Model‐Free International Stochastic Discount Factors

• DOI: http://doi.org/10.1111/jofi.12970
• Date: 01 April 2021
• Published date: 01 April 2021
• Author: MIRELA SANDULESCU,FABIO TROJANI,ANDREA VEDOLIN

THE JOURNAL OF FINANCE •VOL. LXXVI, NO. 2 •APRIL 2021

Model-Free International Stochastic

Discount Factors

MIRELA SANDULESCU, FABIO TROJANI, and ANDREA VEDOLIN∗

ABSTRACT

We provide a theoretical framework to uncover in a model-free way the relationships

among international stochastic discount factors (SDFs), stochastic wedges, and f‌inan-

cial market structures. Exchange rates are in general different from the ratio of in-

ternational SDFs in incomplete markets, as captured by a stochastic wedge. Weshow

theoretically that this wedge can be zero in incomplete and integrated markets. Mar-

ket segmentation breaks the strong link between exchange rates and international

SDFs, which helps address salient features of international asset returns while keep-

ing the volatility and cross-country correlation of SDFs at moderate levels.

THE FINANCIAL CRISIS PROVIDED AT least two insights into the workings of

international f‌inancial markets. First, it was a reminder that despite the pro-

gressive removal of trade barriers such as capital controls and taxes, cross-

border investment activity can be severely disrupted. Indeed, recent empirical

evidence shows that international market segmentation remains a pervasive

feature even in the most liquid and developed markets (see Camanho, Hau,

and Rey (2018) for international equities and Maggiori, Neiman, and Schreger

(2020) for international f‌ixed income markets). Second, the crisis highlighted

the importance of f‌inancial frictions, triggering a vast literature documenting

∗Mirela Sandulescu is with the University of Michigan, USI Lugano, and Swiss Finance Insti-

tute. FabioTrojani is with the University of Geneva and Swiss Finance Institute. Andrea Vedolin is

with Boston University,NBER, and CEPR. We thank Stefan Nagel (the Editor); Caio Almeida; Ric

Colacito; Federico Gavazzoni; Tarek Hassan; Robert Kollmann; Matteo Maggiori; Thomas Mau-

rer; Anna Pavlova; Carolin Pf‌lueger; Alireza Tahbaz-Salehi; Ngoc-Khanh Tran; Raman Uppal;

Adrien Verdelhan; Tony Zhang; the Alphacruncher Team; an Associate Editor; anonymous ref-

erees; and seminar and conference participants at the International Conference of the French

Finance Association (Valence), SoFiE Annual Meeting at NYU, the International Finance Confer-

ence at Cass Business School, the SITE workshop “New Models of Financial Markets,” the Chicago

Booth International Macro Finance Conference 2017, the ECB/Banca d’Italia/Norges Bank “Finan-

cial Determinants of Foreign Exchange Rates” conference, the 2018 Econometric Society Meetings

in Philadelphia, the 2018 European Finance Association Meetings in Warsaw, the 2019 Ameri-

can Finance Associate Meetings in Atlanta, Boston University, University of Tilburg, VU Ams-

terdam/Tinbergen Institute, Université Catholique de Louvain, and McGill University for helpful

comments. We have read TheJournal of Finance disclosure policy and have no conf‌licts of interest

to disclose.

Correspondence: Fabio Trojani, University of Geneva, 42 Bd du Pont d‘Arve, CH-1211 Geneva

4, Switzerland; e-mail: Fabio.Trojani@alphacruncher.com.

DOI: 10.1111/jof‌i.12970

© 2020 the American Finance Association

935

936 The Journal of Finance®

the tight link between asset returns and the health of the f‌inancial intermedi-

ary sector. In particular, most episodes of disruptions in international f‌inancial

markets are viewed as caused by capital frictions in intermediation. Start-

ing from these observations, in this paper, we develop a model-free theoretical

framework for studying the implications of international market segmentation

for asset prices and the role of intermediaries in international f‌inancial mar-

kets.

Canonical models in international f‌inance assume complete and integrated

markets in which stochastic discount factors (SDFs) and exchange rates are

pinned down by the marginal utilities of domestic and foreign households.1Un-

der such assumptions, the rate of appreciation of the real exchange rate (X)is

equal to the ratio of foreign (Mf) and domestic (Md)SDFs,thatis,X=Mf/Md,

an identity referred to as the asset market view of exchange rates. Colacito

and Croce (2011,2013), for instance, rely on recursive preferences and highly

correlated long-term SDF components in a complete market setting to address

many salient features of international asset returns and macroeconomic quan-

tities.2

Despite the success of these models, another strand of the literature ques-

tions the completeness assumption of international f‌inancial markets and asks

whether market incompleteness can help quantitatively match the data. In-

completeness is appealing because it breaks the link between exchange rate re-

turns and SDF ratios. The resulting deviations from the asset market view can

be captured by a stochastic exchange rate wedge (Backus, Foresi, and Telmer

(2001)). However, recent evidence by Lustig and Verdelhan (2019)showsthat

in a no-arbitrage setting, some of the constraints imposed on the wedge to

jointly address international f‌inance puzzles may be diff‌icult to reconcile with

the data.

In this paper, we take a different approach that is motivated by the empirical

observation that markets are likely segmented internationally. Note that mar-

ket completeness and market integration are two distinct concepts. Whereas

market completeness pertains to investors’ ability to hedge relevant risks in

an economy using portfolios of traded payoffs, international f‌inancial market

segmentation corresponds to differences in the sets of traded payoffs across

currency denominations. We adopt a parsimonious framework that allows us

to directly uncover, in a model-free way, the relationships among international

SDFs, stochastic wedges, and international market structures.

1In the following, we take a market to be integrated if domestic investors can trade all foreign

assets via the exchange rate market and vice versa. In contrast, segmented markets imply that

some assets are accessible to investors in one country but not another.

2Other examples include the habit model of Heyerdahl-Larsen (2014) and Stathopoulos (2017)

to generate sizable currency risk premia. Farhi and Gabaix (2016) rely on a complete market econ-

omy with time-additive preferences and a time-varying probability of rare consumption disasters.

Gabaix and Maggiori (2015) provide a theory of exchange rate determination based on capital f‌lows

in segmented f‌inancial markets with heterogenous trading technologies. Hassan (2013), Hassan

and Mano (2019), and Colacito et al. (2018) emphasize the importance of heterogeneous exposure

to global shocks for matching currency risk premia.

Model-Free International SDFs 937

Using this approach, we f‌irst establish theoretically that as long as mar-

kets are integrated, stochastic wedges are always equal to 0 with respect to

a certain pair of international SDFs, irrespective of the extent of market in-

completeness. This result implies that exchange rate risks can be uniquely

determined by the ratio of these SDFs, and thus challenges the notion that

market incompleteness alone may generate deviations from the asset market

view. We next show that market segmentation severs the straightjacket of the

asset market view and leads to volatile stochastic wedges. This result not only

helps address salient features of international asset returns, but also leads to

low SDF volatilities and low cross-country correlations of international SDFs.

Given the multitude of SDFs that price returns in incomplete markets, we

start our analysis by focusing on the family of minimum dispersion SDFs,

each of which minimizes a different notion of variability. This family includes

the well-known Hansen and Jagannathan (1991) SDF, which minimizes the

SDF variance, as well as the minimum entropy SDF. In our main theoretical

contribution, we show that when markets are integrated, minimum entropy

SDFs always imply the validity of the asset market view, that is, the resulting

Backus, Foresi, and Telmer (2001)-type stochastic wedge is equal to 0 even if

markets are incomplete. Hence, in integrated markets, exchange rate risk is

pinned down by the dynamics of international minimum entropy SDFs, as is

the case in complete markets. This result is a consequence of the specif‌ic func-

tional relationship between the minimum entropy SDF and the optimal growth

portfolio in each country: the minimum entropy SDF in each country is equal

to the reciprocal of the optimal growth portfolio return in that country. As a

result, the change in currency denomination, which transforms the return in

one country to that in the other, also transforms the minimum entropy SDF of

the given country to that in the other. Taken together, our results indicate that

market segmentation is the only way to generate a deviation from the asset

market view for the pair of minimum entropy SDFs.

The above f‌indings have important implications for the three asset pric-

ing puzzles that have dominated international f‌inance: (i) the low exchange

rate volatility puzzle of Obstfeld and Rogoff (2001) and Brandt, Cochrane, and

Santa-Clara (2006), (ii) the cyclicality puzzle, or the disconnect between ex-

change rates and macroeconomic variables, of Kollmann (1991) and Backus

and Smith (1993), and (iii) the forward premium anomaly of Hansen and Ho-

drick (1980)andFama(1984). Indeed, if one were to ask “do stochastic wedges

help address exchange rate puzzles in integrated markets?”, the answer would

be a resounding no. First, when we assume that domestic and foreign investors

can trade short-term bonds internationally,we show that the international Eu-

ler equations pin down the currency risk premium by construction, irrespective

of the properties of stochastic wedges. Second, we show that stochastic wedges

can always be interpreted as a measure of the unspanned exchange rate risks

in international f‌inancial markets, that is, risks not insurable by linear portfo-

lios of basic assets, such as stocks and bonds, in each market. Third, we show

that in the data, stochastic wedges are zero or minuscule for all minimum

dispersion SDF pairs when markets are integrated. We therefore extend our