

and Renault (2008), Chabi-Yo (2012), Christoffersen, Heston, and Jacobs

(2013), and Song and Xiu (2016) among others estimate U-shaped pricing

kernel projections using different parametric and nonparametric methods. A

key finding of this research is that the model-implied kernel projection on the

linear span of index and option returns is increasing (decreasing) in return

states that correspond to positive out-of-the-money call (put) payoffs. U-shaped

pricing kernels can be naturally motivated in economies with disagreeing

investors. Bakshi and Madan (2008) obtain a pricing kernel that is a U-shaped