Di¤erences in Beliefs andCurrency Options Markets

ALESSANDRO BEBER, FRANCIS BREEDON, and ANDREA BURASCHI�

ABSTRACT

This paper investigates how heterogeneous beliefs of professional investors impacts on the currency

options market. Using a unique data set with detailed information on the foreign-exchange forecasts

of about 50 market participants over more than ten years, we construct an empirical proxy for dif-

ferences in beliefs. We show that our proxy has a statistically and economically strong e¤ect on the

implied volatility of currency options, on the shape of the implied volatility smile, on the volatility

risk-premia, and on future currency returns. We document that the volatility of macroeconomic

fundamentals explains only a limited part of the di¤erence in beliefs dynamics. Furthermore, our

�ndings cannot be explained by overcon�dence about priors or about a private signal. Our evi-

dence demonstrates that a process largely unrelated to current macroeconomic fundamentals can

have important asset pricing implications.

JEL classi�cation: D9, E3, E4, G12

Keywords: Option pricing, di¤erence in beliefs, incomplete markets, foreign exchange.

Last version: November, 2007.

This paper investigates the effects of heterogeneous beliefs in the foreign

exchange market. Speci�cally, we analyze the impact of the di¤erences in beliefs which

are implicit in currency forecasts of major market participants on the implied volatility of

currency options and, more generally, on the risk premium in the foreign exchange (FX)

market.

�Alessandro Beber is at HEC Lausanne. Andrea Buraschi and Francis Breedon are at Imperial College

London. We thank Karim Abadir, Philippe Bacchetta, Bruno Biais, Michael Brandt, Mike Chernov, Joost

Driessen, Darrell Du¢ e, Bernard Dumas, and conference participants at the CEPR meeting in Gerzensee,

the European Finance Association 2007 meeting in Llubjana, the Swiss Finance Meetings, Imperial College,

London Business School, Queen�s University, and University of Oxford for comments. The usual disclaimer

applies. We would also like to thank Citigroup and Lehman Brothers for supplying market data and Reuters

for their forecast database.

The description of a noteworthy event in the FX markets is useful to illustrate intu-

itively the potential link between exchange rates and di¤erences in beliefs. At the beginning

of 2004, the FX market was approaching an �overwhelming consensus�that the yen would

continue to appreciate against the dollar.1 The combination of a historically small inter-

est rate di¤erential, large and opposing current account imbalances and strong economic

growth in Japan generated the belief that only the Bank of Japan�s persistent currency

intervention was stopping further declines in the dollar/yen exchange rate from becoming

a �one-way bet�.2 At the beginning of 2004 about 75% of forecasters in our sample were

predicting yen appreciation over the coming three months. Figure 1, Panel 1, shows an

histogram of dollar/yen forecasts at the beginning of January 2004 that illustrates the low

disagreement of professional investors. At the same time, dollar/yen implied volatility fell

to a multi-year low. Over the subsequent months, however, record levels of intervention

and a surprise monetary loosening by the Bank of Japan, coupled with G7 communiqué

expressing concerns over the impact of the appreciation of the Yen for economic growth

changed the consensus and created active disagreement in the market over the future of

the dollar/yen exchange rate. The impact of these events on the level of the exchange rate

itself was not dramatic (the Yen weakened about 3% against the dollar during February

having appreciated by almost 2% in January). Its impact on market expectations, how-

ever, was as the Yen appreciation consensus was broken and in March the dispersion of

FX forecasts increased dramatically (see Figure 1, panel 2). At the same time, the implied

volatility on one month options increased by 50% (about 4 percentage points) and the slope

of the implied volatility smile increased substantially, with important implications on the

option-implied state price densities (see Figure 1, panel 3 and 4).3 This example shows

how disagreement amongst market participants is intimately tied to volatility. The aim of

this paper is to investigate systematically the role of disagreement in FX markets and, in

particular, in currency option markets.

The appropriate theoretical framework considers the implications of economic uncer-

tainty in a multiple agent economy. When agents are uncertain about future cash �ows or

dividend growth rates, they eventually disagree on asset expected returns and construct dif-

ferent optimal portfolios. This may ultimately a¤ect equilibrium asset prices and volatilities,

which endogenously re�ect the dispersion in beliefs. In this paper, we �rst brie�y outline a

general equilibrium model in which two agents (one domestic and one foreign) select their

optimal consumption plans based on two di¤erent sets of beliefs (on the probability dis-

1See, for example, Institutional Investor 12.2.2004. For further details on the FX mar-ket in the �rst quarter of 2004, see the report of the Federal Reserve Bank of New York athttp://www.newyorkfed.org/newsevents/news/markets/2004/fxq104.pdf

2See, for example, Financial Times 9.1.2004.3We obtain the option-implied state price density as the second derivative of the option pricing formula

with respect to the strike price (Breeden and Litzenberger, 1978). The option pricing formula is the Black-Scholes formula, where we replace the constant volatility parameter with a cubic-spline interpolated impliedvolatility smile (see, among others, Shimko,1993).

1

tribution dQ), which creates a wedge in equilibrium between their marginal utilities. The

intuition of the main result is simple and comes from the observation that when two agents

have di¤erent beliefs dQ1 6= dQ2, the socially optimal ex-ante allocation between the two

agents must satisfy 0 =R h

u0(c1) + u0(c2)dQ2dQ1

idQ1; which in general depends on the di¤er-

ence of beliefs dQ2dQ1. Under the special case of homogeneous beliefs, that is when dQ1 = dQ2,

we have the standard result that the distribution of consumption depends exclusively on

preference characteristics, such as discount rates, risk aversion, the elasticities of intertem-

poral substitution, and di¤erences in the distribution of initial wealth. From an asset pricing

perspective, one can notice that since changes in dQ2dQ1

a¤ect the relative weights of the two

agents in the representative agent utility function, the equilibrium stochastic discount factor

is a¤ected by two components. The �rst component is aggregate endowment shocks, as in

standard Lucas endowment economies, since this a¤ects the total amount of resources that

can be shared by the two agents. The second component is the di¤erence in beliefs dQ2dQ1,

since this a¤ects the relative consumption share of the agents, given the total amount of

resources available. This second term allows the stochastic discount factor to be volatile

even when the endowment process is smooth. This e¤ect yields potentially important asset

pricing implications. Since exchange rates are equal to marginal rates of substitution be-

tween domestic and foreign consumption goods, di¤erences in beliefs a¤ect foreign exchange

risk premia, thus the spread between FX risk-neutral and realized volatility as well as FX

expected returns (i.e carry trades).

This paper investigates the empirical importance of this link by combining two data

sets. The �rst one is based on over ten years of daily options data for three currency pairs

(EUR/USD, USD/JPY, and GBP/USD), at four di¤erent constant maturities, and di¤erent

moneyness levels. The second data set is a detailed record of currency forecasts made by a

large cross-section of professional market participants that we use to obtain a proxy for the

di¤erences in beliefs. More speci�cally, at each survey date, from the distribution of FX

forecasts we compute measures of cross-sectional dispersion, skewness and kurtosis. These

measures capture di¤erent characteristics of the di¤erences in beliefs in the FX market.

It is important to highlight that a key advantage of our data set is that it allows us to

measure dispersion of beliefs directly on the underlying asset. In contrast, other papers

in the literature construct proxies for the di¤erences in beliefs about factors that are only

related to the underlying asset prices. For example, Anderson et. al. (2006) construct

proxies of di¤erences in beliefs about stock prices using forecasts about earnings, which

require some additional modelling assumptions linking earnings to asset prices.

Our analysis focuses on option markets. A growing body of empirical evidence shows

that options are used for informational driven trades (see, among others, Pan and Potesh-

man (2006), Amin and Lee (1997), Cao, Chen, and Gri¢ n (2005), and Easley, O�Hara, and

Srinivas (1998)). These studies, however, focus on individual stock options and conjecture

2

that some investors have private information on the underlying stock, which they then try

to exploit by trading in the option market. In the case of FX options it is di¢ cult to

argue that private information plays an important role in such a global market. Thus, the

FX market is the ideal laboratory to investigate a di¤erent source of informational driven

trades, such as heterogeneous beliefs in the absence of private information. We use our data

sets to address the following seven questions:

First, do di¤erences in beliefs about future foreign exchange returns a¤ect the level

of implied volatility of currency options? We �nd that they do and that the results are

both statistically signi�cant and economically important. Moreover, our �nding is valid

at four di¤erent horizons and on three di¤erent underlying exchange rates. For instance,

for the USD/JPY exchange rate, our proxy for the di¤erence in beliefs explains up to 64%

of the contemporaneous variation in at-the-money implied volatilities. More speci�cally,

a high dispersion of beliefs (one standard deviation above the mean) increases implied

volatility by about two percent. For the other currencies the e¤ect is slightly smaller but

the coe¢ cients are always signi�cant at the 1% con�dence level. The results support the

model�s implication that the beliefs�structure in the foreign-exchange market a¤ects prices

of derivatives contracts.

Second, do di¤erences in beliefs explain the shape of the implied volatility smile? We

�nd that they do. For instance, our empirical measure of the dispersion of beliefs for the

EUR/USD exchange rates explain, depending on the horizon, between 9% and 14%, of the

contemporaneous variation in the slope of the implied volatility smile. Furthermore, in

periods with high dispersion of beliefs, the slope of the implied volatility smile changes by

about one third. We obtain similar results for the other two currency pairs. This result

is interesting since it implies that agents with di¤erent subjective valuations about market

fundamentals engage in risk sharing trading, thus a¤ecting the relative prices of options

with di¤erent moneyness.

Third, does the dispersion of beliefs a¤ect the volatility risk premia, i.e. the spread

between implied and realized volatility? We �nd that it does: higher di¤erences in beliefs

have a positive impact on the volatility risk premium. For example, for the USD/JPY

exchange rate, the dispersion of the foreign exchange forecasts alone explains between 37%

and 60% of the variation of the volatility risk premia. This �nding suggests that di¤erences

in beliefs directly a¤ect the pricing kernel and thus the di¤erence between the physical and

the risk-neutral probability measure is correlated with di¤erences in beliefs.

Fourth, do di¤erences in beliefs a¤ect option prices because they a¤ect idiosyncratic

volatility or because they are a priced risk factor? We empirically investigate this question

by examining directly the dynamic properties of the underlying asset and investigate the

extent to which di¤erences in beliefs help explain the exchange rate returns. We �nd that

the dispersion in beliefs explains future currency returns beyond what is embedded in the

3

forward premium. This result supports the �ndings in the currency option market, and

is consistent with the interpretation of the di¤erences in beliefs being a component of a

time-varying foreign-exchange risk premium. This is important for at least two reasons:

�rst, it is linked to the extensive literature on the foreign exchange expectation hypothesis

puzzle and it suggests a simple interpretation of classical results. Second, it provides a

simple out-of-sample robustness check of the results based on option data.

Fifth, does the evidence of a risk-premium on the di¤erence in beliefs help to explain

the FX carry-trade expected return puzzle? It is well-known from several empirical studies

that a simple investment strategy based on borrowing in low interest rates currencies and

investing in high interest rates currencies (also known as positive carry trades) yields large

average excess returns. In the last decades, the Sharpe ratios of such a trades have been

several times higher than the Sharpe ratio of an investment in the S&P500 index. Not sur-

prisingly, this strategy has become extremely popular among hedge funds and it eventually

gained center stage during the Summer 2007 when a sharp revaluation of the japanese Yen

revealed the extent of exposure of many hedge funds to these strategies. Our empirical

analysis shows that large di¤erences in beliefs are strongly correlated with high expected

FX returns, consistent with the interpretation of the di¤erence in beliefs being a priced risk

factor. This result is important since it suggests a simple structural explanation that links

contemporaneous changes in FX volatility and carry trades expected returns, which has

been one of the most thought provoking empirical regularities in the FX literature.

Sixth, does our proxy for di¤erences in beliefs simply re�ect the volatility of macroeco-

nomic fundamentals? Theory suggests a separate role for di¤erence in beliefs in generating

volatility of exchange rates even when fundamentals are smooth. However, one could argue

that empirically uncertainty about fundamentals can itself drive the dynamics of the dis-

persion of beliefs. We show that the volatility of the main macroeconomic indicators can

explain only a very small part of the evolution of the di¤erence in beliefs. Clearly, however,

di¤erences in beliefs might capture uncertainty about future fundamentals and the result

suggests the importance of taking explicitly into account di¤erences in beliefs, in addition

to current changes in fundamentals.

Last, does our proxy for di¤erences in beliefs re�ect a behavioral e¤ect? A common

approach to support equilibrium di¤erences in beliefs in economies with no private infor-

mation is to assume that agents have di¤erent degrees of con�dence in either their priors

or signals. We show that a proxy for overcon�dence is indeed statistically signi�cant, but

it does not drive out the independent explanatory power of the di¤erences in beliefs. This

�nding suggests that overcon�dence is not the only mechanism that generates di¤erence in

beliefs.

The results in this paper have broad implications as they demonstrate that a process

that is largely unrelated to current changes in fundamentals can have important asset pricing

4

e¤ects by a¤ecting FX risk premia. Many causes of uncertainty are exogenous and are

naturally resolved over time. Other sources of uncertainty are instead of a strategic nature,

as the timing of their resolution is a strategic variable for some agents with incentives to hide

their information. Financial asset prices re�ect uncertainty and reveal information about its

resolution. Our �ndings are important, because they show that the simple fact that agents

have di¤erent expectations of future market fundamentals generates a number of e¤ects

on asset prices. Although the FX market is an ideal laboratory for our empirical study

because of the nature of information and the unique data we could collect, our evidence

is relevant for asset pricing in other markets as well. Speci�cally, our �ndings on di¤erent

option pricing dimensions show empirically that di¤erences in beliefs have an impact on the

pricing kernel and, as a result, on pricing all �nancial assets.

Literature Review. Our paper draws from a large number of contributions in the litera-

ture that investigate the link between economic uncertainty and asset prices. David (1997),

Veronesi (1999), and Veronesi (2000), study the role of incomplete information in �nancial

markets when agents are uncertain and need to learn about the state of the investment

opportunity set.4 They show that uncertainty has non trivial implications for asset prices,

a¤ecting equilibrium stock market volatility and expected returns. Using a di¤erent mod-

elling approach, Hansen and Sargent have investigated in a series of articles the behaviour

of an agent who regards his model as a reference point, but suspects small speci�cation

errors that are modelled as absolute continuous deviations from the true empirical process.

This agent wants decisions to be insensitive to these errors, thus leading him to focus on the

consequences of his decisions under "worst case scenerios". They show that economically

signi�cant data-implied measures of risk aversion can emerge from a concern about making

small speci�cation errors. An important di¤erence with respect to this literature is that we

introduce heterogeneity among agents.5 When the rate of growth of cash �ows is unknown

and needs to be estimated, agents may form di¤erent expectations (beliefs) depending on

their priors or their interpretation of news. When this occurs, this source of belief hetero-

geneity becomes an independent motive to trade. Buraschi and Jiltsov (2006) show that

options can be used to hedge against unexpected changes in the di¤erences in beliefs. Since

changes in the di¤erence in beliefs a¤ect the relative demands of risk sharing insurance

contracts, it also a¤ects option prices and the shape of the option implied volatility smile.

Our empirical results contribute to the literature that explores the pricing implications

of di¤erences in analysts�forecasts. This literature has focused on �rm�s earnings and stock

4The original asset pricing models of economies with incomplete information include Detemple (1986),Dothan and Feldman (1986), Genotte (1986).

5The implications of heterogeneous beliefs have originally been studied by Detemple and Murthy (1994)and Zapatero (1998), who study how �nancial innovation may a¤ect interest rate volatility, Basak (2000),who studies the existence of sunspot equilibria in economies with extraneous risk, and Buraschi and Jiltsov(2006), who investigate the implications on the joint behaviour of option prices and trading volumes. SeeBasak (2005) for a survey of this literature.

5

market prices, but the results and their interpretation are ambiguous. Diether, Malloy, and

Scherbina (2002) �nd that in the period 1983-1991, the dispersion of analysts short-term

earnings forecasts (obtained from the Institutional Brokers Estimate System (I/B/E/S)) is

negatively correlated with future returns. They interpret this result as evidence of binding

short selling constraints and reject the interpretation of a priced source of risk. Ander-

son, Gyshels, and Juergens (2005), on the other hand, use both short-term and long-term

earnings forecasts and �nd a positive relation between analysts�dispersions and expected

returns, even after controlling for the market risk premium, Fama-French factors and a

momentum factor. Their analysis is based on a di¤erent analysts� forecast data sample

(compiled by First Call/Thompson for all S&P500 �rms), a di¤erent period (1991-1997)

and a di¤erent methodology. We contribute to this literature in several respects: First,

since equity analysts do not provide forecasts of future returns per se, these studies need to

make assumptions on a pricing model to map earnings forecasts to stock price forecasts.6 In

our study, we work directly with price (exchange rate) forecasts. Thus, our results are not

sensitive to these types of modelling assumptions. Second, there is considerable evidence

that analysts�forecasts of earnings are a¤ected by agency issues between �rms and invest-

ment banks (Rajan and Servaes (1997), Daniel, Hirshleifer, and Teoh (2002)), which has

recently led to the introduction of several regulatory measures. Given the size and liquidity

of the FX market, agency biases are much less of a concern in our analysis. Moreover,

the limited role of private information in FX markets also reduces these agency concerns.

Third, we investigate the link between dispersion in beliefs and the volatility risk premium

(the spread between the option-implied risk adjusted volatility and the realized physical

volatility). This allows us to study more directly the extent to which the dispersion in

beliefs is a priced risk factor. Fourth, the absence of short selling constraints in this market

allows us to rule that out as an explanation for our results. Finally, our FX survey features

an average of about 50 market participants�forecasts, which is more than three times larger

than the number of analyst forecasters included in the First Call/Thompson equity survey.

The computation of our dispersion measure is thus likely to be more reliable.

This paper also relates to a broad literature that studies the role of overcon�dence in

agents�decisions and asset prices. Di¤erences in beliefs concerning market fundamentals

can either be the consequence of rational learning in economies with uncertainty and hetero-

geneous priors, or of heterogeneity in the con�dence agents have about their priors or about

their signals. Scheinkman and Xiong (2003) and Dumas, Krushev, and Uppal (2006) study

economies in which di¤erent degrees of con�dence in the accuracy of the signals generate

heterogeneity in posterior beliefs and show how this has direct implications for asset prices

and volatility. We contribute to this literature by documenting the existence of an empirical

relation between the di¤erence in beliefs and implied volatility in a market in which private

6Anderson, Ghysels, and Juergens (2005), for instance, consider a constant dividend growth model andassume that all earnings are distributed as dividends.

6

information is unlikely to a¤ect the results and a direct measure of beliefs on the future

value of the underlying asset is available.

Additionally, our paper is also related to the more recent foreign exchange literature

that has used currency options. Two papers are the most closely related. Driessen and

Perotti (2003) use currency option prices to study the evolution of investor con�dence in

1992-1998 over the probability that individual currencies would converge to the Euro. They

show that convergence risk, which may re�ect uncertainty over policy commitment as well

as exogenous fundamentals, induces a level of implied volatility in excess of actual volatility

(the volatility wedge). Our results are consistent with their �ndings since we also observe

that disagreement has a positive e¤ect on the volatility risk premium. Finally, Carr and Wu

(2006) document that the conditional risk-neutral distribution of currency returns can show

strong asymmetry. Our paper provides an economic explanation for this stylized fact by

showing that di¤erences in beliefs have strong explanatory power for the implied volatility

smile dynamics.

The remainder of the paper proceeds as follows. In Section I, we present a stylized

model to illustrate the link between heterogeneous beliefs and asset prices and to sketch

out testable implications. Section II provides further details about the data and conducts

some preliminary analysis. In Section III, we show the empirical results. In Section IV, we

add some robustness checks and we discuss potential explanation for our �ndings. Section

V concludes.

I. Model and Testable Implications

To illustrate the link between heterogeneous beliefs and asset prices, let us consider a two-

country economy producing two goods �(t) and ��(t) whose aggregate supply is given by

d�(t) = �(t) [��(t)dt+ ��(t)dZt] ; (1)

d��(t) = ��(t) [���(t)dt+ ���(t)dZ�t ] :

Investors can observe �(t) and ��(t), but they must estimate the unknown drifts ��(t) and

���(t) using their information and current beliefs. Let �i�(t) and �

i��(t) be the perceived

growth rates of the two aggregate endowment processes as perceived by agents i = 1; 2. A

large literature explores the rational foundations or psycological biases that might induce

the observed disagreement. Given the objectives in this paper, we abstract from a speci�c

learning process and instead take the disagreement as given and exogenous. Since �i�(t) and

�i��(t) are not known and di¤er across agents, they associate the same observable economic

realizations to di¤erent Brownian innovations. The individual speci�c innovation processes

are given by dZit =1

��(t)

hd�(t)�(t) � �

i�(t)dt

iand dZ�it =

1��(t)

hd��(t)��(t) � �

i��(t)dt

i. The di¤erence

7

in the two innovations processes is proportional to the di¤erence in beliefs

dZ1t = dZ2t +�1�(t)� �2�(t)

��(t)dt and dZ�1t = dZ�2t +

�1��(t)� �2��(t)���(t)

dt: (2)

Thus, let us de�ne �(t) =�1�(t)��2�(t)

��(t)and ��(t) =

�1�� (t)��

2�� (t)

��� (t): These two processes are

the disagreement between the two agents standardized by the local volatility of the two

endowment goods. They can be interpreted as Radon-Nikodym derivatives describing the

change of probability measures between the two agents.

Suppose now that the agents in the two countries can trade in two risky non-dividend

paying assets j = 1; 2 and one riskless asset paying a riskless rate r, so that markets are

dynamically complete. With no loss of generality, the equilibrium perceived dynamics of

the two risky assets follow a di¤usion process

dSij(t) = Sij(t)��ij(t)dt+ �j(t)dZ

it + �

�j (t)dZ

�it

�j = 1; 2: (3)

Since asset prices are observable, agents must agree on the value of Sj(t) at any time t, so

thatdS1jS1j� dS2j

S2j= 0. Thus, using (3), simple algebra shows that the disagreement about the

endowment process must imply a speci�c form of disagreement with regards to the expected

returns of the two �nancial assets:

�1j (t)� �2j (t) = �j(t) �(t) + ��j (t) ��(t): (4)

Clearly, the di¤erence in the perceived expected rates of return must be linked to the di¤er-

ence in the market prices of risk perceived by the two agents. Let mi(t) be the individual-i

stochastic discount factor, such that the price of any tradable asset discounted at mi(t) is a

martingale according to the information set of agent i, mi(t)Sij(t) = Eit

hmi(T )Sij(T )

i. The

stochastic discount factors have a common drift �r(t), otherwise there would be arbitrageopportunities in the riskless asset market, but stochastic components that are a function of

the individual speci�c Brownian innovations:

dmi(t) = �mi(t)�r(t)dt+ �i(t)dZi(t) + ��i(t)dZ�i(t)

�, (5)

As usual, �i and ��i have the interpretation of market prices of risk, which in this economy

are individual-speci�c. For equation (4) to be supported in equilibrium, for any risky asset

j

�ij � r = �j(t)�i + ��j (t)�

�i; (6)

which implies that (�1j � �2j ) = �j(t)(�1 � �2) + ��j (t)(�

�1 � ��2). Together with equation

8

(4), this directly implies that

�1(t)� �2(t) = (t), and (7)

��1(t)� ��2(t) = �(t). (8)

This condition shows that di¤erences of beliefs about market fundamentals, (t) and �(t),

generate di¤erences in the market prices of risks that agents use to price contingent claims,

with immediate implications for exchange rate dynamics.

To obtain speci�c predictions about exchange rates, let us assume that agents�prefer-

ences are separable with respect to the two consumption goods and de�ned as ui(ci(t)) +

u�i(c�i(t)). This is clearly a bold assumption, which we make to obtain reduced form dy-

namics that are easy to interpret. Each individual optimal plan can be described as

maxci;c�i

Ei�Z T

0ui(ci(t)) + u�i(c�i(t))dt

�s:t: Ei

�Z T

0mi(t)[ci(t) + e(t)c�i(t)]dt

�� Ei

�Z T

0mi(t)[�i(t) + e(t)��i(t)]dt

�.

The process e(t) denotes the relative price of the second good and it can be interpreted as

the exchange rate. Let `i be the Lagrange multiplier of the budget constraint, the optimal

consumption plans ci and c�i must satisfy u0(ci(t)) = `imi(t), and u�0(c�i(t)) = `imi(t)e(t).

As usual, the Lagrange multiplier can be obtained by substituting the optimal consumption

plan in the budget constraint and solving for `i.

As it is well known in heterogeneous economies (Basak and Cuoco (1998)), the dynamics

of asset prices and of the exchange rate can be more easily understood by studying the

behavior of a single representative agent endowed with the aggregate supply of securities

and with utility function given by a weighted average of individual agents� preferences

over the domestic and foreign goods U(c; �) = u1(c1)+�(t)u2(c2) and U�(c; �) = u�1(c�1)+

�(t)u�2(c�2). The process �(t) is the relative weight of the second agent, which must be state-

dependent under heterogeneous beliefs for the Pareto optimal allocation to be supported

in equilibrium. This weaker notion of aggregation was introduced by Cuoco and He (1994,

2001) in the context of incomplete markets. From the optimality conditions, it is easy to

see that in equilibrium �(t) = u01(c1(t))=u02(c2(t)) = `1m

1(t)=`2m2(t).

The exchange rate is given by the marginal rate of substitution between the two goods,

that is e(t) = U�0(��(t); �(t))=U 0(�(t); �(t)). By mimicking the arguments in Basak (2005)

and Buraschi and Jiltsov (2006), we can show that the weights �(t) are stochastic with

volatilities proportional to the di¤erences in beliefs:

d�(t) = � (t)dZ1t � �(t)dZ1t : (9)

9

Applying Ito�s Lemma it is possible to show that the exchange rate follows the process

de(t) = e(t)��ie(t) + �e(t)dZ

i(t) + �e�(t)dZ�i(t)

�; (10)

with volatilities

�e(t) = A(t)�(t)��(t) +

�A(t)

A2(t)� A�(t)

A�2(t)

� (t) and (11)

�e�(t) = A�(t)��(t)���(t) +

�A(t)

A2(t)� A�(t)

A�2(t)

� �(t), (12)

where Ai(t) = �U 00(ci(t))=U 0(ci(t)), 1=A(t) = 1=A1(t) + 1=A2(t); and similarly for A�i (t)

and 1=A�(t).

Equations (11) and (12) show that exchange rate volatilities depend on two terms. The

�rst term is the standard component arising in complete information economies and it is

proportional to the volatilities of the endowment process. When = � = 0, exchange rate

dynamics are exclusively driven by endowment shocks. When the di¤erence in beliefs is non

zero, however, exchange rates are a¤ected by a second component that is proportional to the

di¤erence in beliefs. This second term is due to the state-dependent nature of the relative

weights �(t) in (9), which implies that the relative consumption share of the two agents

is stochastic and belief dependent. This second e¤ect makes di¤erences in beliefs a priced

source of risk. The economic intuition is simple: In equilibrium, even in the absence of

frictions, marginal rates of substitution of di¤erent agents are not equal but proportional to

the di¤erence in beliefs. Even if preferences were identical across individuals, the greater the

disagreement over the growth rate of the endowment, the greater the di¤erence in expected

marginal utilities, and therefore the greater the incentive for heterogeneous agents to engage

in state-dependent risk sharing contracts. The agent who is more optimistic about the

endowment process will provide, in equilibrium, insurance to the pessimistic one. This also

implies that changes in di¤erence in beliefs a¤ect the observed distribution of consumption

in equilibrium. Without heterogeneity in beliefs, the distribution of consumption would

depend only on the preference characteristics, such as discount rates, risk aversion, the

elasticities of intertemporal substitution, and di¤erences in wealth.

Equations (11) and (12) show that foreign exchange volatility is a¤ected by , thus

implying that the price of foreign exchange options is a function of the di¤erence in beliefs.

Although it would be feasible, given equation (10), to add additional assumptions and obtain

an explicit model for currency option pricing, the exact functional form of the solution would

depend on the speci�c assumptions made on the utility function and the dynamics of the .

We decide therefore to investigate the broader link between beliefs dispersion and exchange

rates in ways that are relatively insensitive to these assumptions and opt for a reduced-form

approach. A �rst testable implication that comes directly from equations (11) and (12) is

10

thus:

� H1: The level of the option-implied volatility is positively correlated with the di¤erencein beliefs.

Moreover, since marginal rates of substitution across states depend on agents�beliefs,

the subjective valuation of out-the-money options di¤ers across agents with di¤erent beliefs.

In equilibrium, agents engage in state-contingent risk sharing agreements with pessimists

purchasing protection from the optimists. The larger the number of pessimists, the higher

the equilibrium cost of protection required by the optimists to provide insurance. Since

out-of-the-money put options are simple and e¤ective instruments to obtain insurance, this

has immediate implications for their equilibrium price with respect to call options with

the same strike and in-the-money put options. Thus, we investigate the following second

empirical hypothesis:

� H2: The slope of the option implied volatility smile (thus the risk-neutral skewness) iscorrelated with the di¤erence in beliefs.

Di¤erences in beliefs also a¤ect risk premia through the direct e¤ect on the aggregate

stochastic discount factor of the representative agent. The functional form depends on the

relative weight of the two agents �(t), which is state-dependent and a function of di¤erences

in beliefs, as equation (9) shows. In our stylized model, for instance, the di¤erence in the

perceived risk premium by the two agents is exactly equal to their di¤erence in beliefs (see

equation (7)). A simple calibration exercise can easily show that:

� H3: The size of the implied-realized volatility spread (the "volatility risk premium") ispositively correlated with the dispersion in beliefs.

As discussed earlier, di¤erences in beliefs are a priced source of risk since they directly

a¤ect the aggregate stochastic discount factor of the representative agent through the time-

varying stochastic process �(t). As a result, risk premia are time-varying and forward

exchange rates are biased predictors of future spot rate variations. In the economy just

described, a source of deviations from the expectation hypothesis is therefore the level of

the di¤erence in beliefs. We explore this issue by investigating the additional following

testable restriction:

� H4: The extent of the deviation from the expectation hypothesis of exchange rates is

correlated with di¤erences in beliefs.

11

Since this hypothesis refers to the dynamics of the underlying asset itself, as opposed to

the value of the derivative contract, it provides useful independent information about the

link between di¤erences in beliefs and asset prices.

It is important to notice that the previous implications are independent of the basis from

which agents have di¤erent beliefs. This could occur for several reasons, either rational or

irrational.7 As noted before, in this empirical study we use a reduced-form approach that is

independent of the speci�c set of assumptions on how beliefs are updated. However, we can

still empirically investigate the link between di¤erences in beliefs and overcon�dence. Agents

that are overcon�dent about their signals (prior) interpret price innovations di¤erently,

even if the signals are common and observable. This e¤ect, as opposed to heterogeneous

priors, has been used in the literature to model dynamics for the di¤erences in beliefs not

asymptotically converging to zero. For instance, Scheinkman and Xiong (2003) assume

the existence of an overcon�dent investor to explain speculative bubbles, while Dumas,

Kurshev, and Uppal (2006) use it to study the dynamic implications of overcon�dence on

excess volatility.

Our dataset contains trader-speci�c information on future exchange rate beliefs and

therefore on the extent of updating of those beliefs. Thus, if overcon�dence of some traders

is the main reason for the existence of di¤erences in beliefs, one would expect to �nd

a correlation between the dynamics of updating beliefs, given the observation of public

signals, and option prices. Thus, we investigate as an additional testable restriction:

� H5: The implied volatility is positively correlated with aggregate measures of averageovercon�dence.

Obviously, the model is agnostic about the reasons why people disagree. Therefore, this

hypothesis is not a test of the model itself but rather a study of the reasons and dynamics

of disagreement.

II. Data and Preliminaries

In this section, we describe our data sets in detail. We also show our approach to obtaining

an empirical proxy for di¤erences in beliefs and perform some preliminary analysis.

Currency option data. Our dataset contains the daily currency option prices traded

in the over-the-counter (OTC) market over the period April 1992 through December 2006,

kindly supplied by Citigroup. The options are European-style and they are written on three

di¤erent currency pairs: the Japanese yen price of the U.S. dollar (USD/JPY), the dollar

7Clearly, if one were to specify the way in which individuals update their beliefs, it would be possibleto restrict the cross-sectional and time-series properties of option prices uniquely, as Buraschi and Jiltsov(2006) show in the case of stock index options.

12

price of the British pound (GBP/USD), and the dollar price of the euro (EUR/USD).8

Options on each pair have four constant times to maturity, at one month, three months, six

months, and one year. The constant time-to-maturity feature of the OTC currency option

market is useful as it allows us to sidestep the issue of interpolating between adjacent

maturities that typically occurs with the preset maturity dates of exchange-traded options.

The OTC currency option market has peculiar trading conventions.9 Unlike bond or

equity options that are typically traded in terms of option premiums at di¤erent strike prices,

currency options trade in terms of implied volatilities. More speci�cally, implied volatility

quotes are most commonly available in the form of three types of option combinations: the

delta-neutral straddle, the risk reversal, and the strangle. A straddle is a portfolio of a

call option and a put option with the same strike price and maturity. For the straddle

to be delta-neutral, the strike price needs to be su¢ cently close to the forward price to

make the implied volatility quote of the straddle an at-the-money implied volatility (ATM).

The risk reversal (RR) measures the di¤erence in implied volatility between an out-of-the-

money call option and an out-of-the-money put option. The moneyness level is de�ned

in terms of the Black-Scholes delta of the option and is conventionally set at 25-delta.10

The strangle (STR) corresponds more precisely to a butter�y spread and measures the

di¤erence between the average volatility of the two 25-delta options and the delta-neutral

straddle implied volatility. In summary:

�ATM = �50�Call (13)

�RR = �25�Call � �25�Put (14)

�STR =�25�Call + �25�Put

2� �ATM : (15)

Equations (13), (14), and (15) show that the straddle is a measure of the level of the implied

volatility, the risk reversal is a measure of the slope of the implied volatility smile, and the

strangle is a measure of the curvature of the implied volatility smile. From the three quotes,

we can derive the level of implied volatilities at the three levels of delta:

�25�call = �ATM + �STR +1

2�RR (16)

�25�put = �ATM + �STR �1

2�RR: (17)

8 In the remainder of the paper, we will refer to the EUR/USD exchange rate as the currency pairrepresented by the USD/DEM exchange rate before 1999 and the actual EUR/USD exchange rate thereafter.

9We describe the trading conventions of the currency option market that are key to understanding theproperties of our dataset. For further details on the currency option market practices see, among others,Malz (1997).10A 25-delta call corresponds to a call option with Black-Scholes delta of 0.25, and a 25-delta put corre-

sponds to a put option with Black-Scholes delta of -0.25. Another level of moneyness that is typically tradedin the currency option market is the 10-delta.

13

For our empirical analysis, we use implied volatilities from market quotes and obtained

through equations (16) and (17). As a result, we do not need to match the currency option

data with underlying FX rates and interest rates, thus avoiding potential asynchronicity

biases.

Table 1 contains summary statistics for our sample of currency options related to the

three option combinations de�ned by equations (13), (14), and (15). The sample averages

for at-the-money implied volatilities show that USD/JPY is the most volatile currency pair

in our sample, followed by EUR/USD and GBP/USD. The average term structure of at-

the-money implied volatility is upward-sloping for all the underlyings. The analysis of the

risk reversal combination reveals that the sample average for the USD/JPY is negative

at all maturities, implying that the out-of-the-money put options are on average more

expensive than the corresponding out-of-the-money call options during our sample period.

In contrast, for the EUR/USD, the sample average of the risk reversal is positive and, for

the GBP/USD, is close to zero. In all cases, the standard deviation is large, suggesting

that risk reversals vary greatly over time, as documented also in Carr and Wu (2006). The

strangle combination is signi�cantly positive for all currency pairs and maturities, peaking

for the USD/JPY currency pair. This is an indication that the risk-neutral distribution of

all three currencies exhibits fat tails. The strangle spreads are much more stable than the

risk reversals, as the lower standard deviations demonstrate.

The fx forecasts: Our dataset on di¤erences in beliefs is based on the full set of forecasts

that make up the Reuters survey of FX forecasts. The data is sampled at a monthly

frequency from May 1993 to December 2006. At the beginning of each month, about 50

market participants provide their forecasts of future exchange rates. The forecast horizons

are set to be one month, three months, six months, and twelve months respectively. Table 2

contains summary statistics for the FX forecasts. For all forecasting horizons and currency

pairs, we have more than 6,000 individual forecasts over 140 monthly surveys. The average

number of forecasters in each survey is almost 50 in all cases and we always have at least

27 major market participants in the survey. This fares well against surveys on company

earnings that typically feature less than one-third of this number of respondents.

Besides o¤ering a meticulous archive of individual forecasts (the longest uninterrupted

sample available), the Reuters survey has a number of advantages over other FX forecast

surveys such as those undertaken by Consensus Economics, WSJ, ZEW, Blue Chip and

Forecasts Unlimited (formerly the FT currency forecasts and the Currency Forecast Digest).

First, since it is conducted by the key FX news provider, it is very much focussed on FX

market participants whereas other surveys often include many other forecasters such as

professional forecast �rms, corporations and academic institutions. We estimate that around

95% of contributors to the Reuters survey are active market participants compared to 85%

for Consensus Economics and even less for the other major surveys. This is important since,

14

as Ito (1990) �nds, these other forecasters are not comparable with those actively trading

in foreign exchange. Second, the pool of forecasters is relatively constant. Other surveys

have both gaps in coverage (missing individuals months and in some cases years) and a

relatively rapid turnover of contributors. Third, it is the only survey that collects 1,3, 6

and 12 months ahead forecasts, thus o¤ering the most complete short-term coverage.

FX forecast data has been already used to help understand market expectations. Frankel

and Froot (1987) use FX forecasts and �nd that expectational errors play a signi�cant role

in explaining the forward premium puzzle. More recently, Bacchetta, Mertens and Van

Wincoop (2006) �nd a strong relationship between forecasts errors and excess returns, con-

�rming the link between FX forecasts and underlying market behaviour. On the other

hand, studies that focus on the predictive power of the forecasts (e.g. Bo�nger and Schmidt

(2004)) generally �nd no informational content in FX forecasts that help in predicting future

exchange rates.

A maintained assumption in our proposed interpretation of the link between beliefs

heterogeneity and currency markets is that the average agent has no private information

on the future dynamics of the exchange rate. We brie�y investigate the robustness of this

assumption by estimating the following regression:

ST � St = �+ ��St;T � St

�+ et; (18)

where St represents the foreign exchange rate at time t and St;T represents the average

foreign exchange forecasts on day t with horizon T . We also compute the Root Mean

Squared Error (RMSE) of the average forecast and compare it with the RMSE of a random

walk (RMSErw).

We �nd that professional forecasters do not predict future exchange rates better than a

random walk (Table 3). The explanatory power of the forecasting regression is generally very

low, with the di¤erence between the average forecast and the spot rate never explaining more

than 6% of the change in future exchange rates. The slope coe¢ cient is signi�cantly di¤erent

from zero for only one currency pair, GBP/USD, and is always signi�cantly di¤erent from

one. The RMSE is always higher than the RMSE of a random walk forecasting model.11

Not surprisingly for such a liquid market, these results lend support to the conjecture that

currency prices are mainly determined by the �ow of public information about cash �ows

and discount rates and by the mechanism of price discovery, i.e. by the aggregation of

heterogeneous private information (or heterogeneous interpretation of public information)

through trading. Clearly, even if exchange rates are not predictable by the average trader,

the observed heterogeneity in beliefs may still a¤ect, as the model demonstrates, the risk

neutral measure and derivative asset prices.

11The results of estimating equation (18) do not change when we use the median foreign exchange rateforecast instead of the average foreign exchange rate forecast.

15

An empirical proxy for differences in beliefs: We construct an empirical measure

for the di¤erence in beliefs implicit in the cross-sectional distribution of FX forecasts. This is

consistent with the model of Section I. Speci�cally, we compute the mean absolute deviation

(MAD), i.e. the average absolute deviation of all forecasts from the mean forecast. We

express the forecast in terms of log returns of the forecast with respect to the spot exchange

rate.12 In the following regressions, we empirically explore the robustness of the link between

the dispersion in beliefs and asset prices when we use alternative popular measures of

dispersion, such as entropy and standard deviation.

Table 4 shows summary statistics. Since the average number of forecasters in each

survey is about 50 (see Table 2), the sample mean absolute deviation is a reliable estimate

of the dispersion of FX forecasts. The maximum and minimum values for the mean absolute

deviation show a relatively wide range around its mean. This indicates a signi�cant time-

variation in the dispersion of beliefs and that the dispersion is larger at longer forecast

horizons. The average sample skewness is negative in most cases, but the large maximum

values indicate that it is positive for some periods in our sample. Summary statistics for

kurtosis suggest that extreme forecasts are more likely at shorter horizons and that they

occur more often for the GBP/USD currency pair.

Figure 2 illustrates the time-series dynamics of the mean absolute deviation, skewness,

and kurtosis of beliefs at di¤erent forecast horizons. The time variation of the mean absolute

deviation of beliefs is signi�cantly di¤erent across our sample currency pairs, suggesting that

they are not driven by a single factor and that a panel of data for di¤erent exchange rates

adds important information. The skewness of beliefs switches sign over our sample period,

and is less correlated at di¤erent forecast horizons. The kurtosis of beliefs is subject to

frequent extreme peaks.

III. Empirical Results

In our empirical analysis, we investigate the implications of the stylized model presented

in Section I. First, we analyze the relation between at-the-money implied volatilities and

the di¤erences in beliefs of foreign-exchange forecasters. Second, we extend this analysis

to the entire implied volatility smile. Third, we provide evidence on the links between the

currency volatility risk premium and the foreign exchange di¤erences in beliefs. Finally, we

show that the dispersion of beliefs is indeed a priced risk factor since it explains the returns

on the underlying currencies beyond the forward premium.

12Expressing the forecast in log returns rather than levels has the desirable property of removing levele¤ects from the time-series of the FX forecasts.

16

A. At-the-money implied volatility and di¤erences in beliefs

One of the main testable implications of the model in Section I is the link between the

implied volatility of currency options and di¤erences in beliefs. We measure the di¤erence in

beliefs t;T by computing the mean absolute deviation (MAD) of foreign-exchange forecast

returns, as shown in the previous section and estimate the following regression in which we

control for other characteristics that may a¤ect the implied volatility:

IVt;T = �+ � t;T + �skSKt;T + �kuKUt;T + �ivIVt�30;T�30 + et;T ; (19)

where IVt;T represents the implied volatility of at-the-money (ATM) options on day t with

maturity T .13

In order to account for a comprehensive description of market participants�beliefs, we

also include the skewness and the kurtosis of the cross-sectional distribution of the foreign

exchange forecasts on day t with horizon T , SKt;T and KUt;T respectively. Moreover, since

there is evidence that stock market implied volatility has predictive power for the future

level of volatility, beyond past realized volatility (e.g., Christensen and Prabbhala, 1998),

we also include a control variable the lagged at-the-money implied volatility. This addresses

the possibility that t;T could simply pick up the e¤ect of the persistent implied volatility

shocks. Table 5 shows the results of estimating equation (19). In the �rst three columns,

we only use the mean absolute di¤erence of foreign exchange forecast returns as a regressor.

The coe¢ cient is positive and highly statistically signi�cant in all cases. The results for

USD/JPY are striking. The MAD of foreign exchange forecasts in isolation explains between

54% and 64% of the variation of at-the-money implied volatilities. These results are also

economically important. A one-standard deviation increase in di¤erences in beliefs implies

an increase in implied volatilities between 1.9% and 2.1%. The explanatory power for

GBP/USD is lower and decreasing for longer horizons. However, all the coe¢ cients are

statistically signi�cant at the one percent level. The economic signi�cance is also weaker,

ranging between 0.7% and 0.3% for one standard deviation shock in the di¤erences in

beliefs, although the underlying currency is unconditionally less volatile. For the EUR/USD

exchange rate, the results are again highly signi�cant at all horizons. The increase in implied

volatility for a one standard deviation increase in the standard deviation of foreign exchange

forecasts is between 1.4% at short horizons and 0.7% at long horizons.

In the next three columns of Table 5, we estimate equation (19) including the skew-

ness and the kurtosis of the distribution of foreign-exchange forecasts, as well as the mean

absolute deviation. In fact, it could be argued that part of the relevant characteristics of

the di¤erences in beliefs is given by the asymmetry of market forecasts or by the frequency

13The implied volatility is the daily implied volatility prevailing precisely on the day in which the FXsurvey is conducted. The results do not change if implied volatility is measured on the day in which the FXsurvey results are released.

17

of extreme forecasts, as re�ected in the thickness of the tails of the distribution. We �nd

that the mean absolute deviation is still statistically signi�cant. In general, however, the

explanatory power of the regression increases with respect to the univariate case. The role

of the asymmetry is particularly relevant for EUR/USD. The skewness coe¢ cient is statis-

tically signi�cant at least at the 5% level at all forecast horizons, generating a substantial

increase in explanatory power and indicates that the asymmetry of foreign-exchange views

re�ects some priced features in the di¤erence in beliefs. The kurtosis of the foreign exchange

forecasts explains implied volatility with somewhat lower statistical signi�cance levels for

both EUR/USD and USD/JPY. This �nding suggests that information in the tails of the

distribution can constitute an additional priced source of dispersion of beliefs.

The last three columns of Table 5 show the results of estimating the full speci�cation

of equation (19). This includes controlling for the at-the-money implied volatility of the

previous month. As expected, due to the persistence of implied volatility, its coe¢ cient is

highly signi�cant and the level of signi�cance is increasing in the horizon. However, the

statistical signi�cance of the slope coe¢ cient of t;T is always preserved and, in most cases,

this is the case also for the signi�cance of the higher-order moments of the foreign exchange

forecasts. In summary, the di¤erence in beliefs explains the current level of at-the-money

implied volatility beyond the information in lagged at-the-money implied volatility.

As a series of robustness checks, we consider speci�cations of equation (19) with di¤erent

lags for the regressors. For example, we lag by one week the MAD, the higher-order moments

of the distribution of the foreign exchange forecasts, and implied volatility. The results

are virtually unchanged. Another interesting example is the use of dispersion of beliefs

characteristics lagged one month. With this speci�cation, the empirical exercise becomes

more a forecasting exercise of future implied volatility. The �ndings with lagged beliefs (not

reported) are only slightly weaker and broadly con�rm the earlier results.

In summary, a simple empirical proxy of the di¤erence in beliefs of foreign-exchange

market participants, t;T , signi�cantly explains the level of at-the-money implied volatility

in currency options markets. This result holds for di¤erent currency pairs, at di¤erent

forecasting horizons, and controlling for the traditional variables used in the literature

to explain the level of implied volatility. The presented evidence gives strong empirical

support to the main testable implication of the stylized model of Section I and documents

an economically and statistically important driver of volatility in FX markets.

B. The option-implied volatility smile

The second set of empirical implications of the model suggests that the relative prices of

derivative contracts are a¤ected by the structure of beliefs. More speci�cally, in the case of

options, this should occur in terms of the cross-section of implied volatilities for di¤erent

moneyness levels, i.e. the implied volatility smile. This theoretical prediction is especially

18

relevant, given that the characteristics of the option-implied volatility smile are closely

related to the characteristics of the foreign-exchange state price density (e.g., Bates, 2001;

Bakshi, Kapadia, and Madan, 2003).

We obtain the implied volatility of at-the-money options directly from market quotes

and compute implied volatilities of options with other moneyness levels from risk reversal

and strangle combinations, as described in Section II. Then, we investigate the link between

di¤erences in beliefs and the slope of the implied volatility smile by estimating the following

regression: IV 25�deltat;T

IV 75�deltat;T

!� 1 = �+ � t;T + �skSKt;T + �kuKUt;T + et;T ; (20)

where IV x�deltat;T represents the implied volatilities at x � delta moneyness for options on

day t with maturity T .

Table 6 (�rst three columns) shows these results in the univariate regression case. We

observe that higher dispersion of beliefs in�uences the slope of the volatility smile sig-

ni�cantly at three-, six-, twelve-month horizons for USD/JPY, and at all horizons for

EUR/USD. This evidence con�rms the theoretical predictions of the model. These re-

sults are also economically important. For example, the unconditional slope of the implied

volatility smile for USD/JPY is about -7% for longer horizons. However, the slope drops to

about -4% when the di¤erences in beliefs are one-standard deviation above the mean. The

economic signi�cance is similar for the EUR/USD exchange rate.

In the remainder of Table 6, we estimate the full speci�cation of equation (20). The

results are very similar to the univariate speci�cation for USD/JPY and GBP/USD, except

that the higher-order moments of the distribution of beliefs become statistically signi�-

cant when the mean absolute deviation is not signi�cant (e.g., the one-month horizon for

USD/JPY). For EUR/USD, the higher-order moments improve explanatory power signif-

icantly. The R2 doubles on average, with the largest increase at the one-month horizon.

Both skewness and kurtosis of beliefs a¤ect the slope of the implied volatility smile.

In summary, the results show that the di¤erences in beliefs are an important deter-

minant of the shape of the option-implied volatility smile, and thus, of the shape of the

risk-neutral foreign-exchange state price density. This is important evidence that the di¤er-

ences in beliefs drives di¤erential pricing of state-contingent claims, precisely as theoretical

models with heterogeneous beliefs would predict. It also suggests that the dispersion of

beliefs has a likely impact on the stochastic discount factor and thus on the pricing of all

�nancial assets. This last aspect will become apparent in the empirical tests of the volatility

risk premium in the next subsection.

19

C. Volatility risk premium and di¤erence in beliefs

The model in Section I shows that beliefs directly a¤ect the stochastic discount factor and

thus the di¤erence between the risk-neutral and physical probability measure. A natural

test for this theoretical prediction is thus to investigate whether measures of the di¤erences

in beliefs impact the volatility risk premium, i.e. the spread between implied and realized

volatility.

The risk-neutral expected value of the return quadratic variation from time t to time

T can be approximated from the time t prices of options maturing at time T . A measure of

implied volatility for these options is readily available from the currency option market. As

an empirical proxy for the physical expected value of the return quadratic variation from

time t to time T , we use an exponential moving average of the quadratic returns on the

underlying according to the following speci�cation:

Et [RVt;T ] = (1� �T�t)(r2t�1 + �T�tr2t�2 + �2T�tr2t�3; :::); (21)

where rt is the log return on the underlying asset on day t and �T�t is the smoothing factor

that depends on the horizon.14 Measures of realized volatility based on squared returns

have strong theoretical underpinnings and good empirical performance (see, among others,

Andersen et al. 2003). The use of exponentially decaying weights is a useful device in

our setting to increase the number of observations, since we use daily rather than intraday

returns, and corresponds to a widely used approach among pratictioners (see J.P. Morgan�s

Riskmetrics, 1997).15

We then regress the volatility risk premium on the characteristics of the distribution of

foreign exchange forecasts:

IVt;T � E [RVt;T ] = �+ � t;T + �skSKt;T + �kuKUt;T + et;T ; (22)

where IVt;T represents the at-the-money implied volatilities for options on day t with ma-

turity T , E [RVt;T ] is an exponential moving average of realized volatility obtained with

the formula in equation (21) for a horizon matching the maturity of the option, and t;T ,

SKt;T , KUt;T are de�ned as before.16

14The exponential moving average can also be calculated in a recursive fashion as Et [RVt;T ] = �T�tr2t�1+(1� �T�t)Et�1 [RVt�1;T�1] :The smoothing parameter decreases with the horizon and is equal to 0.03, 0.01, 0.005, and 0.003 for a

one-month, three-month, six-month, and one-year horizon, respectively.15Our results are robust to the choice of the speci�c model to proxy for expected realized volatility. As

alternative estimators, we use the exponential moving average of equation (21) with di¤erent smoothingparameters, a centered moving average with squared di¤erences between returns and mean returns, and aGARCH(1,1) model.16We also estimate a version of equation (22) where the volatility risk premium is the ratio of implied

to realized volatility instead of the di¤erence between them. We obtain similar results, suggesting that our�ndings are not driven by the positive correlation between the general level of volatility and the volatility

20

Table 7 shows the results of estimating equation (22). As before, the �rst three columns

show our �ndings for the univariate speci�cation with the mean absolute deviation of the

foreign exchange forecasts on the right-hand side. For USD/JPY and EUR/USD, t;T has

a signi�cantly positive e¤ect on the volatility risk premium. In particular, for USD/JPY,

the dispersion of beliefs explains a striking average 50% of the variation of the volatility

risk premium across forecasting horizons. Unconditionally, the di¤erence between implied

and realized volatility is about 3.3%, but it increases to about 5.3% when the dispersion of

beliefs is one standard deviation above average. For EUR/USD, the volatility risk premium

is unconditionally lower, about 1.4%, but it increases to about 1.8% with high dispersion

of beliefs.

The remainder of Table 7 shows the results of estimating the full speci�cation of equa-

tion (22). The �ndings of the previous analysis are con�rmed. There is only some evidence

that the higher-order moments of foreign-exchange forecasts distribution explain the volatil-

ity risk premium beyond t;T . More speci�cally, for EUR/USD, the kurtosis of the distri-

bution of beliefs has systematically a signi�cantly positive e¤ect on the di¤erence between

implied and realized volatility.

In the literature, the volatility risk premium is frequently assumed to be linear in

volatility (e.g., Heston, 1993). Bakshi and Kapadia (2003), among others, �nd empirical

support for this hyphothesis for the volatility risk premium in the stock market. We thus

estimate a more general version of equation (22), where we also control for the lagged level of

implied volatility on the right-hand-side. The �ndings of Table 7 on the e¤ect of dispersion

of beliefs on the volatility risk premium are unchanged (results not reported).

As an additional robustness check, we also test for the e¤ects of di¤erences in beliefs on

our empirical proxy for realized volatility used as a dependent variable in isolation. This test

has the purpose to make sure that the e¤ects on the volatility risk premium documented in

Table 7 are not simply the upshot of the e¤ects on implied volatility documented in Table

5. This would be the case if realized volatility had no statistical relation with dispersion of

beliefs. We �nd that dispersion of beliefs is indeed impacting positively realized volatility

of di¤erent currency pairs and at di¤erent horizons (results not reported).

In summary, the evidence in Table 7 shows empirically that beliefs directly a¤ect the

stochastic discount factor and thus the di¤erence between the risk-neutral and physical

probability measure, as predicted by the model of Section I. This �nding has broad im-

plications. For instance, since di¤erences in beliefs are time-varying, this can generate

time-varying risk-premia and expected returns, which could help to explain several stylized

facts in the FX markets.

risk premium expressed as a di¤erence.

21

D. Carry trades, exchange rates predictability and dispersion of beliefs

The evidence presented so far corroborates the model predictions that the dispersion of

beliefs is a priced source of risk in FX markets. The �rst papers in the literature to

investigate risk factors in currency markets are related to the so called forward premium

puzzle. See, for example, Engel (1996) and Hodrick (1987) for surveys on the extent to which

the forward premium puzzle can be explained by time-varying risk premia. More recently,

research has focused on carry trades: borrowing a low interest rate currency and investing in

a high interest rate one. These strategies are widely used by market practitioners to exploit

the predictable excess returns implied by the forward premium puzzle. In the last decades,

the Sharpe ratios of such trades have been several times higher than the Sharpe ratio of

an investment in the S&P500 index. For example, the USD/JPY carry trade, which is

arguably the most popular strategy, has generated positive returns of more than 90% over

the last ten years. The academic literature has tried to reconcile these puzzling returns

with risk-based explanations. Lustig and Verdelhan (2007), for example, show that U.S.

consumption growth risk can explain predictable returns in currency markets. Burnside et

al (2006), in contrast, show that the high carry trade returns have little or no correlation

with standard risk factors. Given our empirical evidence that dispersion of beliefs is a priced

risk factor, it becomes natural to ask how heterogeneity of beliefs a¤ects carry trade returns

and, more generally, the forward premium puzzle.

Given the sample of currencies for which we can compute measures of dispersion of

beliefs, we focus on carry trade returns for the USD/JPY currency pair, which is the most

popular (and discussed) carry trade strategy among pratictioners.17 We proceed as follows.

At each monthly survey date, we borrow JPY and invest in USD and close out the position

on the day before the following survey release. This approach allows us to relate precisely

carry trade returns to dispersion of beliefs and implied volatility. We decompose total carry

returns into returns generated by the interest rate di¤erentials and the returns generated

by exchange rate appreciation/depreciation. Figure 3 plots average returns in high/low

dispersion of beliefs states, high/low implied volatility states, and high/low volatility risk

premium states. We can readily see that the interest rate di¤erential is larger when dis-

persion of beliefs, implied volatility, or the volatility risk premium are high. This �nding

suggests that the high carry can be seen as a compensation for risks associated with higher

dispersion of beliefs (and higher implied volatility and volatility risk premiums). However,

when we look at total carry trade returns, they seem to be virtually una¤ected by low/high

implied volatility states. In contrast, we notice that total returns are much lower when

the volatility risk premium is high and they even turn negative when dispersion of beliefs is

high. Whenever market participants disagree about the future value of currencies, the carry

compensation seems to be insu¢ cient to make up for adverse spot currency movements.

17The interest rate di¤erential is lower and switches sign for the EUR/USD and GBP/USD. Furthermore,these are not typical carry trade strategies among pratictioners.

22

We test more formally for the relation between USD/JPY carry trade returns and the

dispersion of beliefs by regressing total carry trade returns on the dispersion of beliefs, im-

plied volatility, and the volatility risk premium. Table 8, panel A, shows the results. The

graphical intuition from Figure 3 is con�rmed. Carry trade returns are virtually una¤ected

by the level of implied volatility. The coe¢ cients are not statistically di¤erent from zero.

In contrast, carry returns are signi�cantly lower when dispersion of beliefs is high. The eco-

nomic signi�cance is striking: a one-standard deviation shock to dispersion of beliefs leads

to a negative 7.7% carry trade total annual return. The returns generated by changes in

the exchange rates largely o¤set the positive average 3.7% interest rate di¤erential between

USD and JPY in our sample. The e¤ect of the volatility risk premium is not statistical

signi�cant at conventional levels, but economically a one-standard deviation shock to the

volatility risk premium generates a negative 5% annual return. These �ndings emphasize

again that the degree of disagreement about future exchange rates is a key variable that

a¤ects asset returns, volatilities, and risk premiums.

The adverse e¤ect of high dispersion of beliefs on carry trade returns could be easily

rationalized if the distribution of beliefs had some forecasting power for the large negative

returns that periodically hit the carry trade strategy. Obviously, a potential shortcoming of

this empirical exercise is the availability of a sample of currency returns and FX forecasts

that is long enough to contain a su¢ cient number of FX jumps for statistical inference. In

our sample, we identify only 18 instances in which FX returns are larger than two standard

deviation bands. The dispersion of beliefs at one-month horizon shows some predictive

power for these negative large returns, albeit not at the conventional statistical levels of

signi�cance.

It would also be interesting to study the link between currency returns and beliefs

beyond the USD/JPY exchange rate and the one-month horizon. To do this we extend our

analysis in to expectation hypothesis regressions and the extensive literature on the forward

premium puzzle. We thus estimate the following speci�cation:

st+n � st = �+ �fwd (ft;t+n � st) + � t;t+n + et;t+n; (23)

where st represents the logarithm of the spot rate at time t, ft;t+n represents the logarithm

of the forward rate at time t with maturity t + n, t;t+n is the mean absolute deviation

of foreign exchange forecasts at time t for the horizon t + n, and n is either one-month,

three-months, six-months, or one-year. This analysis is clearly related to the carry trade

strategy. The dependent variable is one of the two components of the carry trade returns.

Furthermore, the interest rate di¤erential (the other component of carry trade returns) is

controlled for on the right-hand-side through the forward premium.

Table 8, panel B, shows the results of estimating equation (23). The �rst column

for each currency and horizon just replicates the �ndings in previous literature and shows

23

that the forward premium puzzle is generally present in our sample period, for our sample

currency pairs, and at our sampling frequency. In the second column, we add our empirical

proxy for dispersion of beliefs. We observe that for all currencies and at almost all horizons,

the dispersion of beliefs has signi�cant predictive power for subsequent currency returns

beyond the forward premium. More speci�cally, when we also include the MAD among the

regressors, we obtain a substantial increase in the R2 . For example, the explanatory power

at the one-month horizon increases on average by more than two times across currencies.

The R2 for the USD/JPY exchange rate across horizons increases on average by more than

100%. The sign is negative for the USD/JPY and GBP/USD exchange rates, two currency

pairs trading at a forward discount in our sample period, and is positive for the EUR/USD,

the currency pair trading mostly at a forward premium. In terms of a carry trade strategy,

this means that higher dispersion of beliefs has a negative e¤ect on carry trade returns,

consistent with the results presented in Panel A for the USD/JPY.

In summary, we show that the dispersion of beliefs is a risk factor in currency mar-

kets with important asset pricing e¤ects that go beyond the strong e¤ects documented for

currency options. More speci�cally, the degree of dispersion of beliefs explains carry trade

returns for the USD/JPY currency pair and, more generally, explains spot currency returns

beyond the forward premium.

IV. Robustness and Discussion

In this Section, we investigate the results�robustness by exploring an alternative measure of

dispersion that has been used widely in the information theory literature and more recently

in the ambiguity literature. We then explore the link between uncertainty about macroeco-

nomic fundamentals and di¤erence in beliefs. Finally, we study the role of overcon�dence,

one of the main mechanisms known to support di¤erences in beliefs.

A. Entropy and Di¤erence in Beliefs

Our measure of di¤erences in beliefs is based on the dispersion of FX forecasts by market

participants. In the �nance literature, it has become common practice to use as a proxy

for the di¤erences in beliefs the standard deviations of the forecasts (Diether, Malloy, and

Scherbina (2002); Anderson, Ghysels, and Juergens (2005); Buraschi and Jiltsov (2006)).18

In the case of di¤erences in beliefs models, however, the standard deviation does not have a

clear cut theoretical link with the properties of . In what follows, we explore an alternative

measure of uncertainty that connects more naturally with the notion of di¤erence in beliefs

as a Radon-Nikodym derivative between two distributions. In the information theory lit-

18To allow a comparison of our results to the �ndings in the literature, we repeat all of our analysis usingthe standard deviation in place of the mean absolute deviation. We obtain similar results, both in terms ofstatistical signi�cance and economic magnitude.

24

erature entropy is widely accepted as a measure of uncertainty in information systems. In

economics, entropy has been related to the accuracy of predictions (Sandroni, 2000) and

model uncertainty (e.g., Anderson, Hansen, Sargent, 2003). According to Shannon�s (1949)

de�nition, the entropy of a distribution P is:

H(P ) = �Ep(log p(X)):

The Shannon measure can be used to obtain a measure of relative entropy between two

sets of beliefs. Consider two probability measures dP and dQ, absolutely continuous with

respect to a measure d�. Taking the di¤erence of H for P and Q under the assumption

that P is the true distribution, we obtain the relative entropy:

Hkl(P jQ) = Ep

�log

p(X)

q(X)

�=

Zp log(

p

q)d�:

Using dP = pd�:

Hkl(P jQ) =Zlog(

dP

dQ)dP;

where dPdQ is the Radon-Nikodym derivative between the two distributions. Since in our

economy the di¤erences in beliefs can be interpreted as the Radon-Nikodym derivatives

of the two agents�beliefs distributions, a simple and natural way to obtain a proxy for is

from the entropy of the belief distribution.

In the discrete case H(P ) = �P

i p(Xi) log(p(Xi)), which has its maximum value when

all probabilities are equal (uniform distribution, in the case of bounded supports), and the

resulting value for entropy is the logarithm of the number of states. The intuition here is

that when any random event can occur with equal probability, one faces a situation with the

highest uncertainty. Shannon (1949) shows that H(P ) is the only continuous, symmetric,

additive measure of dispersion such that

H(p1; :::; pn) � H(1

n; :::;

1

n) � H(

1

n+ 1; :::;

1

n+ 1):

That is, if all the outcomes are equally likely, the entropy is maximal and, in this case, the

entropy increases with the number of outcomes. This property is generally not satis�ed by

the standard deviation. Standard deviation and entropy measure di¤erent properties of a

distribution. The standard deviation is proportional to the average squared di¤erences of

the realizations of a random variable with respect to the mean. The entropy is an average

of the log-probabilities of these realizations. For this reason the standard deviation is known

to be very sensitive to the tail behavior of the distribution.

25

A simple example can illustrate the intuitive di¤erence between entropy and standard

deviation for our purposes. Consider four banks forecasting the USD/JPY exchange rate

at two di¤erent points in time t1 and t2. At t1, banks come up with the set of forecasts 104,

105, 106, 107, and at t2 with the set of forecasts 104, 104, 107, 107. The following diagrams

represent the situation:

102 103 104 105 106 107 108 1090

0.5

1

1.5

2

2.5

3

3.5

4

102 103 104 105 106 107 108 1090

0.5

1

1.5

2

2.5

3

3.5

4

Entropy and standard deviation would rank the degree of dispersion of beliefs in oppo-

site ways: the entropy at t1 and t2 is 1.3863 and 0.6931, respectively, while the standard

deviation is 1.2910 and 1.7310. Although the mean forecast is the same in the two cases,

it is reasonable to expect that the extent of ex-ante risk-sharing and the equilibrium cost

of insurance protection could be substantially di¤erent in the two cases. The result may

depend on several structural characteristics of the underlying economy that could be the

focus of empirical investigation. It thus becomes important to test whether all the results

presented in the previous section of the paper are robust to this alternative measure for the

di¤erence in beliefs.

We construct an empirical measure of entropy of the cross-sectional distribution of FX

beliefs by computing H(P ) = �P

i p(Xi) log(p(Xi)); where the probability p(Xi) is the

empirical frequency of the forecast Xi. We express the forecast Xi as log returns of the

forecast with respect to the spot exchange rate. As before, the higher-order moments are

used to control for the FX forecasts�departures from normality, since asymmetry in the

distribution or extreme forecasts may also be priced in the currency option market. We

again analyze how the dispersion of beliefs impacts the level of implied volatility (equation

(19)), the implied volatility smile (equation (20)), the volatility risk premium (equation

(22)), and currency returns (equation (23)), using our empirical measure of entropy as the

of dispersion of beliefs de�nition. The results, both in terms of signi�cance and economic

magnitude, are similar to those reported in Table 5 to Table 8. This shows that di¤erences

in beliefs has important asset pricing implications, regardless of the way in which it is

empirically measured.

26

B. Di¤erence in Beliefs: The Role of Fundamentals

The model developed in Section 2 takes the di¤erences in beliefs as given and shows that it

can generate volatility of exchange rates independently from the volatility of fundamentals.

However, one could argue that the di¤erences in beliefs themselves could be a¤ected by

the volatility of fundamentals. For example, higher uncertainty about in�ation could cause

wider disagreement about the evolution of exchange rates. To investigate this idea, we

collect a comprehensive data set of economic indicators for the U.S., Japanese, European,

and British economy from the OECD. The unique source for this data is important to

guarantee homogeneity of measurement and accuracy across countries. More speci�cally,

we obtain indicators for the gross domestic product (GDP), industrial production (IP),

consumer price index (CPI), unemployment rate (UR), broad money (BM), and balance

of payments (BP). To estimate the volatility of the macroeconomic time-series, we use the

Schwert (1989) procedure. We �rst estimate a 12th-order autoregression for the returns,

including dummy variables Djt to allow for di¤erent monthly returns:

Rt =

12Xj=1

�jDjt +

12Xi=1

�iRt�i + "t: (24)

We then estimate a 12th-order autoregression for the absolute values of the errors from

(24),

j"tj =12Xj=1

jDjt +12Xi=1

�i j"t�ij+ ut: (25)

The �tted values from (25) estimate the conditional standard deviation of Rt, given

information available before month t . We now test to what extent the time-series of the

di¤erences in beliefs for each currency is explained by the volatility of fundamentals that

are relevant for that currency. Speci�cally, we estimate

t = �+

2Xh=1

�h

���~"GDP;ht

���+ 2Xh=1

�h

���~"IP;ht

���+ 2Xh=1

�h

���~"CPI;ht

���++

2Xh=1

�h

���~"UR;ht

���+ 2Xh=1

�h

���~"BM;ht

���+ 2Xh=1

�h

���~"BP;ht

���+ et; (26)

where we explain the mean absolute deviation t for each currency and each horizon

with estimates of the volatility of economic indicators for the home economy (h = 1) and

for the foreign economy (h = 2). For example, we explain the t at a 1 month horizon for

the USD/JPY currency pair with the volatility of GDP, IP, CPI, UR, BM, and BP of both

27

the U.S. and Japan.19 Table 9 reports the R2 of these regressions. In all cases, at least 85%

of the variation in our empirical proxy for the di¤erences in beliefs is not explained by the

volatility of current fundamentals. Although we experimented with additional macroeco-

nomic indicators and other econometric speci�cations for macroeconomic volatility without

any relevant change to the results of Table 9, it is still likely that we are missing some

macroeconomic factor that is important for exchange rates (or that we did not experiment

with the best econometric model). However, it is hard to think that these changes would

substantially alter the main implication of this analysis, that is, the di¤erences in beliefs

seems to re�ect to a greater extent dynamics that are not related to uncertainty about

fundamentals. Obviously, it is still possible that our measure of dispersion of beliefs re-

�ects uncertainty about future fundamentals that is not re�ected in the current and past

realizations of the economic variables used in the econometric analysis.

C. Di¤erence in Beliefs: The Role of Overcon�dence

Di¤erences in beliefs can be supported either in a rational learning economy when agents

have heterogeneous priors (Buraschi and Jiltsov (2006)), or in an economy when some

agents are overcon�dent about the precision of a signal. This latter modelling approach is

explored in Scheinkman and Xiong (2003) and Dumas, Kurshev, and Uppal (2006) and is

motivated by signi�cant empirical evidence of cognitive biases. Barber and Odean (2000) use

a large dataset from a U.S. discount broker and �nd that a substantial fraction of individuals

overestimates the precision of their forecasts. These agents have unrealistically rosy views of

their abilities and prospects and are ultimately observed to trade excessively while realizing

lower than average performance. Are our results suggestive of overcon�dence?

We can think about overcon�dent forecasters as irrational investors who persistently

believe that there is a correlation between a signal and the innovations in unobserved un-

derlying variables when, in fact, there is no correlation (e.g., Dumas, Kurshev, and Uppal,

2006). This class of investors is thought to be overcon�dent about the signal. We can

empirically model the signal to each forecasters as the di¤erence between the realization

in the underlying variable and the individual forecast. Hence, we obtain an overcon�dence

measure for each professional forecaster j, each forecast horizon H, and each currency pair

S, as the ratio of the average forecast update in absolute value to the average forecast error

in absolute value. Speci�cally:

OV ERCj;H;S =1

N�1PN�1

t=1 jbSj;t+1;t+1+H � bSj;t;t+H j1

N�1PN�1

t=1 jSt+H � bSj;t;t+H j ; :::H = f1; 3; 6; 12g ; (27)

19 In the case of the EUR/USD currency pair, we use the volatility of the fundamentals of the U.S. andof the euro area. Replacing the volatility of the fundamentals of the euro area with the fundamentals ofGermany before 1999 does not change the results.

28

where bSj;t;t+H is the forecast of professional forecaster j for the underlying currency S at

time t for the forecast horizon t+H, St+H is the realized underlying currency rate at time

t + H, and N is the total number of forecasts for each j, H, and S. The overcon�dent

forecasters are those that change their forecast too much and too often given a signal.

For each foreign-exchange forecast, we obtain a measure of average overcon�dence in

the marketplace, computed as the average overcon�dence of the professional forecasters:

OV ERCt;H;S =1

Jt

JtXj=1

OV ERCj;H;S ; (28)

where Jt is the number of professional forecasters at each time t:We investigate whether

the dispersion in beliefs has di¤erent implications for option pricing depending on how

con�dent traders are with respect to their (di¤erent) beliefs. For example, at one point

in time, we could measure the dispersion of beliefs of extremely overcon�dent forecasters

expressing their views on one currency for a speci�c horizon. At another point in time, we

could measure the dispersion of beliefs of extremely non-con�dent forecasters. One may

argue that the dispersion of beliefs could have di¤erent e¤ects on option pricing in these

two cases. Based on this reasoning, we estimate the following regression:

IVt;T = �+ � t;T + �overcOV ERCt;T + et;T ; (29)

where IVt;T represents the implied volatility of at-the-money (ATM) options on day t with

maturity T , t;T represents the mean absolute deviation of foreign exchange forecasts, and

OV ERCt;T is the average overcon�dence obtained, as in equation (28).

Table 10 shows the results. In general, we observe that the measure of overcon�dence

never drives out our proxy for di¤erence of beliefs. The mean absolute deviation of FX

forecast returns is signi�cant at the one percent level for all currencies. This is evidence that

di¤erence in beliefs is a priced risk factor that does not re�ect a behavioral e¤ect, at least

not an e¤ect related to overcon�dence. This result is important, because option markets

are considered particularly vulnerable to behavioral phenomena (e.g., Shefrin, 1999).

Whenever our measure of overcon�dence has a statistically signi�cant impact on cur-

rency implied volatility beyond the di¤erence in beliefs, the e¤ect is positive: stronger

overcon�dence leads to higher implied volatility. This e¤ect does not seem to be clearly

related to a speci�c forecasting horizon or underlying currency, since we obtain a highly

signi�cant coe¢ cient for di¤erent currencies at di¤erent horizons. The positive sign is con-

sistent with the empirical predictions of some theoretical models in the literature featuring

di¤erent classes of investors. For example, in the general equilibrium model of Dumas,

Kurshev, and Uppal (2006), there are two classes of agents and both stock and bond prices

are excessively volatile because one class is overcon�dent about a public signal. As a result,

29

this class of irrational agents changes its expectations too often, sometimes being excessively

optimistic, sometimes being excessively pessimistic, consistent with (28).20

However, we could think about an alternative de�nitions of overcon�dence. For exam-

ple, the overcon�dent agents could be forecasters that stick with their priors, independently

of any news and forecasting errors. In fact, as discussed in Odean (1998b) and Daniel,

Hirshleifer, and Subrahmanyam (2001), overcon�dent investors believe that the precision

of their private information is higher than its true precision. In this case, the measure

de�ned by equation (28) would actually be the inverse of overcon�dence, because little

updating relative to individual forecast errors would identify forecasters that stick with

their priors. The theoretical model of Vanden (2007) about information quality and option

prices is consistent with this de�nition of overcon�dent agents. In his setting, the presence

of overcon�dent investors would increase information quality and would decrease the level

of implied volatility. This prediction is consistent with our empirical results where higher

information quality - low OV ERCt;H;S - implies lower implied volatility.

20Although the model of Dumas, Kurshev, and Uppal (2006) does not consider foreign currencies andtherefore does not have predictions for currency volatility, we can still think that the main intuition of themodel would follow through. In particular, the positive relation between overcon�dence and volatility isvalid not only for stocks, but also for bonds that can be thought of as more similar to currencies.

30

V. Conclusions

Financial prices re�ect uncertainty and reveal information about its resolution. Many causes

of uncertainty are exogenous and are naturally resolved over time. Some other sources are

instead of a strategic nature, as the timing of their resolution is a strategic variable for some

agents with incentives to hide their information. Independent of the nature of uncertainty,

however, when agents have di¤erent expectations of future market fundamentals, they can

engage in dynamic trading either for speculative reasons or for dynamic hedging. In this

paper, we test the link between uncertainty and asset prices using a measure of dispersion

in beliefs that is based directly on the distribution of forecasts of future exchange rates

provided by market participants. Combining this with an extensive data set on OTC

currency options, we test a number of economic restrictions.

We �nd that the di¤erences of beliefs derived from currency forecasts help explain

the level of implied volatility of currency options. The result is statistically signi�cant,

economically important, and valid at di¤erent horizons. The most striking result is obtained

in the case of the USD/JPY exchange rate, for which the di¤erence in beliefs can explain up

to 60% of the contemporaneous variation in at-the-money implied volatilities. For the other

currency pairs the e¤ect is somewhat less striking, but the coe¢ cients are always signi�cant

at the 1% con�dence level.

Moreover, we �nd that di¤erences in beliefs helps to explain the shape of the implied

volatility smile. This set of results is related to the derivative pricing literature. For example,

Bakshi, Kapadia, and Madan (2003) show the relation between the option-implied volatility

smile and the skewness of the risk-neutral distribution. Since it is known that, in economies

with uncertainty, the skewness in the distribution of beliefs may a¤ect the skewness in the

risk neutral distribution (Buraschi and Jiltsov (2006)), our �ndings lend empirical support

to the importance of modelling heterogeneous information to help explain the dynamics of

derivative prices.

We also obtain important results when we relate our empirical proxy for di¤erences

in beliefs to the volatility risk premium and to FX returns. Both results are consistent

with the interpretation that the di¤erences in beliefs are a component of a time-varying

foreign-exchange risk premium.

Our �ndings demonstrate that di¤erences in beliefs have pervasive e¤ects on derivatives

pricing and risk premiums in the FX markets. Our evidence is striking, since the degree

of beliefs dispersion seems to be largely unexplained by the volatility of macroeconomic

fundamentals.

31

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35

Table 1: Currency options summary statistics

This table presents summary statistics for our sample of currency options. We report the

mean, standard deviation, and number of observations on the straddle (�ATM ), on the 25-

delta risk reversal (�RR), and on the 25-delta strangle (�STR), for four option maturities.

Data are daily from May 1993 to December 2006.

Usd/Jpy Gbp/Usd Eur/Usd Usd/Jpy Gbp/Usd Eur/Usd

One-month maturity Three-month maturity

Mean 10.79 8.41 10.13 10.94 8.75 10.35

�ATM Std.Dev. 2.90 1.68 2.05 2.62 1.52 1.78

Obs. 3597 3597 3597 3597 3597 3597

Mean -0.75 -0.02 0.24 -0.75 0.01 0.28

�RR Std.Dev. 0.90 0.33 0.45 0.82 0.28 0.38

Obs. 3122 3122 3122 2515 2515 2515

Mean 0.35 0.21 0.24 0.36 0.21 0.24

�STR Std.Dev. 0.13 0.06 0.08 0.13 0.05 0.06

Obs. 3122 3122 3122 2515 2117 2515

Six-month maturity One-year horizon

Mean 11.07 8.95 10.47 11.18 9.15 10.56

�ATM Std.Dev. 2.56 1.45 1.66 2.50 1.44 1.57

Obs. 3597 3597 3597 3597 3597 3597

Mean -0.72 0.02 0.29 -0.74 0.03 0.30

�RR Std.Dev. 0.86 0.27 0.36 0.92 0.26 0.34

Obs. 2515 2515 2515 2515 2515 2515

Mean 0.37 0.22 0.24 0.37 0.22 0.25

�STR Std.Dev. 0.12 0.04 0.06 0.12 0.04 0.06

Obs. 2515 2515 2028 2515 2421 2515

36

Table 2: Foreign-exchange forecasts summary statistics

This table presents summary statistics for our sample of foreign-exchange forecasts. For each forecasting

horizon, we show the number of individual forecasts, the number of monthly surveys, and the number of

forecasters for each survey.

Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

Individual forecasts No. 7566 6987 7679

Monthly surveys No. 156 142 156

max No. 66 65 66

Forecasters ave No. 47.8 48.6 49.2

min No. 27 30 27

Three-month horizon

Individual forecasts No. 7640 7006 7729

Monthly surveys No. 156 142 156

max No. 67 66 67

Forecasters ave No. 48.2 48.6 49.5

min No. 30 29 27

Six-month horizon

Individual forecasts No. 7562 6948 7643

Monthly surveys No. 156 142 156

max No. 66 65 66

Forecasters ave No. 47.7 48.2 49.0

min No. 27 29 27

One-year horizon

Individual forecasts No. 7521 6896 7581

Monthly surveys No. 156 142 156

max No. 65 64 65

Forecasters ave No. 47.4 47.8 48.6

min No. 27 29 27

37

Table 3: Do foreign exchange forecasts have predictive power?

This table shows the results of estimating the following regression:

ST � St = �+ � (Mt;T � St) + et;

where St represents the foreign exchange rate at time t and Mt;T represents the average foreign exchange

forecasts on day t with horizon T . We also compute the Root Mean Squared Error (RMSE) of the average

forecast and compare it with the RMSE of a random walk (RMSErw).

Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

constant 0.2730 0.0002 0.0006

� 0.0655 0.3075 0.1651

R2 0.0015 0.0120 0.0155

RMSE/RMSErw 1.3045 1.0494 1.3816

Three-month horizon

constant 0.1985 0.0055 0.0079

� -0.2675 0.4667�� -0.0618

R2 0.0139 0.0272 0.0008

RMSE/RMSErw 1.3000 1.0139 1.2315

Six-month horizon

constant 0.1644 0.0107 0.0213

� -0.0538 0.3597 -0.2350

R2 0.0005 0.0151 0.0086

RMSE/RMSErw 1.2089 1.0467 1.2067

One-year horizon

constant 0.6375 0.0287�� 0.0434��

� -0.2013 0.7904�� -0.3935

R2 0.0090 0.0629 0.0186

RMSE/RMSErw 1.3156 0.9678 1.1704

� � �, ��, and � denote statistical signi�cance at the one, �ve, and ten percent levels, respectively.

38

Table 4: Di¤erence in beliefs in foreign-exchange forecasts

This table presents summary statistics for the mean absolute deviation (MAD), the skew-

ness, and the kurtosis of the cross-sectional distribution of forecasts.

Usd/Jpy Gbp/Usd Eur/Usd

max 0.0311 0.0263 0.0248

MAD ave 0.0142 0.0101 0.0134

min 0.0058 0.0059 0.0070

max 1.5594 6.7005 1.8880

One-month horizon Skewness ave -0.2151 -0.0478 -0.1495

min -2.7184 -2.2878 -1.2899

max 14.0450 46.6212 7.7532

Kurtosis ave 3.7382 4.4633 3.2631

min 1.9067 1.8743 1.8088

max 0.0530 0.0322 0.0373

MAD ave 0.0256 0.0175 0.0236

min 0.0147 0.0102 0.0124

max 1.1200 6.2487 2.0457

Three-month horizon Skewness ave -0.1059 -0.1131 -0.2482

min -2.8092 -5.5905 -2.1533

max 13.1263 42.0335 10.4946

Kurtosis ave 3.3926 4.1744 3.2151

min 1.7834 2.0986 1.7866

max 0.0654 0.0382 0.0499

MAD ave 0.0375 0.0241 0.0333

min 0.0228 0.0165 0.0181

max 2.8892 5.6869 1.6808

Six-month horizon Skewness ave 0.1300 -0.1263 -0.2189

min -1.7103 -3.7290 -1.6646

max 15.8295 37.3594 8.2769

Kurtosis ave 3.3414 3.8324 3.2045

min 1.8746 2.0347 1.4880

max 0.0836 0.0470 0.0769

MAD ave 0.0518 0.0324 0.0448

min 0.0264 0.0221 0.0246

max 1.8026 5.0023 2.1723

One-year horizon Skewness ave 0.4185 -0.2039 -0.2398

min -1.3730 -2.1288 -1.7149

max 8.6445 31.8274 11.5625

Kurtosis ave 3.5670 3.9591 3.5135

min 1.6948 2.2310 1.8886

39

Table 5: At-the-money implied volatility and di¤erence in beliefs

This table shows the results of estimating the following regression:

IVt;T = �+ � MADt;T + �skSKt;T + �kuKUt;T + �ivIVt�30;T�30 + et;T ;

where IVt;T represents the implied volatility of at-the-money (ATM) options on day t with

maturity T , MADt;T , SKt;T , KUt;T , represent the mean absolute deviation, the skewness,

and the kurtosis of foreign exchange forecasts on day t with horizon T , respectively.

Usd/ Gbp/ Eur/ Usd/ Gbp/ Eur/ Usd/ Gbp/ Eur/

Jpy Usd Usd Jpy Usd Usd Jpy Usd Usd

One-month horizon

� 4.32��� 5.44��� 4.91��� 3.78��� 5.28��� 3.34��� 1.25�� 1.51��� 1.07�

� 466.73��� 283.45��� 407.25��� 458.11��� 332.24��� 420.45��� 241.83��� 209.32��� 246.78���

�sk 0.27 0.05 -0.74��� 0.07 -0.13 -0.35�

�ku 0.19� -0.07 0.39��� 0.07 -0.02 0.24���

�iv 0.55��� 0.57��� 0.50���

R2 0.54 0.23 0.46 0.55 0.27 0.53 0.72 0.59 0.68

Three-month horizon

� 3.45��� 6.39��� 5.95��� 2.14��� 6.43��� 3.98��� 0.09 1.34��� 0.60

� 294.79��� 123.71��� 192.02��� 296.34��� 129.34��� 213.78��� 118.83��� 63.38��� 93.53���

�sk -0.14 0.05 -0.89��� 0.04 -0.01 -0.24�

�ku 0.37��� -0.03 0.37��� 0.13� -0.01 0.16���

�iv 0.68��� 0.72��� 0.67���

R2 0.58 0.14 0.30 0.63 0.15 0.48 0.85 0.66 0.76

Six-month horizon

� 2.74��� 6.81��� 6.76��� 2.63��� 6.82��� 5.34��� 0.37 1.11�� 0.87�

� 225.08��� 78.87��� 113.77��� 222.01��� 82.26��� 124.82��� 57.97��� 29.17��� 29.83���

�sk -0.53�� 0.01 -0.82��� -0.09 -0.04 -0.18

�ku 0.08 -0.03 0.27� 0.01 0.01 0.06

�iv 0.77��� 0.79��� 0.80���

R2 0.64 0.10 0.21 0.65 0.11 0.35 0.88 0.70 0.77

One-year horizon

� 3.36��� 7.13��� 7.25��� 2.81��� 7.35��� 6.56��� 0.23 1.03�� 0.49

� 152.92��� 53.83��� 75.61��� 149.92��� 52.67��� 78.77��� 21.40��� 11.45� 13.90��

�sk -0.42 0.09 -0.33�� -0.06 0.04 -0.08

�ku 0.24� -0.04 0.14� 0.02 -0.01 0.07��

�iv 0.88��� 0.84��� 0.87���

R2 0.57 0.09 0.22 0.58 0.10 0.27 0.91 0.77 0.84

obs. 156 142 156 156 142 156 155 141 155� � �, ��, and � denote statistical signi�cance at the one, �ve, and 10 percent levels, respectively.

40

Table 6: The implied volatility smile and di¤erence in beliefs

This table shows the results of estimating the following regression: IV 25�deltat;T

IV 75�deltat;T

!� 1 = �+ � MADt;T + �skSKt;T + �kuKUt;T + et;T ;

where IV x�deltat;T represents the implied volatilities at x�delta moneyness for options on dayt with maturity T and MADt;T , SKt;T , KUt;T , represent the mean absolute deviation, the

skewness, and the kurtosis of foreign exchange forecasts on day t with horizon T , respectively.

Usd/Jpy Gbp/Usd Eur/Usd Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

� -0.06��� -0.01 0.05��� -0.06��� -0.02 0.06���

� -0.52 0.92 -1.60� 0.38 1.29 -2.20��

�sk -0.03��� -0.01�� -0.02���

�ku -0.01 0.01� -0.00

R2 0.01 0.01 0.02 0.06 0.05 0.09

obs. 138 137 138 138 137 138

Three-month horizon

� -0.11��� 0.02 0.09��� -0.13��� 0.05�� 0.05��

� 1.49� -0.86 -2.38��� 1.63� -1.55 -2.04���

�sk -0.02 0.00 -0.00

�ku 0.01 -0.01�� 0.01��

R2 0.02 0.01 0.09 0.04 0.07 0.14

obs. 112 94 112 112 94 112

Six-month horizon

� -0.18��� 0.01 0.08��� -0.19��� 0.02 0.11���

� 3.01��� -0.36 -1.46��� 2.94��� -0.73 -1.72���

�sk -0.01 0.01 -0.02�

�ku 0.01 0.00 -0.01�

R2 0.15 0.00 0.09 0.16 0.02 0.14

obs. 112 112 91 112 112 91

One-year horizon

� -0.19��� 0.01 0.05��� -0.17��� 0.06��� 0.012

� 2.23��� -0.26 -0.52� 2.21��� -1.18��� -0.21

�sk 0.03 0.02��� 0.03���

�ku -0.01 -0.00��� 0.01���

R2 0.12 0.00 0.02 0.13 0.18 0.10

obs. 112 107 112 112 107 112

� � �, ��, and � denote statistical signi�cance at the one, �ve, and 10 percent levels, respectively.

41

Table 7: The volatility risk premium and di¤erence in beliefs

This table shows the results of estimating the following regression:

IVt;T � E [RVt;T ] = �+ � MADt;T + �skSKt;T + �kuKUt;T + et;T ;

where IVt;T represents the at-the-money implied volatilities for options on day t with ma-

turity T , E [RVt;T ] is an exponential moving average of realized volatility over the previous

month, and MADt;T , SKt;T , KUt;T , represent the mean absolute deviation, the skewness,

and the kurtosis of foreign exchange forecasts on day t with horizon T , respectively.

Usd/Jpy Gbp/Usd Eur/Usd Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

� -2.90��� 0.54 0.76� -2.98��� 0.65 -0.57

� 443.05��� 23.18 50.08� 431.71��� -6.74 59.01�

�sk 0.37 -0.02 -0.38�

�ku 0.09 0.04� 0.35���

R2 0.37 0.002 0.01 0.38 0.02 0.09

Three-month horizon

� -4.54��� 1.26��� 0.67 -5.85��� 1.31��� -0.59

� 305.32��� -18.77 31.41� 306.86��� -25.63 45.15���

�sk -0.13 0.07 -0.43��

�ku 0.37 0.02 0.25���

R2 0.47 0.004 0.01 0.51 0.02 0.13

Six-month horizon

� -6.13��� 1.31��� 0.62 -5.88��� 1.25��� -0.21

� 251.47��� -11.57 22.00� 252.04��� -13.94 29.24��

�sk -0.20 -0.07 -0.18

�ku -0.08 0.03 0.17��

R2 0.60 0.003 0.01 0.60 0.01 0.05

One-year horizon

� -5.53��� 0.85� 0.37 -5.52��� 0.47 -0.82

� 168.63��� 6.17 19.05�� 161.47��� 12.20� 29.62���

�sk 0.80��� -0.22�� -0.03

�ku 0.02 0.03 0.21���

R2 0.55 0.01 0.02 0.57 0.03 0.10

obs. 156 142 156 156 142 156

� � �, ��, and � denote statistical signi�cance at the one, �ve, and ten percent levels, respectively.

42

Table 8: Carry trade returns, currency returns, and dispersion of beliefs

Panel A shows the results of estimating the following regression:

carryt;t+1 = �+ � MADt;t+1 + �ivIVt;t+1 + �rpRPt;t+1 + et;t+n;

where carryt;t+1 represents USD/JPY carry trade returns between time t and t + 1,

MADt;t+1 is the mean absolute deviation of USD/JPY one-month forecasts at time t,

IVt;t+1 is the implied volatility of USD/JPY one-month options, and RPt;t+1 is the volatil-

ity risk-premium with one-month horizon.

Panel B shows the results of estimating the following regression:

st+n � st = �+ �fwd (ft;t+n � st) + � MADt;t+n + et;t+n;

where st represents the logarithm of the spot rate at time t, ft;t+n represents the logarithm

of the forward rate at time t with maturity t+n,MADt;t+n is the mean absolute deviation of

foreign exchange forecasts at time t for the horizon t+n, and n is one-month, three-months,

six-months, and one-year.

Panel A

� 0.0169� 0.0083 0.0069� 0.0046

� -0.9519� -1.3905�

�iv -0.0004 0.0021

�rp -0.0009 -0.0013

R2 0.0151 0.0016 0.0088 0.0267

43

Panel BUsd/Jpy Gbp/Usd Eur/Usd

One-month horizon

� -0.0049 0.0055 -0.0006 -0.0074� 0.0038� 0.0074

�fwd -1.9377 -3.0161�� -1.3137 -1.4692 -4.5970��� -4.0266��

� -1.0099�� -0.6869� 0.8052�

R2 0.0058 0.0185 0.0027 0.0080 0.0388 0.0487

Three-month horizon

� -0.0168� 0.0180 -0.0025 -0.0209 0.0081 0.0119

�fwd -2.3323� -3.6999��� -2.0814 -2.3989 -3.4795�� -3.5397��

� -1.8992�� -1.4011�� -0.1532

R2 0.0271 0.0607 0.0245 0.0438 0.0758 0.0760

Six-month horizon

� -0.0549��� -0.0054 -0.0026 0.0496 0.0235��� -0.0353

�fwd -6.3486��� -8.2463��� -4.1094 -4.7260 -10.5510��� -9.3788���

� -1.7592�� -2.2483� 1.6443�

R2 0.0970 0.1215 0.0513 0.0918 0.2953 0.3180

One-year horizon

� -0.0850��� -0.0331 -0.0173 -0.0165�� 0.0414� -0.0845�

�fwd -2.4804�� -3.4377��� -2.7106� -2.7128�� -4.6206��� -3.7687���

� -2.9331�� -0.0243 2.5645��

R2 0.1451 0.2341 0.1687 0.1687 0.4737 0.5060

� � �, ��, and � denote statistical signi�cance at the one, �ve, and ten percent levels, respectively, using Newey-Weststandard errors (12 lags).

44

Table 9: Di¤erence in beliefs and fundamentals

This table shows the R2 from the estimation of the following regression:

t = �+2X

h=1

�h

���~"GDP;ht

���+ 2Xh=1

�h

���~"IP;ht

���+ 2Xh=1

�h

���~"CPI;ht

���++

2Xh=1

�h

���~"UR;ht

���+ 2Xh=1

�h

���~"BM;ht

���+ 2Xh=1

�h

���~"BP;ht

���+ et;where t represents the mean absolute deviation of foreign exchange forecasts, and

���~"ECO;ht

���estimates of the volatility of economic indicators ECO = fGDP; IP;CPI; UR;BM;BPgfor the home economy (h = 1) and for the foreign economy (h = 2).

Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

R2 0.1554 0.0379 0.1259Three-month horizon

R2 0.1558 0.0461 0.1239Six-month horizon

R2 0.1462 0.0827 0.0972One-year horizon

R2 0.1049 0.1158 0.1312

45

Table 10: Di¤erence in beliefs, overcon�dence, and implied volatility

This table shows the results of estimating the following regression:

IVt;T = �+ �psiMADt;T + �overcOV ERCt;T + et;T ;

where IVt;T represents the implied volatility of at-the-money (ATM) options on day t with

maturity T , MADt;T represents the mean absolute deviation of foreign exchange forecasts,

and OV ERCt;T represents the average ratio of forecasting update on forecasting error.

Usd/Jpy Gbp/Usd Eur/Usd

One-month horizon

� -11.9545��� 9.9275��� 13.6029���

� 398.6172��� 273.496��� 394.4274���

�overc 19.7108��� -4.5324 -9.1213

R2 0.5736 0.2445 0.4709

observations 156 142 156

Three-month horizon

� -3.2522��� 6.1664��� 4.4953��

� 256.3822��� 125.1418��� 199.2984���

�overc 10.6990��� 0.2468 1.6626

R2 0.6116 0.1383 0.3013

observations 156 142 156

Six-month horizon

� -2.6515��� 4.9463��� 4.5751���

� 177.1870��� 93.0773��� 120.7616���

�overc 12.8612��� 2.0684 3.1137�

R2 0.7176 0.1135 0.2274

observations 156 142 156

One-year horizon

� -5.1284��� 3.8800��� 5.2511���

� 109.6251��� 77.3221��� 84.3924���

�overc 22.4235��� 4.0155��� 3.5383��

R2 0.7506 0.1541 0.2358

observations 156 142 156

� � �, ��, and � denote statistical signi�cance at the one, �ve, and ten percent levels, respectively.

46

102 104 106 108 110 112 1140

2

4

6

8

10

12

14

16

18

20January 2004

USD/JPY forecast

Mean     = 106.01Std.Dev. = 0.78Skewness = 0.54Kurtosis = 3.16

102 104 106 108 110 112 1140

2

4

6

8

10

12

14

16

18

20March 2004

USD/JPY forecast

Mean     = 107.54Std.Dev. = 2.53Skewness = 0.14Kurtosis = 2.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

Implied Volatility Smile

delta

impl

ied 

vola

tility

Jan 2004

Mar 2004

95 100 105 110 115 120 125 1300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Jan 2004Mar 2004

Figure 1: Beliefs, implied volatility smiles, and risk-neutral densities. The �rst andsecond plot show histograms of the one-month foreign exchange forecasts for the USD/JPY at the

beginning of January 2004 and March 2004. The plots also report the mean, standard deviation,

skewness and kurtosis of the forecasts. The third panel is the cubic-spline interpolated implied

volatility of one-month currency options as a function of the delta for January and March 2004.

The fourth panel shows the option-implied state price density of the USD/JPY in January and

March 2004.

47

1994

 199

6 1

998

 200

0 2

002

 200

40

0.02

0.04

0.06

0.08

EUR

/USD

Sam

ple 

Perio

d

MAD

1994

 199

6 1

998

 200

0 2

002

 200

4­3­2­10123

Sam

ple 

Perio

d

Skewness

1994

 199

6  19

98 2

000  

2002

 200

424681012

Sam

ple 

Perio

d

Kurtosis

1mo

3mo

6mo

12m

o

1994

 199

6 1

998

 200

0 2

002

 200

40

0.02

0.04

0.06

0.08

USD

/JPY

Sam

ple 

Perio

d

MAD

1994

 199

6 1

998

 200

0 2

002

 200

4­3­2­10123

Sam

ple 

Perio

d

Skewness

1994

 199

6 1

998

 200

0 2

002

 200

4

24681012

Sam

ple 

Perio

d

Kurtosis

1mo

3mo

6mo

12m

o

1995

 199

7 1

999

 200

1 2

003

 200

50

0.02

0.04

0.06

0.08

GBP

/USD

Sam

ple 

Perio

d

MAD

1995

 199

7 1

999

 200

1 2

003

 200

5­3­2­10123

Sam

ple 

Perio

d

Skewness19

95 1

997

 199

9 2

001

 200

3 2

005

24681012

Sam

ple 

Perio

dKurtosis

1mo

3mo

6mo

12m

o

Figure 2: Measures of Foreign Exchange Di¤erences in Beliefs. These plots show thedynamics of the mean absolute deviation (MAD), skewness, and kurtosis of foreign exchangeforecasts for the three currencies and four forecast horizons in our sample.

48

Figure 3: USD/JPY carry trade returns. This plot shows average total carry tradereturns (annualized) and interest rate di¤erential when the MAD for USD/JPY one-monthforecast is low or high, USD/JPY one-month implied volatility is low or high, the volatilityrisk premium (implied - realized one-month volatility) is low or high. For all measures, low(high) is bottom (top) quartile.

49