(Almost) Model-Free Recovery

54 Pages·2015·1.85 MB·English

(Almost) Model-Free Recovery∗
Paul Schneider†and Fabio Trojani‡
August 10, 2015
Abstract
Based on mild economic assumptions, we recover the time series of
conditional physical moments of market index returns from a model-
free projection of the pricing kernel on the return space. These mo-
ments identify the minimum variance pricing kernel projection and
are supported by a corresponding set of physical distributions. The
recoveredmomentspredictS&P500returns,especiallyforlongerhori-
zons, give rise to refined conditional versions of Hansen-Jagannathan
bounds, andcanbetradedusingdelta-hedgedoptionportfolios. They
alsoimplyconditionalpricingkernelprojectionsthatareoftenfarfrom
being uniformly monotonic and convex.
1 Introduction
A seminal finding in Breeden and Litzenberger (1978) shows that in an
arbitrage-free market the density of the conditional forward-neutral distri-
bution (Q ) of an asset return coincides with the second derivative of the
T
∗WearethankfulforhelpfuldiscussionswithGianlucaCassese,DamirFilipovic,Patrick
Gagliardini, Peter Gruber, Olivier Scaillet, Christian Wagner and seminar participants of
theUSIbrownbagworkshop. FinancialsupportfromtheSwissFinanceInstitute(Project
”Termstructuresandcross-sectionsofassetriskpremia”),andtheSNF(Project“Trading
Asset Pricing Models”) is gratefully acknowledged.
†Boston University, University of Lugano, and Swiss Finance Institute. Email:.
[email protected]
‡University of Geneva, University of Lugano, and Swiss Finance Institute. Email:.
[email protected]
1
price function of European call options with respect to the option’s strike.
Thereisnosuchgeneralandmodel-freeresultfortheconditionaldistribution
of asset returns under the physical probability measure (P).
To learn more about physical distributions, researchers often have made
use of parametric modelling approaches (e.g., Jones, 2003; Eraker, 2004).
In these models, probabilities P and Q are usually linked by a paramet-
T
ric (forward) stochastic discount factor dQ /dP, which can be estimated
T
from historical return information and allows to uniquely recover P probabil-
ities from Q probabilities using this additional information. More recently,
T
some authors have derived recovery theorems based on a different set of non-
parametric assumptions. In a complete market setting with Markovian and
stationary state dynamics on a (bounded) finite state space, Ross (2015)
uniquely recovers a path-independent stochastic discount factor dQ /dP us-
T
ing Perron Frobenius theory. Boroviˇcka et al. (2014) discuss in more detail
the implications of path-independence and emphasize that in general only a
misspecified probability measure can be recovered, different from the phys-
ical probability, which incorporates long-run risk adjustments. Hence, the
physical probability remains unidentified without introducing additional re-
strictions or using additional data.
Weproposetoidentifythemaincharacteristicsofthephysicalprobability
P with a conceptually different approach from the one adopted in existing
recovery theorems. Without making stringent assumptions about the under-
lying economy or price processes, we start from a set of plausible economic
assumptions on the sign of the risk premia for trading particular nonlinear
risks in option markets. While theoretically the forward equity premium,
the first conditional P moment of forward market returns, needs to be posi-
tive in equilibrium, additional natural assumptions can be motivated for the
risk premia on higher moments. For instance, it is widely recognized in the
theoretical and empirical literature that the price of market variance risk,
and more generally even market divergence risk, is negative; see Carr and
Wu (2009), Martin (2013) and Schneider and Trojani (2014), among others.
Similarly, the risk premium for exposure to odd market divergence risk, such
as skewness risk, is naturally positive, because it is generated by risks that
2
are monotonic transformations of market returns; see Kozhan et al. (2013)
and Schneider and Trojani (2014), among others.
A sign restriction on an asset risk premium is a constraint on the co-
variance between the pricing kernel and the return of that particular asset.
Given a set of observed prices of suitable option portfolios and a model-free
no-arbitrage condition, we show that this restriction implies useful model-
free constraints on the physical conditional moments of market returns. In
this way, we obtain a family of model-free upper and lower bounds on differ-
ent conditional moments of market returns, which extend the lower bound in
Martin (2015) for the market equity premium. We show that these bounds
are highly time-varying, reflecting a rich conditional distribution of market
returns, and that they imply relatively tight intervals for the unknown phys-
ical moments of market returns.
Our model-free bounds on the physical conditional moments effectively
constrain the set of physical probabilities that in abitrage-free markets can
support (i) our economic risk premium constraints and (ii) the observed
prices of suitable option portfolios. We characterize the admissible moments
supportedbyaprobabilitymeasuresatisfyingconditions(i),(ii),usingknown
results on the solution of (truncated) moment problems.1 In this way, we
obtain a parsimomious description of the family of physical moments for
which a model-free recovery result can be motivated.
We obtain recovery based on a model-free L2−projection of the pric-
ing kernel on market returns. This projection is parameterized by forward-
neutral and physical moments alone. Therefore, any parameterization con-
sistent with our physical moment constraints and with the prices of suitable
option portfolios defines an admissible physical measure P in our incomplete
market setting.2 We focus on model-free recovery of the minimal variance
physicalmeasure, whichimpliesthetightestupperboundontheSharperatio
1Momentproblems(truncatedmomentproblems)dealwiththequestionofwhetherfor
a given countable (finite) sequence of numbers there exists a probability measure having
these numbers as its moments.
2 As the number of constrained physical and forward-neutral moments goes to infinity,
our L2projection parameterization converges to the physical expectation of the pricing
kernel conditional on forward market returns.
3
of any portfolio of asset returns that are exactly priced by the projection.
Usingoureconomicallymotivatedmodel-freerecovery, weavoidanumber
of strong technical assumptions on the underlying economy, which might be
difficult to motivate or to test in practice. For instance, we do not need
strong assumptions on the economy state space or the underlying stochastic
processes, such as the Markovianity or the stationarity of asset returns, nor
do we need to assume path-independent pricing kernels. We can assume a
weak model-free definition of arbitrage opportunities to invoke model-free
versions of the fundamental theorem of asset pricing (Acciaio et al., 2013)
and ensure existence of a forward-neutral measure in our setting. For the
existence of our L2 pricing kernel projection, we need the existence of all
polynomial moments of market returns, which is ensured, for example, if the
state space of market returns is conditionally bounded. Boundedness of the
state space is assumed in virtually all recovery theorems in the literature.
Moreover, from our treatment based on truncated moment problems, the
recovered physical probability in our incomplete market setting can indeed
be ensured to have bounded support. Clearly, the cost of the generality of
our approach in terms of weak technical conditions arises from the economic
assumptions about the risk premia of particular asset returns, which might
however by easier to motivate and test in some cases.
We find that the conditional moments implied by our recovered pricing
kernel projections are highly time-varying and informative about future mar-
ket realizations, especially for longer horizons. Our technology also naturally
recovers second conditional moments of nonlinear minimum variance pricing
kernel projections, which extend the linear projection approach in Hansen
and Jagannathan (1997). We document large Sharpe ratios from option
strategies trading the different moments of the pricing kernel projection. The
conditional projections themselves suggest frequent departures from mono-
tonicity and convexity. Precisely, higher-order projections frequently exhibit
a u-shape at short maturities of 1 month, in line with Beare and Schmidt
(2014) and Bakshi et al. (2010), implying an average unconditional projec-
tion that is concave (convex) in regions of low (large) returns. For longer
maturities, the average unconditional projection is concave everywhere.
4
Our paper borrows from several strands in the literature. Chapman
(1997) investigates consumption-based asset pricing models unconditionally
using technology similar to ours. A¨ıt-Sahalia and Lo (1998) estimate the
state price density (the product of the pricing kernel projection and the phys-
ical probability measure) using kernel regression. Their approach requires a
choice of regressors and uses information from the entire sample history of
option prices. Song and Xiu (2014) exploit also the information contained in
VIX options. Jackwerth (2000) investigates the marginal rate of substitution
in a complete market. A¨ıt-Sahalia and Duarte (2003) and Birke and Pilz
(2009) develop a polynomial kernel projection which maintains convexity of
option prices in strike. A¨ıt-Sahalia and Lo (2000) learn about the marginal
rate of substitution from time series data and option prices.
There is also a related literature on inequalities for moments of asset
prices and the pricing kernel. Alvarez and Jermann (2005) develop bounds
on the pricing kernel along with a decomposition under the maintained as-
sumption that it is a stationary stochastic process. Hansen and Scheinkman
(2009) obtain a similar decomposition under an additional Markov assump-
tion. Martin (2015) derives a lower bound of the equity premium from a
negative covariance condition, a joint restriction on the pricing kernel and
the market return. Julliard and Ghosh (2012) develop an empirical like-
lihood estimator of a consumption-based pricing kernel. This approach is
extended in Gosh et al. (2013) along with entropy bounds for the pricing
kernel. Within the same methodological framework, Almeida and Garcia
(2015) compute a family of discrepancy bounds for pricing kernels. Carr and
Yu (2012) trade in the discrete state assumption in Ross (2015) mentioned
above for the family of bounded diffusion processes. Boroviˇcka et al. (2014)
show that Ross (2015) recovery reveals the physical conditional density only
under additional technical conditions.
ThispaperfirstdevelopsassetpricingboundsonmomentsoftheS&P500
inSection2. Subsequentlyitputstheseboundstousetoparameterizepricing
kernel projections in Section 3. An empirical study uses these projections
in an empirical study in Section 4. Section 5 concludes and the Appendix
contains additional computations B, figures and tables in Section C.
5
2 Bounds on Conditional Polynomial Mo-
ments of the Market
In this Section we develop upper and lower bounds on polynomial moments
under the true, unobserved physical P measure of gross forward returns
R := FT,T ∈ D ⊂ R , where F is the forward price at time t of the
Ft,T + t,T
spot S&P 500 for delivery at time T ≥ t. We refer to R as the gross market
return. Under no-arbitrage the true and unobserved forward pricing kernel
and its expectation conditional on a time-T−measurable random variable R
are denoted by
dQ
M = T, and M (R) := EP[M | R], (1)
P dP P P
where Q denotes the forward martingale measure associated with the zero
T
couponbondnumeraire. Weintroducethisnotationanticipatingourfocuson
P, and that a representative Q of forward-neutral measures can be inferred
T
from option prices. This paper is based on model-independent arguments
in the sense that we do not assume an underlying stochastic process for R.
In our context it is therefore instructive to take D to be a compact subset
of R , as this ensures, together with sufficiently many options written on
+
R, a model-free fundamental theorem of asset pricing (Acciaio et al., 2013,
Remark 2.4).
We next introduce notation that will help us in the context of financial
markets equipped with European options. The time-dependent operator J
t
takes a function f, twice differentiable almost everywhere, and approximates
it in a piece-wise linear fashion inside a certain corridor, and linearizes the
6
function outside of the corridor3
(cid:90) 1 (cid:90) bt
J f(R) := f(cid:48)(cid:48)(K)(K −R)+dK + f(cid:48)(cid:48)(K)(R−K)+dK
t
at 1

f(at)+f(cid:48)(at)(R−at) R < at (2)

= f(R) a ≤ R ≤ b
t t


f(b )+f(cid:48)(b )(R−b ) R > b .
t t t t
Note that J R = R. We introduce this linearization operator to use the
t
spanning results from Carr and Madan (2001) in option markets with lim-
ited moneyness. There is a strict ordering between J f(R) and f(R) which
t
helps relating unobserved forward-neutral expectations of f(R) to observed
forward-neutral expectations of J f(R).
t
Lemma 2.1 (Observed and Theoretical Moments). The difference f(R) −
J f(R) is positive (negative) for f strictly convex (concave).
t
Importantly, operator J is curvature-preserving in the sense that if f is
t
convex (concave) also J f(R) is convex (concave). We will in the empirical
t
section work with a state space D = [a −(cid:15) ,b +(cid:15) ] for some (cid:15) ,(cid:15) > 0,
t l,t t u,t l,t u,t
not too big, in which case J f(R) ≈ f(R). It is tempting to assume that
t
the state space of R agrees exactly with the observed option moneyness, but
then the price of the farthest out-of-the-money put and call options would
need to be zero.
Ourmethodologyisbasedonmoments. Toensurethatwhatwedoiswell-
defined we need moments of all orders. Denote by P the time-t-conditional
t
physical probability measure (and likewise Q the T-forward measure con-
t,T
3In practice we will compute Eq. (2) from a finite out-of-the-money option portfolio
with strikes a =K <K <···<K =F <K <···<K =b as
t 1 2 n t,T n+1 N t
n−1 N
(cid:88) (cid:88)
f(cid:48)(cid:48)(K )(K −F )+∆K + f(cid:48)(cid:48)(K )(F −K )+∆K
i i T,T i i T,T i i
i=1 i=n
with
∆K :=K −K , ∆K :=1/2(K −K ) for 1<i<N, and ∆K :=K −K .
1 2 1 i i+1 i−1 N N N−1
7
ditional on time t information) generating the conditional expectations EP[·],
t
respectively EQT [·].
t
Lemma 2.2 (Existence of Moments). The moment-generating functions
EP(cid:2)eu·R(cid:3), and EQT (cid:2)eu·R(cid:3) (3)
t t
exist for u ∈ R .
+
The compactness of the state space D guarantees integrability even for
fat-tailed distributions.4 A computation shows that Lemma 2.2 also guaran-
tees existence of moments of discrete returns R := R−1, and corridor mo-
e
ments J Rn. Denote the conditional monomial moments by µP := EP[Rn]
t t,n t
and µQT := EQT [Rn], respectively. The next assumption is on expected
t,n t
profits of trading strategies with exposure to nonlinear functions of R.
Assumption 1 (Negative Divergence Premium (NDP)). With power diver-
gence function
Rp −pR+p−1
D (R) := ,
p p2 −p (4)
D (R) := Rlog(R)−R+1, and D (R) := R−log(R)−1,
1 0
we define the negative n-power divergence premium NDP(p,n) assumption
of orders p and n as the inequality
−CovP[M,J D (Rn)] = −CovP[M(R),J D (Rn)] ≤ 0. (5)
t t p t t p
Power divergence swaps introduced by Schneider and Trojani (2015) pay-
ing the difference between realized and implied divergence5 can be replicated
4Standard models on unbounded state spaces such as Black-Scholes or Heston (1993)
donotsatisfytherequirementofacompactstatespacenecessaryforLemma2.2,butthey
are relatively easy to compactify. We use them in Figure 4 to illustrate model likelihood
ratios.
5In their construction realized divergence depends on the path of the forward price
from time t to time T, but all payoffs arise at time T, such that there is no difference
between pricing with MP, or MP(R).
8
from option data. Together with the identity
n
D (Rn) = [(np−1)D (R)−(n−1)D (R)], with
p pn n
p−1
n
D (Rn) := lim [(np−1)D (R)−(n−1)D (R)] (6)
1 pn n
p→1 p−1
= nRnlog(R)−Rn +1,
Assumption 1 is therefore empirically testable unconditionally, since both
D (R) and D (R) are tradeable quantities. From Assumption 1 and
pn n
Jensen’s inequality we directly get
Proposition 2.3 (Upper Bounds on Conditional P Moments). NDP(p,n)
holds if and only if
S(J D (Rn)) := EQT [J D (Rn)] ≥ EP[J D (Rn)] ≥ J D (EP[Rn]). (7)
t p t t p t t p t p t
From Proposition 2.3 above we define implicitly the upper bound on the
n-th moment of R, µPupper, as the solution to
t,n
S(J D (Rn)) = J D (µPupper). (8)
t p t p t,n
The NDP from Assumption 1 establishes relations between Q and P mo-
T
ments of different orders by varying p and n and therefore entails economic
information beyond Proposition 2.3. The next assumption is harder to test
empirically.
Assumption 2 (Negative Covariance Condition (NCC)). For p,q ∈ R we
define the negative covariance condition NCC(p,q) as the inequality
CovP[MRq,Rp] = CovP[M(R)Rq,Rp]
t t
(9)
= EQT (cid:2)Rp+q(cid:3)−EQT [Rq]EP[Rp] ≤ 0.
t t t
To adapt Assumption 2 to finite option markets we employ Lemma 2.1
9
and define for q ∈ (0,1] and p+q > 1
EQT [J Rp+q] EQT [Rp+q]
L(p,q) := t t ≤ t ≤ EP[Rp]. (10)
EQT [J Rq] EQT [Rq] t
t t t
This inequality is not binding, whenever the equity premium is assumed
positive, for q ≤ 0. In contrast, it provides informative bounds for q > 0.
In the empirical investigation we employ L(1,q), for which there is ample
evidence for its validity from a battery of economic models and empirical
tests.6 A weaker lower bound, assuming only the NCC for p = 1, is given by
Proposition 2.4 (Lower Bounds on Conditional P Moments). Suppose
NCC(1,q) holds for q ∈ (0,1], then
L(1,q)p ≤ EP[Rp]. (11)
t
Analogously to the definition of the upper bound, we define the lower
boundµPlower correspondingtoProposition2.4basedonL(1,1). Whileupper
t,n
and lower bounds on gross returns of the market are intrinsically interesting,
we have one particular application in mind in the context of pricing kernel
projections that we will elaborate in the Section below.
3 Nonlinear Pricing Kernel Projections
This Section explains how the pricing kernel can be made visible through the
moment bounds and option prices and how this projection relates to the ex-
tant literature. Here we adopt the framework of Filipovi´c et al. (2013) to ex-
pand the unobservable pricing kernel in terms of the moments of the forward-
neutral density Q and a candidate physical measure M. This method is
T
preferable over Taylor expansions in the context of approximating likelihood
6Martin (2015) mentions for the NCC(1,1): 1) a jointly log-normal pricing kernel and
market return and a Sharpe ratio on the market that is greater than its volatility, 2) a
representative agent who maximizes expected utility and whose risk aversion is at least 1
3) an Epstein-Zin representative agent with risk aversion greater than 1, and elasticity of
inter-temporal substitution greater than 1. From Schmidt (2003) these results carry over
for R-measurable pricing kernels also to NCC(p,1), p>1.
10