FX Options and Excess Returns

Yu-chin Chen and Ranganai Gwati ∗

September 3, 2013

Abstract

The empirical failure of the uncovered interest rate parity (UIP) condition are

commonly attributed to time-varying risk premia and market expectation biases, yet

neither concept has been reliably measured in empirical data. This paper proposes to

use prices of traded foreign exchange (FX) options to capture common market assess-

ment for the probability distributions of future exchange rate realizations. Using daily

options data for six major currency pairs, we show that the options-implied FX risk

measures - standard deviation, skewness, and kurtosis - consistently explain subsequent

excess currency returns for horizons between one week to twelve months. This robust

empirical pattern is consistent with a representative expected utility-maximizing in-

vestor who, in addition to the mean and variance of asset returns, also cares about

the skewness and kurtosis of the return distribution. Pushing beyond analyses that

consider only the conditional mean of future expectation, we use quantile regressions to

demonstrate that the options-based measures of higher moment risks can predict the

full distribution of subsequent excess returns. The term structure and cross-correlation

of the options-implied moments add further predictive power, supporting previous lit-

erature emphasizing term-structure dynamics and global risk.

Keywords: exchange rates; excess returns; options pricing; volatility smile; risk; term struc-

ture of implied volatility;quantile regression JEL Code: E58, E52, C53

∗We are indebted to Keh-Hsiao Lin and Joe Huang for pointing us to the relevant data sources. Wewould also like to thank Samir Bandaogo, Nick Basch, David Grad, Abe Robison, and Eric Zivot for theirhelpful comments and assistance. All remaining errors are our own. Chen’s and Gwati’s research aresupported by Gary Waterman Scholarship and the Buechel Fellowship at the University of Washington.Chen: yuchin@uw.edu; Gwati: gwatir@uw.edu; Department of Economics, University of Washington, Box353330, Seattle, WA 98195

1 Introduction

The forward exchange rate is well-known to be a biased predictor of the future spot rate, an

empirical regularity commonly referred to as the forward premium puzzle or the uncovered

interest parity (UIP) puzzle. One manifestation of this empirical (ir)regularity is that coun-

tries with higher interest rates tend to see their currencies appreciate subsequently, and a

“carry-trade” strategy exploiting this pattern, on average, delivers excess currency returns.1

This violation of the UIP condition is commonly attributed to time-varying risk premium

and biases in (measured) market expectations. However, empirical proxies based on surveyed

forecasts or standard measures of risk -e.g. ones built from consumption growth, stock mar-

ket returns, or the Fama-French (1993) factors - have been unsuccessful in explaining it.2

As such, while recognizing the presence of risk, macro exchange rate modeling efforts often

ignore it in empirical testing (e.g. Engel and West (2005); Mark (1995)). On the finance side,

efforts aiming to identify portfolio return-based “risk factors” offer some empirical success in

explaining the cross-sectional distribution of excess FX returns, but have little to say about

bilateral exchange rate dynamics (e.g. Lustig et al. (2011); Menkhoff et al. (2012); Verdelhan

(2012)).3

This paper proposes to use information in the currency (FX) option prices to construct

measures of expected risk, and test whether option-implied higher-moment risks explain

subsequent excess returns. We show that these moments have a huge predictive ability for

both the conditional mean and entire conditional distribution of excess returns.

Conceptually, since payoffs of option contracts depend on the uncertain future realization

of the price of the underlying asset, option prices must reflect market sentiment and beliefs

1A carry trade strategy is to borrow low-interest currencies and lend in high-interest currencies, or sellingforward currencies that are at a premium and buying forward currencies with a forward discount.

2See, for example, Engel (1996) for a survey of the forward premium literature, as well as recent studiessuch as Bacchetta and van Wincoop (2009); Burnside et al. (2011).

3Lustig et al. (2011) and Verdelhan (2012) for example, identify a “carry factor” based on cross sectionof interest rate-sorted currency returns, and a “dollar factor” based on cross-section of beta-sorted currencyreturns.

1

about the probability of future payoffs. Using prices of a cross section of option contracts

(at-the-money, risk reversals and vega-weighted butterflies at 10- and 25 deltas) which de-

liver payoffs under differential future realizations, we uncover ex-ante standard deviations,

skewness, and kurtosis of expected future exchange rate changes. With daily options data

for six major currency pairs and seven tenors, we show that these market-based measures of

FX variance, crash, and tail risk can explain subsequent FX excess returns between horizons

of one week and 12 months. We then extend the approach pioneered by Hansen (1980), and

in Clarida and Taylor (1997), Chen and Tsang (2013) that use the forward rates or interest

differentials over time and across currency pairs to model excess returns.

We note that simple derivatives such as the forwards and futures have been used exten-

sively in explaining excess returns (e.g. Hansen (1980), Clarida and Taylor (1997) among

many others). However, payoffs from forward contracts are linear in the return on the un-

derlying currency and as such do not contain as useful a set of information as the non-linear

contracts we examine. Indeed, FX options have been used to proxy variance or tail risk in

various specific but limited context, such as testing the portfolio balance model of exchange

rate determination (Lyons (1998)), measuring announcements effects (Grad (2010)), evalu-

ating rare events theory (Farhi et al. (2009)), or conducting density forecasts (Christoffersen

and Mazzotta (2005)). To the best of our knowledge, however, there has been no system-

atic and comprehensive testing of whether the ex-ante information contained in FX options

indeed predicts ex-post excess currency returns. Our paper aims to bridge this gap.

Our use of options price data and related empirical methodologies is motivated by a

number of reasons. First, options are forward-looking by construction. Campa et al. (1998a)

point out that this forward-looking property means option prices should incorporate informa-

tion such as forthcoming regime switches or the presence of a peso problem.4 Second, option

prices are rooted in market participant behavior because they are based on what market

4The peso problem refers to the effects on inferences caused by low-probability events that do not occurin the sample,which can lead to positive excess return.

2

participants do instead of what they say.5 Furthermore, cross sections of option-prices imply

a subjective probability distribution of future spot exchange rates, which captures both mar-

ket participants’ beliefs and their preferences. (This distribution is commonly referred to as

the “risk-neutral distribution”, though it does NOT imply it’s derived under risk-neutrality.

On the contrary, it incorporates both the expected physical probability distribution of fu-

ture exchange rate realization as well as the risk premium, or compensation required to

bear the uncertainty.) Lastly, Chang et al. (2013) argue that the options approach avoids

the traditional trade-off problem with estimates of higher moments from historical returns

data:longer windows are required to increase precision, but shorter windows are required to

obtain conditional rather than unconditional estimates.

Modern techniques such as the Vanna-Volga method (see, e.g. Castagna and Mercu-

rio (2005)) and the methodology of Bakshi et al. (2003) facilitate elegant computation of

options-implied higher order moments of future exchange rate changes. These methods are

particularly attractive because the options-implied moments are extracted without having

to impose any assumptions on investor preferences and expectation-formation mechanisms,

or to specify the stochastic process driving the underlying spot exchange rate. Lastly, op-

tion prices are an attractive source for extracting market expectations when compared to

other sources commonly considered for the same purposes. For example, Chen and Good-

hart (1998) argue that FX options markets are likely to contain cleaner measures of market

expectations than interest rate differentials because there is less government intervention in

these markets. Survey data are based on what market participants say, not what they do,

and their coverage is confined to a subset of survey participants instead of being an aggregate

of the whole market. 6

5Contrast with say, survey data.6As discussed earlier, forward contracts are forward-looking by construction, but for a given currency pair

and tenor, there is only one forward price with linear dependency on future spot realization. The multipleoption prices for options with different strikes offer a much richer information set.Lastly, standard, standardconstructions of market expectations and perceived risks based on macro fundamentals or finance factor donot work well.

3

Our main empirical result is that risk-neutral moments appear to be quite robust in pre-

dicting not only the conditional means but also the entire distributions ex-post FX excess

returns. We also find that information extracted from a panel of currency pairs across coun-

tries and across tenors also help predict bilateral excess returns.Finally,there are significant

breaks in the relationship between currency excess returns and option-implied risk-neutral

moments of exchange rate changes around late 2008 and early 2009.

2 International Portfolio Optimization

2.1 Forward Premium Puzzle and Excess Currency Returns

The efficient market condition for the foreign exchange markets, under rational expectations,

equates cross border interest differentials it − i∗t with the expected rate of home currency

depreciation, adjusted for the risk premium associated with currency holdings 7, ρt:

iτt − iτ ,∗t = Et∆st+τ + ρt+τ (2.1)

This condition is expected to hold for all investment horizon τ , with interest rates are at

matched maturities. Ignoring the risk premium term, numerous papers have tested this

equation since Fama (1984), and find systematical violations of this UIP condition:

st+1 − st = α + β(it − i∗t ) + εt+1

H0 : β = 1(2.2)

with an estimated β < 0. This is the so-called uncovered interest rate parity puzzle or the

forward premium puzzle (see Engel (1996), for a survey of the literature). To see the connec-

7In this paper, we define the exchange rate as the domestic price of foreign currency.A rise in the exchangerate indicates a depreciation of the home currency.However, “home”does not have a geographical significancebut follow the FX market conventions. See table (1A)

4

tion with forward rates, we note that the covered interest parity condition, an empirically

valid no-arbitrage condition, equates the forward premium f t+τt − st, with interest differen-

tials. The UIP condition above thus implies that the forward rate should be an unbiased

predictor for future spot rate: Etst+τ = f t+τt or st+τ = f t+τt + ut+τ .

We should next define FX excess returns as the rate of return across borders net of cur-

rency movement, and one can see that the UIP or forward premium puzzle can be expressed

as a non-zero averaged excess return over time:

xrt+τ = f t+τt − st+τ = (iτt − iτ ,∗t )−∆st+τ = ρt+τ + ut+τ (2.3)

It is natural then to note that the empirical failure of the risk-neutral UIP condition can be

attributable to either the presence of a time-varying risk premium,ρt+τ , or that expectation

error, ut, may not be i.i.d normal over time. If the distribution of either of these are not

normal mean zero over the time series, empirical estimates of the slope coefficient would likely

suffer omitted variable bias or other complications. The difficulty faced by the literature is

that standard proposed measures of risk, such as ones based on real per-capita consumption

growth (CAPM), excess returns in the equity markets, the Fama and French (1993) factors

(excess returns in the stock market, the size premium, and the value premium), and so on

do not appear to have strong correlation with excess returns.

2.2 General Asset Allocation Problem with n risky assets 8

In this subsection we show that in addition to risk neutrality and rational expectations

assumptions, the UIP condition also hinges on the rather restrictive assumptions of normality

of returns and constant absolute absolute risk aversion (CARA) utility. These two additional

assumptions have the joint implication of reducing the representative investor’s optimal asset

allocation problem to a mean-variance optimization problem.

8Material in this section is mostly from Jondeau et al. (2010)

5

The empirical failure of the UIP condition and the documented departure of FX returns

from normality suggest that the calibration of shocks to UIP may need to be more sophisti-

cated than the normal distribution commonly used in standard open-economy models.

We start with the problem of an investor who,in each period, allocates her portfolio

among assets with the goal of maximizing the expected utility of next period wealth. In

each period, the investor has n risky assets to choose from. The vector of gross returns is

given by rt+1 = (r1,t+1, ..., rn,t+1). If we suppose Wt is arbitrarily set to 1, then Wt+1 = α′trt+1

, where α is an n by 1 vector of portfolio weights.

The investor’s problem is to choose αt to maximize the expression

Et[U(Wt+1)] = Et[U(α′trt+1)]

=∫...∫U(Wt+1)f(rt+1)dr1,t+1dr2,t+1...drn,t+1

(2.4)

subject to the condition that∑n

i=1 αi,t = 1, where f(rt+1) is the joint probability distribution

of rt+1.

2.2.1 CARA and Normality reduce problem to mean-variance optimization

Let’s further assume that the investor has CARA utility and that returns are conditionally

normally distributed. This preference assumption means the utility is given by

U(Wt+1) = −e−γWt+1 , where γ ≥ 0 is the coefficient of absolute risk aversion. The

assumption that rt+1 ∼ N(µt+1,Σt+1) implies that Wt+1 ∼ N(µp,t+1, σ2p,t+1), where µp,t+1 =

α′tµt+1 and σ2

p,t+1 = α′tΣt+1αt

With these two assumptions, expression (2.4) reduces to9

Et[U(Wt+1)] = −Et[e−γWt+1 ] = γµp,t+1 −1

2γ2σ2

p,t+1 (2.5)

Equation (2.5) show that under the assumptions of CARA function and conditional

9The second equality follows from the fact that e−γWt+1 ∼ LN(−γµp,t+1, γ2σ2p,t+1) , so Et[e−γWt+1 ] =

−γµp,t+1 + γ2σ2p,t+1

6

normality of returns, the general portfolio allocation problem (2.4) reduce to the standard

mean-variance optimization problem.

2.2.2 2-asset international bond portfolio assumption yields UIP condition

If we further assume that our investor has a 2-asset portfolio made up of a nominally safe

domestic bond and a foreign bond, and that she allocate α to domestic bond, then

next period wealth expressed in local currency units is given by

Wt+1 =

[α(1 + it) + (1− α)(1 + i∗t )

St+1

St.

]Wt (2.6)

In this 2-asset case and CARA and conditionally normal returns,

µp,t+1 =[α(1 + it) + (1− α)(1 + i∗t )

EtSt+1

St

]Wt,

σ2p,t+1 =

(1−α)2(1+i∗t )2V art(St+1)W 2t

S2t

(2.7)

Plugging in the expressions in equation (2.7) into objective function (2.5) and rearranging

the first order condition gives us the following expression, which implicitly determines the

optimal α:

(1 + it)− (1 + i∗t )EtSt+1

St=−γWt(1− α)(1 + i∗t )

2V art(St+1)

S2t

(2.8)

Equation (2.8) reduces to the UIP condition if we assume that all investors are risk-

neutral (γ = 0)10:

1 + it1 + i∗t

=EtSt+1

St.

2.2.3 Asset allocation under higher order moments

Jondeau et al. (2010) emphasize that under the general cases in which the distribution

of portfolio returns is asymmetric, the investor’s utility function of higher order than the

10UIP will also hold if α = 1, regardless of investors’ degree of risk aversion.

7

quadratic and the mean and variance do not completely determine the distribution, then

higher order moments and their signs must be taken into account in the portfolio asset

allocation problem. In this subsection we present a framework for incorporating higher

order moments into the asset allocation problem due to Jondeau et al. (2010).

Jondeau et al. (2010) note that the objective in (2.4) can be intractable and it is usual to

focus on approximation of (2.4) based on higher order moments. They consider a Taylor’s

series expansion of the utility function around expected utility up to the fourth order:

U(Wt+1) = U(EtWt+1) + U (1)(Wt+1)(Wt+1 − EtWt+1) + 12!U (2)(Wt+1)(Wt+1 − EWt+1)2+

13!U (3)(Wt+1)(Wt+1 − EtWt+1)3 + 1

4!U (4)(Wt+1)(Wt+1 − EtWt+1)4

(2.9)

This means

Et[U(Wt+1)] ≈ U(EtWt+1) + U (1)(Wt+1)(Wt+1 − EtWt+1) + 12!U (2)(Wt+1)(Wt+1 − EtWt+1)2+

13!U (3)(Wt+1)(Wt+1 − EtWt+1)3 + 1

4!U (4)(Wt+1)(Wt+1 − EtWt+1)4,

(2.10)

which reduces to

Et[U(Wt+1)] ≈ −e−γµp[1 + γ2

2σ2p −

γ3

6s3p + γ4

24k4p

], (2.11)

under the assumption of CARA utility. In equation (2.11), s3p and k4

p are the skewness and

kurtosis of portfolio return. Optimal portfolio weights can then be obtained by maximizing

expression (2.10) instead of the exact objective function shown in expression (2.4).

2.2.4 Investor’s Preferences for moments

For CARA and CRRA utility, the weight the investor puts on the higher order moments

depend on the degree of risk aversion parameter γ. For these utility functions, the investor

has a preference for positive skewness and a negative preference for high kurtosis. In more

8

general settings, however, the weight on the nth moment depends on the nth derivative of the

utility function, and the signs of sensitivities of utility function to changes in higher moments

cannot be easily pinned down. Jondeau et al. (2010) point out that if the moments are not

orthogonal to each other , then the effect of utility of increasing one moment might not be

straight forward .

Scott and Horvath (1980) establish some general conditions for investor preference for

skewness and kurtosis. They show that if investors have positive marginal utility of wealth,

consistent risk aversion at all levels and strict consistency of moment preference, then they

will have preference for positive skewness: U (3) > 011. These authors also show that consis-

tent risk aversion,strict consistency of moment preference and preference for positive skew-

ness imply a negative preference for kurtosis : U (4) < 0.

3 Information Content of Currency Options

3.1 Information Content of Currency Option Prices

3.1.1 Information Contained in Volatility Smile

The starting point for understanding the information content of the volatility smile is the

result by Breeden and Litzenberger (1978), who show that in complete markets, the call

option pricing function (C) and the exercise price K are related as follows:

∂2C

∂K2= e−r

dτπQt (St+τ ), (3.1)

where rd and rf are the domestic and foreign risk-free interest rates and πQt (St+τ ) is the

risk-neutral probability density function (pdf) of future spot rates. Equation (3.1) implies

that, in principle, we can estimate the whole pdf of time St+τ spot exchange rate from time

11Define consistent and strict consistency.

9

t volatility smile. In addition to the Breeden and Litzenberger (1978) result in equation

(3.1), we note that although market participants can be treated as if they are risk-neutral

for the purpose of option-pricing, option prices theoretically contain information about both

investors’ beliefs and their risk preferences, as shown from the following formula for the price

of a European-style call option:

C(t,K, T ) =

∫ ∞K

Mt,T (ST −K)πPt (ST )dST = e−rdτ

∫ ∞K

(ST −K)πQt (ST )dST . (3.2)

In equation (3.2), Mt,t+τ is the pricing kernel, which captures the investor’s degree of risk

aversion and πPt (St+τ ) is the physical probability density function of future spot exchange

rates. 12

Both forward contracts and currency options are forward-looking by construction. Fur-

thermore, the risk-neutral density implied by the pricing equations for both derivatives is

the same. For the purpose of making inferences about this density, however, the option price

equation (3.2) is more useful than (3.3), the one for forward prices. This is because for a

given tenor, a cross section of option prices with different strike prices are observed, whereas

there is only one forward price observed. A forward contract can in fact be viewed as a

European-style call option with a strike price of zero. To see this, we recall that, on one

hand, the theoretical forward exchange rate is given by the formula:

Ft,T = e−rdτ

∫ ∞0

STπQt (ST )dST . (3.3)

On the other hand, evaluating equation (3.2) at K=0 yields:

C(t, 0, T ) = e−rdτEQt (ST −K)+ = e−r

dτ

∫ ∞0

STπQt (ST )dST = Ft,T , (3.4)

12As we mentioned earlier, in the second expression, the pricing kernel is performing both the risk-adjustment and discounting functions, while in the third expression these functions are divided between

πQt and e−rdτ .

10

where (x)+ means max(x, 0). Currency option prices should therefore, at a minimum, contain

as much information about investors’ beliefs and preferences as that contained in forward

prices.

3.1.2 Information contained in the term structure of option prices

If the expectations hypothesis holds in the FX market, then the implied volatility for long

dated options should be consistent with the implied volatility of short dated options quoted

today and in the future.For example, if the current six month implied volatility is 10% and

the current three month implied volatility is 5%, then, under the expectation hypothesis,

then the three month implied volatility three months from now should be 13.2% because

0.5(0.1)2 = 0.25(0.05)2 + 0.25(0.132)2

Starting from the Hull and White (1987) stochastic volatility model, Campa et al. (1998b)

derive the following relationship between long-dated volatility and short-dated volatility:

σ20,km =

1

kE0

[k−1∑

0

σ2im,(i+1)m

](θkmθm

)2

, (3.5)

where m is the time expiration for the short-dated option,k is the number of periods of length

m and σt,T is implied volatility quoted at time t for an option expiring at time T .(θkmθm

)2

< 1

is a constant. The expectation hypothesis therefore suggests that the term structure of

option-implied volatility contain information about the market’s perception about the future

dynamics of short term implied volatility. Using data for four developed country currency

pairs all involving the USD,Campa et al. (1998b) test the expectation hypothesis for FX

implied volatility and generally fail to reject the hypothesis. Mixon (2007) shows that the

term structures of implied volatilities for a number of national indexes have predictive power

for future future short dated implied volatility, but this predictive ability is not as high as

11

that would be obtained under the expectation hypothesis. For the SP 500 index options with

1M and 2M maturity for 1983-1987, (Stein (1989)) find that long-dated options “overreact”

to volatility shocks. while (Poteshman (2001)) uses 1988-1997 data to extend Stein’s result.

3.1.3 Information Contained in Measures of FX Co-movement

Option-implied correlations arise from three way arbitrage arguments. For example: if the

exchange rates at time t are given by SAB,t , SAC,t and SBC,t and assuming they follow

stationary processes, we have that

ln(SAB,t) = ln(SAC,t)− ln(SBC,t) = sAC,t − sBC,t (3.6)

Equation (3.6) above implies that

V ar(sAB) = V ar(sAC) + V ar(sBC)− 2Corr(sAC , sBC)V ar(sAC)12V ar(sBC)

12 , (3.7)

which can be rearranged to give:

Corr(sAC , sBC) =V ar(sAC) + V ar(sBC)− V ar(sAB)

2V ar12 (sAC)V ar(sBC)

12

(3.8)

If we use option-implied variance to estimate the right hand side of equation (3.8), then the

resulting estimate of ρ(sAC , sBC) is option-implied correlation. Siegel (1997) points out that

this option-implied correlation reveals market sentiment regarding how closely the currencies

are expected to move in the future. The average option-implied correlation can be interpreted

as capturing global FX risk. An attractive feature of these option-implied correlations is that

we only need to calculate option-implied variances for the relevant currency pairs, and we

can get the implied correlation as a by-product.

12

3.2 Extracting Option-Implied Risk-Neutral Moments

We use the methodology of Bakshi et al. (2003) (henceforth BKM) to extract model-free

option-implied standard deviation, skewness and kurtosis from the volatility smile. Grad

(2010) and Jurek (2009) also use the BKM methodology to extract FX options-implied

higher order moments. In this section we closely follow Grad’s exposition and notation.

The extracted moments using the BKM methodology are model-free because we make no

assumptions regarding the time series process governing the underlying spot exchange rate.

The model-free nature of the methodology is attractive because it means the methodology

is equally applicable to all exchange rate regimes. Campa et al. (1998a) argue that having a

methodology that does not presuppose a stochastic process followed by the underlying spot

exchange rate is especially useful in situations where the FX regime is unknown or changing,

or when the degree of government intervention is unclear. The BKM methodology rests on

the results of Carr and Madan (2001), which show that if we have an arbitrary claim with

a pay-off function H[S] with finite expectations, then H[S] can be replicated if we have a

continuum of option prices. They also show that if H[S] is twice-differentiable, then it can

be spanned algebraically by the following expression

H[S] = (H[S̄] + (S − S̄HS[S̄]) +

∫ ∞S̄

HSS[K](S −K)+ +

∫ S̄

0

HSS[K](K − S)+dK, (3.9)

where HS = ∂H∂S

and HSS = ∂2H∂S2 . Assuming no arbitrage opportunities, the price of a claim

with pay-off H[S] is given by the expression

pt = (H[S̄]−S̄HS[S̄])e−rdτ +HS[S̄]Se−r

dτ +

∫ ∞S̄

HSS[K]C(t, τ ,K)+

∫ S̄

0

HSS[K]P (t, τ ,K)dK

(3.10)

where K is the exercise price, C(t, τ ,K) and P (t, τ ,K) are, respectively, the prices of a

European-style call and put options. S̄ is some arbitrary constant, usually chosen to equal

current spot price.

13

Equation (3.10) indicates that any pay-off function H[S] can be replicated by a position

of (H[S̄] − S̄HS[S̄]) in the domestic risk-free bond, a position of H[S̄] in the stock, and

combinations of out-of-the-money calls and puts, with weights HSS[K].

H[S] =

[Rt(St+τ )]

2, Volatility Contract

[Rt(St+τ )]3, Cubic Contract

[Rt(St+τ )]4, Quartic Contract

(3.11)

BKM show that the the variance, skewness and kurtosis of the distribution of Rt+τ can be

calculated using the following formulas:

Stdev(t, τ) =√erdτV (t, τ)− µ(t, τ)2

Skew(t, τ) =erdτW (t, τ)− 3V (t, τ)µ(t, τ)er

dτ + 2µ(t, τ)3

[erdτV (t, τ)− µ(t, τ)2]32

Kurt(t, τ) =erdτX(t, τ)− 4er

dτµ(t, τ)W (t, τ) + 6erdτµ(t, τ)2V (t, τ)− 3µ(t, τ)4

[erdτV (t, τ)− µ(t, τ)2]2,

where the expressions for µ(t, τ), V (t, τ), V (t, τ)andV (t, τ) are given in the online appendix.

The BKM methodology described above requires a continuum of exercise prices. However,

in the OTC FX options market, implied volatility are observed for only a discrete number

of exercise prices. We therefore need a way to estimate the entire volatility smile from a

few (K − σ) pairs by interpolation and extrapolation. To this end, we use the VV method

described in Castagna and Mercurio (2007). The procedure allows us to build the entire

volatility smile using only three points. Castagna and Mercurio (2007) note that if we have

three options with implied volatility σ1,σ2, σ3 and corresponding exercise prices K1,K2 and

K3 such that K1 < K2 < K3, then the implied volatility of an option with arbitrary exercise

14

price K can be accurately approximated by the following expression:

σ(K) = σ2 +−σ2 +

√σ2

2 + d1(K)d2(K)(2σ2D1(K) +D2(K))

d1(K)d2(K), (3.12)

where

D1(K) =ln[K2

K

]ln[K3

K

]ln[K2

K1

]ln[K3

K1

]σ1 +ln[KK1

]ln[K3

K

]ln[K2

K1

]ln[K3

K2

]σ2 +ln[KK1

]ln[KK2

]ln[K3

K1

]ln[K3

K2

]σ3 − σ2,

D2(K) =ln[K2

K

]ln[K3

K

]ln[K2

K1

]ln[K3

K1

]d1(K1)d2(K1)(σ1 − σ2)2 +ln[ K

K1]ln[ K

K2]

ln[K3

K1]ln[K3

K2]d1(K3)d2(K3)(σ3 − σ2)2

and

d1(x) =log[St

x] + (rd − rf + 1

2σ2

2)τ

σ2

√τ

, d2(x) = d1(x)− σ2

√τ , x ∈ K,K1, K2, K3.

Expression (3.12) allows us to find the implied volatility of an option with an arbitrary

strike price. We use K1 = K25δp, K2 = KATM and K3 = K25δc. The VV methodology has a

number of attractive features, which are explained in Castagna and Mercurio (2007). First,

it is parsimonious because it uses only three option combinations to build an entire volatility

smile. This is the minimum number that can be used if one wants to capture the three most

prominent movements in the volatility smile: change in level, change in slope, and change in

curvature.13 The VV method also has a solid financial motivation: Castagna and Mercurio

(2007) show that it is based on a replication argument in which an investor constructs a

portfolio that, in addition to hedging against movements in the price of the underlying asset,

a being delta-neutral, is also Vega-neutral. In situations where volatility is stochastic, it

might be useful to construct portfolios that, in addition to hedging against changes in the

13The three particular option combinations that we use, the ATM straddle,VWB and the Risk Reversalcapture these movements. See discussion in Castagna (2010) and Malz (1998)

15

price of the underlying asset, the investor also hedge against for the Vega(∂C∂σ

), the Vanna

(∂2C∂σ2 ) and the Volga( ∂2C

∂σ∂S) as might be necessary in situations when volatility is stochastic.

3.3 Data Description

In the o-t-c market,the exchange rate is quoted as the “domestic” price of “foreign” currency,

which means a fall in the reported exchange rate represents an appreciation of domestic

currency.“Domestic” and “foreign” , however, do not have any geographic significance, but

are in accordance to market quoting conventions. Table (1A) contains details of the market

quoting conventions for the six currency pairs that we use in this paper.

Compared to exchange-traded options, there are several advantages that come with using

o-t-c data in our empirical analysis. First, most of the FX options trading is concentrated

in the o-t-c market. This means o-t-c currency options prices are more competitive and

therefore more likely to be representative of aggregate market beliefs compared to prices in

the less liquid exchange market. Table (1C) below, obtained from the 2010 BIS Triennial

Survey, show that although the o-t-c options market is small relative to the overall FX

market, it is quite liquid and rapidly growing when we look at it in absolute terms.

[INSERT TABLE (1C) HERE]

A second advantage of using o-t-c option price data is that fresh options for standard

tenors are quoted each day, making it possible to obtain a time series of FX option prices

with constant maturities. This can be contrasted with exchange traded options, whose prices

are quoted for a spefic expiry date, such that as we approach the expiry date, the option

prices also incorporate the fact that the tenor is changing. O-t-c options makes it possible to

disentangle term structure,cross-sectional and time series aspects embedded in option prices.

Our third and final motivation for o-t-c option data is that European-style options are

traded in this market, whereas exchange traded FX options are usually American-style.

16

When analyzing option prices of a given tenor, American-style options have to be adjusted

for the possibility of early exercise.

We next explain some important OTC currency market quoting conventions. First, op-

tion prices are given in terms of implied volatility instead of currency units while “moneyness”

is measured in terms of the delta of an option. The delta of an option is a measure of the

responsiveness of the option’s price with respect to a change in the price of the underlying

asset. If the prices of call and put options are given by Ct and Pt, then option price and

implied volatility are linked using the formula from Garman and Kohlhagen (1983) extension

of Black-Scholes model to FX.

Ct = e−rdτ[F t+τt Φ(d1)−KΦ(d2)

]Pt = e−r

dτ[KΦ(−d2)− F t+τ

t Φ(−d1)]

where

d1 =log[St

K] + (rd − rf + 1

2σ2

2)τ

σ2

√τ

, d2 = d1 − σ2

√τ ,

There is a one-to-one relationship between option price and implied volatility when using

the (Black and Scholes (1973)) formula. 14 The expressions for call and put deltas are given

delta are given by the expressions:

δc = e−rf

Φ(d1)δp = e−rf

Φ(−d1), (3.13)

where Φ(.) is the standard normal cumulative density function(cdf). The absolute values of

δc and δp are therefore between 0 and 1, while put-call parity implies that δp = δc − 1. The

market convention is to quote a delta of magnitude x as a 100 ∗ x delta. For example, a put

option with a delta of -0.25 is referred to as a 25δ put.

14 Use of the Black-Scholes formula does not, however, mean traders agree with the assumptions underlyingthe Black-Scholes model.

17

Lastly, in the FX OTC option market, prices are quoted in combinations rather than sim-

ple call and put options. The most common option combinations are at-the-money (ATM)15

straddle, risk reversals (RR), and Vega-weighted butterflies (VWB). An ATM straddle is the

sum of a base currency call and a base currency put, both struck at the current forward rate.

This is the most liquid structure in the OTC FX options market. A RR is set up when one

buys a base currency call and sells a base currency put with a symmetric delta. The most

liquid RR is the 25δ, in which both the call and put have a delta of 25 percent. Finally, the

VWB,is built by buying a symmetric delta strangle and selling an ATM straddle. The 25δ

combination is the most traded options VWB. 16

The definitions of the three option combinations are as follows1718 :

σATM,τ = σ0δc,τ = σ50δc + σ50δp (3.14)

σ25δRR,τ = σ25δc,τ − σ25δp,τ

σ25δvwb,τ =σ25δc,τ + σ25δp,τ

2− σATM,τ

Equations (3.14) can be rearranged to get the implied volatility for 0δ call , 25δ call and 25δ

put. Expressions for backing out implied volatility of these“plain-vanilla” options from the

15“ATM here means the delta of the option combination is zero. That is, the option combination is“delta-neutral”

16A positive value for the VWB means out-of-the-money options are on average relatively more expensivethan at-the-money options. The high implied volatility of these out-of-the-money options suggests that themarket perceives the underlying distribution of the spot rate to have fat tails. The price of a VWB cantherefore be considered a short indicator of the kurtosis of the distribution of the future spot rate.

17The sign and size of the RR volatility is related to the market’s view on the skewness of the distributionof the future spot rate. To see this, note that the underlying asset is the base (foreign) currency. The payoffof a call option increases when the value of the underlying asset increases, while the value of a put optionincreases when the underlying asset’s price goes down. Therefore, a negative RR volatility indicates that themarket thinks the base currency is more likely to depreciate, or, equivalently, the domestic currency is morelikely to appreciate. We therefore expect negative RR volatility to be associated with a negatively skewedextracted density of the future spot rate.

18Table (1B) contains sample option price quotes for standard combinations and standard maturities.

18

prices of traded option combinations are given below:

σ0δc,τ = σATM = σ50δc,τ + σ50δp,τ (3.15)

σ25δc,τ = σATM + σ25δvwb,τ +1

2σ25δRR,τ

σ25δp,τ = σATM + σ25δvwb,τ −1

2σ25δRR,τ .

Finally, K25δp,KATM ,K25δc ,the exercise prices corresponding to σATM,τ , σ25δc,τ and σ25δp,τ

can be backed out by using the expression for option deltas given in equation (3.13 ). For

example, to get KATM we use the fact that the ATM straddle has a delta of zero:

e−rf τΦ

(ln[ St

KATM] + (rd − rf + 1

2σ2ATM)τ

σATM√τ

)−e−rf τΦ

(−ln[ St

KATM] + (rd − rf + 1

2σ2ATM)τ

σATM√τ

)= 0.

(3.16)

Since Φ(.) is a monotone function, we can solve equation (3.16) above for KATM to get:

KATM = Ste(rd−rf+ 1

2σ2ATM )τ = F t+τ

t e12σ2ATM . (3.17)

Similar arguments to the above 19, one can show that the expressions for K25δc and K25δp

K25δc = Ste[−Φ−1( 1

4erdτ )σ25δc,τ

√τ+(rd−rf+ 1

2σ225δc)τ ] (3.18)

K25δp = Ste[Φ−1( 1

4erdτ )σ25δp,τ

√τ+(rd−rf+ 1

2σ225δp)τ ]. (3.19)

Castagna and Mercurio (2007) point out that for tenors up to two years, the following

relationship generally holds: K25δp < KATM < K25δc.

Our options data consists of over the counter (o-t-c) option prices for the six currency

pairs listed in table (1A) and covering the period 1 January 2007 to April 19 2011. Table(1C)

contains the average daily turnovers of each currency pair that we use as a percentage of

19see appendix

19

average daily turnover in the FX market.

[INSERT TABLE (1B) HERE].

The spot rates, forward rates and risk-free interest rates are obtained from Datastream.

4 Empirical Strategy and Main Results

4.1 Empirical properties of extracted option-implied moments

First, The extracted moments are generally very persistent,as shown in the time series plots

for 1M tenor in Figures (1)- (6) and the high AR(1) coefficients in the summary statistics

table (2)20 21. Zivot and Andrews (1992) unit root tests , however, suggest that the implied

moments are stationary, with structural breaks in the mean on dates around late 2008 and

early 2009.

INSERT TABLE (2) HERE

INSERT FIGURES (1)- (6)HERE

Second,the time series plots in figures (1)- (6) and the online appendix also show that

option-implied risk-neutral moments of ln(St+τSt

)are quite dynamic and capture changes in

market perceptions. For example, all three moments show sharp movements in late 2008 to

early 2009, days which coincide with the height of the global financial crisis. The standard

deviation and kurtosis increasing for majority of currency pairs and tenors, indicating in-

creases in both the market perceived general level of uncertainty and the probability of large

movements. The summary statistics also suggest the presence of outliers in the extracted

option-implied moments,especially for 9M and 12M tenor.

20 Summary statistics for the option-implied moments from the other five currency pairs are in the onlineappendix, provide similar information.

21We only have time series plots for 1M implied moments. The rest of the time series plots for τ =1WK, 2M, 3M, 6M, 9M and 12M are in the online appendix. The general message in these additional seriesplots is the same as the 1M we present.

20

Third, table (3) show that the correlations between option-implied moments and their

respective short-cut indicators can be very low for skewness and kurtosis indicators. If that

is is the case, option-implied moments are preferable to the short cut indicators since they

incorporate information in the entire distribution of ln(St+τSt

). Using the short cut indicators

as proxies for volatility, skewness and kurtosis would not be ideal if the two risk measures

are not highly correlated.

INSERT TABLE (3) HERE

Lastly, the moments appear to be highly correlated with standard measures of macroeco-

nomic and financial risk. Cameron (2013) documents a consistent ability of risk indicators

such as VIX, Nelson and Siegel (1987) relative yield curve factors and TED spread to explain

variation in the option-implied moments over the currency pairs and tenors considered in

the current paper.

INSERT SOME WITH OTHER RISK MEASURES HERE.

4.2 Can option-implied moments forecast FX excess returns?

4.2.1 Matched Frequency Analysis

We begin by investigating the predictive information content of the volatility smile. For each

currency pair i and tenor τ , we estimate the following baseline regression :

f i,t+τt −Et(sit+τ ) = γ0,τ +γ1,τSTDEVi,t+τt +γ2,τSKEW

i,t+τt +γ3,τKURT

i,t+τt +ui,t+τ . (4.1)

We note that Et(st+τ ) is not observable.If we assume that market participants have

rational expectations, then Et(st+τ ) and st+τ will only differ by a forecast error νt+1 that is

21

uncorrelated with all variables that use information at time t 22:

st+τ = Et(st+τ ) + νt+1 (4.2)

Plugging equation (4.2) into equation (4.1) and rearranging gives us the following estimable

regression equation:

xrt+τ = γ0,τ + γ1,τSTDEVt+τt + γ2,τSKEW

t+τt + γ3,τKURT

t+τt + εt+τ (4.3)

where the error term εt+τ = ut+τ + νt+τ .

In equation (4.3), the LHS variable is τ -period currency excess returns while the RHS vari-

ables are options-based proxies for volatility,crash and tail risk for same τ . The specification

in equation (4.3)can be loosely motivated from ICAPM and higher moment CAPM intu-

itions. The link to ICAPM comes the fact that regression (4.3) expresses an inter-temporal

trade-off between FX risks and return. However, while ICAPM assumes that the returns

are normally distributed which reduces the analysis to mean-variance anlysis, we allow ex-

cess returns to be compensation for higher moments risks. Another difference between our

empirical model and the ICAPM set up is that the risk measures considered in ICAPM are

market risks while we look at the inter-temporal trade-off between bilateral excess returns

and measures of bilateral FX risk.

The link to higher moment CAPM follows from the idea that,similar to the motivations

in higher moment CAPM models. we investigate whether observed currency excess returns

can be understood as compensation for exposure to higher moments risks. The difference

between the higher order CAPM and our analysis in regression model (4.3) is that our

measures of risk are bilateral and we set up a predictive framework, while the higher order

CAPM involves cross sectional analysis and invesgta eexposure to market risk.

22in particular, stdevt+τt , skewt+τt and kurtt+τt .

22

Gereben (2002) , Malz (1997) and Lyons (1988) motivate regression equation (4.3) in

light of the time-varying risk premia explanation of the UIP puzzle. In particular,Gereben

(2002) argues that if the forward bias is due to time-varying risk premia, then its size should

be explain by variable that capture the nature of that risk.

We interpret the coefficient signs from the point of view of a domestic investor who invests

in domestic bonds using money borrowed from abroad. As shown in equation (2.3), such

an investor benefits from higher domestic interest rates and appreciation of the domestic

currency .

To enable discussing expected signs for the coefficients on STDEV and KURT, we assume

that the home currency is riskier, such that our investor would demand higher excess returns

for higher stdev and kurtosis in the exchange rate. If inverstors are averse to high variance

and kurtosis, they would require higher excess returns for holding bonds denominated in

units of the riskier domestic and we would expect the coefficients on stdev and kurtosis to

be both positive. We also expect the SKEW coefficient to be positive for investor’s with

preference for positive skewness. Such an investor will require higher compensation for an

increase in SKEW, which represents an higher likelihood of domestic currency depreciation.

Based on the discussion in subsection (2.2.4), however, we note that pinning down the signs

of the coefficients apriori is impossible without making further assumptions on the utility

function or when the moments are not orthogonal.

We first estimate regression equation (4.3) using OLS. Sub-sample analyses suggest the

presence of structural breaks in the matched-frequency regression relationship. We therefore

use the Bai and Perron (2003) and Bai and Perron (1998) structural break test to identify

the date for the most prominent break for each currency pair i and tenor τ . 23. Our final

matched frequency regression model modifies equation (4.3) to include the interactions with

23We only focus on the major breaks, and therefore do not choose the number of breaks according toinformation criteria such as AIC.

23

structural break indicator variables:

xrit+τ = γ0,τ +D1i,τ +D1i,τ ∗ γ1,τSTDEVi,t+τt +D1i,τ ∗ γ2,τSKEW

i,t+τt +D1i,τ∗

γ3,τKURTt+τt + γ4,τSTDEV

i,t+τt + γ5,τSKEW

i,t+τt + γ6,τKURT

i,t+τt + εi,t+τ ,

(4.4)

Regression results for all currencies and tenors are shown in tables (4) to (9).

[INSERT TABLES (4) to (9) HERE ]

The adjusted R2s are consistently high and coefficients on non-intercept terms are con-

sistently jointly significant across tenors and currency pairs, indicating a consistent ability

of option-implied higher order moments to explain subsequent currency excess returns or

deviations from uncovered interest parity.

The pre-break coefficients on STDEV and KURTOSIS are generally positive across tenors

and currency pairs, consistent with investor aversion for higher variance and higher kurtosis.

The skewness coefficients are also generally positive, consistent with preference for positive

skewness. We also see several cases of coefficients switching signs for pre-break to post

break and across tenors. We interpret pre/post break sign switches as representing time-

varying investor perceptions perceptions regarding which side of a currency pair is riskier. We

interpret the sign switches across tenors as suggesting at any given time, market participants’

expectations depend on the time horizon and thus another motivation for studying the term

structure dynamics of option implied volatilities more closely.

We next go beyond OLS regression, which models the conditional mean of the the depen-

dent variable given the covariates, to investigating the predictive ability of options-based risk

measures for the entire distribution of f t+τt − st+τ using quantile regression analysis(QR).

By modeling the entire distribution of the dependent variable, QR allows us to get a more

complete picture of the predictive ability of the option-implied moments. QR also have the

further advantages over OLS that it is robust to outliers in the dependent variable and does

not impose restrictive distributional assumptions on the error terms.

24

We estimate the estimate the following linear quantile regression model24:

Qxri (θ|.) = γ0,τ +D1i,τ +D1i,τ ∗ γ1,τSTDEV

i,t+τt +D1i,τ ∗ γ2,τSKEW

i,t+τt +D1i,τ ∗ γ3,τ

KURT t+τt + γ4,τSTDEVi,t+τt + γ5,τSKEW

i,t+τt + γ6,τKURT

i,t+τt + εi,t+τ ,

(4.5)

where Qxri (θ|.) is the θth quantile of excess returns given information available at time t.25

The 3M quantile regression results in tables (11)-(16) 26 show a consistent ability of option-

implied moments to explain the entire distribution of subsequent excess returns as shown by

Wald tests for joint significance across quantiles an across currency pairs as well adjusted

Rs as high as 50%.

A consistent pattern across currency pairs and tenors is that option-implied moments

have more predictive ability for lower and upper quantiles of excess returns than than the

middle quantiles.

• Interpretation: Better ability of option-implied moments to predict crashes and ex-

treme events?

There are also several situations where the coefficient signs swiches across quantiles.

• Interpretation

INSERT TABLES (11) -(16) HERE

24The quantile regression parameter estimates are obtained the following linear programming problem.25We estimate the quantile regression model using the same break dates obtained in the OLS analysis261M results are in the online appendix, and show similar strong results

25

4.2.2 Term structure and global risk regressions

We next study the predictive information content of the term structure of implied volatility.

For 3M tenor, we first regress excess returns on 1M, 3M and 12M option-implied moments:

f i,t+3Mt − si,t+3M = γ0,3M + γ1,1MSTDEV

i,t+1Mt + γ2,1MSKEW

i,t+1Mt + γ3,1MKURT

t+1Mt

+γ1,3MSTDEVi,t+3Mt + γ2,3MSKEW

i,t+3Mt + γ3,3MKURT

i,t+3Mt + γ1,12MSTDEV

i,t+12Mt

+γ2,12MSKEWi,t+12Mt + γ3,12MKURT

i,t+12Mt + εi,t+3M ,

(4.6)

We also estimate a variation of regression equation (4.6) with the moments replaced by the

principal components. OLS term structure results are shown in tables (17) and (18), while

the results with principal components are in the appendix.

INSERT TABLES (17) AND (18) HERE.

Finally, we investigate the predictive ability of option-implied co-movements among cur-

rency pairs by estimating a variant of equation (4.3) with principal components of implied

moments of all the six currency pairs. These regressions are aimed at evaluating the predic-

tive information content of the information in co-movements of currency pairs, as described

in subsection (3.1.3).

Tables (19) and (20)show the global risk regressions.

INSERT TABLES (19) and (20) HERE

4.2.3 Can implied moments forecast components of currency excess returns?

In this subsection we investigate whether information extracted from the volatility smile can

explain the two components of excess returns. As shown in equation (2.3), currency excess

returns can be decomposed into two components: the interest rate differential component,it−

i∗t -which, under CIP, is equal to the forward premium f t+τt −st - and FX return component,

26

st+1 − st. Della Corte et al. (2013) argue that although the literature has made significant

progress in understanding the determinants and predictability of currency excess returns,

little progress has been made in understanding FX return predictability since Meese and

Rogoff (1983).

Clarida and Taylor (1997) and Clarida et al. (2003) find that the term structure of forward

premia has predictive power for the FX returns. Della Corte et al. (2013) show that volatil-

ity risk premia-defined as the difference between realized and option-implied volatilities-have

predictive power for FX returns. We add to this line of research by investigating the predic-

tive ability of information extracted from the volatility smile for FX returns. We estimate

a variant of equation (4.1) in which the RHS variable is now ∆st+τ . Similar to the analysis

with excess returns on the RHS, we use the Bai-Perron test to identify major breaks in the

linear regression relationship between option-implied moments and FX returns. We estimate

the following regression model for 1M, 3M and 12M tenors:

st+τ − st+τ = γ0,τ +D1 +D1 ∗ γ1,τSTDEVt+τt +D1 ∗ γ2,τSKEW

t+τt +D1 ∗ γ3,τKURT

t+τt +

γ4,τSTDEVt+τt + γ5,τSKEW

t+τt + γ6,τKURT

t+τt + εt+τ

(4.7)

Lastly, we study the explanatory power of option-implied moments for the forward premium.

We estimate regressions of the form

f t+τt − st = γ0,τ +D1 +D1 ∗ γ1,τSTDEVt+τt +D1 ∗ γ2,τSKEW

t+τt +D1 ∗ γ3,τKURT

t+τt +

γ4,τSTDEVt+τt + γ5,τSKEW

t+τt + γ6,τKURT

t+τt + εt+τ

(4.8)

Condensed regression results are found in table (21), while detailed results can be found in

the online appendix.

27

INSERT TABLE (21) HERE

The results in table (21) show that option-implied moments have strong predictive power

for FX returns across currency pairs and across tenors, as evidenced by high adjusted R2s

and Wald tests for joint significance. For the majority of currency pairs and tenors, we also

have that option-implied moments have explanatory power for the forward premium, again

as evidenced by adjusted R2s and tests for joint significance of the moments.

5 Further Discussion and Robustness Checks

5.1 Robustness Checks

Non-overlapping data As a robustness check , we rerun the matched-frequency regressions

for 1M tenor using non-overlapping observations, with results shown in table (10) 2728

[INSERT TABLE (10) HERE ]

5.1.1 Sub-sample Analysis

Table (22) shows condensed29 matched frequency regression results when we run separate

regressions for each sub-sample, with the break being the break dates being those shown in

tables (4) - (9).

[INSERT TABLE (22) HERE]

5.1.2 Using 10δ Options to Extract Implied Moments

For 1M and 3M tenor, we use the BKM methodology and the Vanna-Volga method, but use

10δ RRs and 10δ VWBs in addition to the ATM straddle to construct a continuous volatility

27 We pick the second trading of each month. We also try the 15th day of the month, there isn’t muchdifference in.

28Because of the overlapping nature of our data, we refrain from interpreting the generally higher R2 forlong horizon regressions as suggesting higher predictive ability of option-implied moments at longer horizons

29Detailed results can be found on the online appendix

28

smile. Using 10δ options combinations involve a trade-off since 25δ combinations are more

actively traded than 10δ options, while using 10δ options to construct a continuous volatility

smile means the amount of extrapolation done on the tails is less than what would be the

case if we use 25δ options.

Using arguments similar to those in subsection (3.2), one can show that the expressions

for strikes corresponding to 10δ call and 25δ put are gives by the following expressions:

K10δc = Ste[−Φ−1( 1

10erdτ )σ10δc,τ

√τ+(rd−rf+ 1

2σ210δc)τ ] (5.1)

K10δp = Ste[Φ−1( 1

10erdτ )σ10δp,τ

√τ+(rd−rf+ 1

2σ210δp)τ ]. (5.2)

Fitting a continuous volatility smile and extracting options-implied proceeds exactly as de-

scribed in subsection (3.2) . We carry out analyses for 1M and 3M data. The break dates

using 10δ options are similar to those obtained using 25δ options. Condensed regression

results are shown in table (23), while detailed results can be found in the online appendix.

[INSERT TABLE (23) HERE]

5.1.3 Regression Analysis using trimmed option-implied moments

We re-do the matched-frequency analysis using trimmed option-implied moments.

[INSERT TABLES ( ??)-( ??) HERE]

5.2 Higher order moments through asset pricing motivations

In this subsection we present an asset pricing framework as an alternative to the asset alloca-

tion motivation for the including higher order moments is risk-return relationships considered

in subsection (2.2). We consider the higher moment CAPM of Harvey and Siddique (2000).

We start from the standard asset pricing equation, Under the assumption of no-arbitrage,the

29

standard asset pricing equation for each asset i is given by

P it = Et[Mt+1P

it+1], i = 1, 2, ..., n (5.3)

where Mt+1 is the pricing kernel . 30Equation (5.3) can equivalently be written as

Et[Mt+1Rit+1] = Covt[R

it+1,Mt+1]+Et[Ri

t+1]Et[Mt+1] = 1 =⇒ Et[Rit+1)] =

[1

Et[Mt+1]−Covt[R

it+1,Mt+1]

Et[Mt+1]

].

(5.4)

where Rit+1 =

[Pt+1

Pt

]. Equation (5.4) can equivalently be written as

Et[(1 +Rit+1)] =

[1

Et[Mt+1]−Covt[(1 +Ri

t+1),Mt+1]

Et[Mt+1]

], (5.5)

We can use the pricing equation (5.3) to price a risk free asset with gross return Rf , to get

Et[RfMt+1] = 1 =⇒ Et[Mt+1] =1

Rf. (5.6)

Plugging in (5.6 ) into 5.4: yields the standard CAPM result that the returns to asset i is a

linear function of Rwt+1 , and that assets that are highly correlated with the portfolio return

require higher return :

Et[Rit+1] = Rf

[1− Covt[Ri

t+1,Mt+1]]

(5.7)

The four moment CAPM assumes that Mt+1 is a cubic function of Rwt+1 , the return to the

wealth portfolio and the existence of a risk-free asset with return Rf :

Mt+1 = at + btRwt+1 + ct(R

wt+1)2 + dt(R

wt+1)3, (5.8)

30In consumption/macro -based asset pricing models,the pricing kernel is tightly linked to the representa-tive investors preferences/utility function. The pricing kernel therefore provides a link between macro andfinance in macro-finance models (Alvarez and Jermann (2005))

30

which reduces to 5.7 to

Et[Rit+τ −R

ft = β1,tCovt[R

it+1, R

Mt+1]+β2,tCovt[R

it+1, (R

Mt+1)2]+β2,tCovt[R

it+1, (R

Mt+1)3] (5.9)

6 Conclusion

This paper has documented a consistent ability of option-implied measures of FX volatility,

crash and tail risk risk to explain both the conditional mean and the entire conditional

distribution of subsequent currency excess returns or ex-post deviation from UIP. We show

that the set of useful information is quite broad, with the volatility smile, the term structure

of implied volatility and information from a broad set of currency pairs all have predictive

ability.

We have also documented some empirical properties of these option-implied risk mea-

sures, showing that they are responsive to The options-based measures of risk are corre-

lated with other measures of macroeconomic and financial risk, such as yield curves, TED

spread,....

6.1 Addressing the “so what” question

1. ) The UIP condition is widely used in standard open-economy macro-economics models.

An implication of the analysis i this paper is that the calibration of UIP shocks may

need to be more sophisticated and possible include a role for higher order moments.

The UIP condition hinges too much on the restrictive assumptions that investors have

exponential utility and returns are normal, which reduces the investor’s asset allocation

problem to mean-variance.

2. ) Related to the above point, OLS regression-based tests of equation (2.2) do not make

too much sense since they are based on the assumption of normality. The regressions

31

could be made more sensible by adding higher order moments of exchange rate returns

(or their proxies) and their term structures as additional controls so as to improve

prediction.

3. ) As highlighted in section (2.2) unless we make the joint restrictive assumptions of

normality of returns and CARA utility, higher order moments should feature in the

solution to the asset allocation problem. Option prices give us estimates of higher

order moments by allowing us to extract the entire (risk-neutral) distribution of future

exchange rate returns, for various tenors. While short-hand proxies of these moments

can be used, they are not always great substitutes given that, as shown in table (3),

the correlations between extracted moments and the short-cut indicators of FX risk

can be very low.

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251–271.

37

Tab

le1:

O-T

-Cm

arke

tquot

ing

conve

nti

ons

and

stat

isti

cs

(A)

Quoti

ng

Conventi

ons

inover-

the-c

ounte

rF

XO

pti

ons

Mark

et

Sym

bol

Definit

ion

Base

curr

ency

Dom

est

iccu

rrency

Posi

tive

Skew

ness

means

AU

DU

SD

USD

per

AU

DA

UD

USD

USD

dep

reci

atio

nE

UR

JP

YJP

Yp

erE

UR

EU

RJP

YE

UR

dep

reci

atio

nE

UR

US

DU

SD

per

EU

RE

UR

USD

USD

dep

reci

atio

nG

BP

US

DU

SD

per

GB

PG

BP

USD

USD

dep

reci

atio

nU

SD

CA

DC

AD

per

USD

USD

CA

DC

AD

dep

reci

atio

nU

SD

JP

YJP

Yp

erU

SD

USD

JP

YJP

Ydep

reci

atio

n

(B)

Sam

ple

Annuali

zed

Impli

ed

Vola

tili

ties;

Jan

24,

2007,

AU

SU

SD

Ten

or

AT

M25D

RR

25D

VW

B10D

RR

10D

VW

B1

Week

7.35

2-0

.495

0.13

1-0

.847

0.37

91

Month

6.85

1-0

.347

0.13

6-0

.584

0.38

92

Month

6.85

1-0

.366

0.15

7-0

.619

0.44

93

Month

6.85

1-0

.396

0.16

2-0

.663

0.48

56

Month

6.90

1-0

.426

0.18

7-0

.703

0.54

9M

onth

7.05

1-0

.446

0.19

7-0

.743

0.57

112

Month

6.90

1-0

.426

0.18

7-0

.703

0.54

(C)

Avera

ge

Dail

yT

urn

over

inF

Xm

ark

et

(bil

lions)

1998

2001

2004

2007

2010

Sp

otF

XT

ransa

ctio

ns

568

386

631

1005

1490

Per

centa

geC

han

geN

/A-3

263

.559

.348

.3F

XD

eri

vati

ves

128

130

209

362

475

Outr

ight

For

war

ds

734

656

954

1714

1765

FX

Sw

aps

107

2131

43O

pti

on

san

doth

er

pro

duct

s87

60

119

212

207

Per

centa

geC

han

geN

/A-3

198

.383

-2.4

Exch

ange

Tra

ded

Der

ivat

ives

1112

2680

166

Note

:“A

TM

”is

at-

the-

mon

eyst

rad

dle

,25

DR

Ran

d10

DR

Rar

e25

%-

and

10%

-d

elta

risk

reve

rsal

sre

spec

tive

ly;

and

25D

VW

Ban

d10

DV

WB

are

25%

-an

d10

%-

del

taV

ega-

wei

ghte

db

utt

erfl

ies

resp

ecti

vely

.S

eeS

ecti

on(3

.3)

for

mor

ed

etai

ls.

38

Table 2: SUMMARY STATISTICS OF OPTION-IMPLED MOMENTS:AUDUSD

STDEV1WK 1M 2M 3M 6M 9M 12M

Mean 0.022 0.044 0.044 0.074 0.099 0.112 0.122Median 0.020 0.041 0.042 0.072 0.098 0.112 0.121

Maximum 0.080 0.114 0.099 0.144 0.170 0.211 0.224Minimum 0.010 0.019 0.019 0.031 0.039 0.025 0.010Std. Dev. 0.010 0.018 0.016 0.025 0.034 0.044 0.055

AR(1) 0.970 0.987 0.990 0.999 0.993 0.988 0.986

SKEWMean -0.371 -0.636 -0.704 -0.979 -1.394 -1.880 -2.081

Median -0.352 -0.631 -0.693 -0.936 -1.196 -1.513 -1.695Maximum 0.576 0.043 -0.087 -0.159 -0.540 6.386 371.902Minimum -1.461 -2.710 -2.777 -2.568 -3.360 -5.371 -13.301Std. Dev. 0.214 0.268 0.259 0.367 0.610 1.144 13.235

AR(1) 0.818 0.931 0.927 0.963 0.982 0.939 0.038

KURTMean 3.602 4.424 4.553 5.634 8.088 16.254 164.917

Median 3.528 4.197 4.312 5.045 7.161 8.791 10.176Maximum 12.881 21.148 19.917 17.426 34.967 1186.148 72716.840Minimum 2.384 3.481 3.666 3.964 3.918 4.117 4.622Std. Dev. 0.661 1.191 1.183 1.629 3.610 45.786 2627.969

AR(1) 0.628 0.788 0.781 0.912 0.958 0.419 0.032

XRMean -0.002 -0.008 -0.016 -0.023 -0.044 -0.059 -0.052

Median -0.005 -0.016 -0.028 -0.042 -0.078 -0.103 -0.104Maximum 0.177 0.317 0.348 0.441 0.418 0.383 0.400Minimum -0.113 -0.130 -0.195 -0.266 -0.324 -0.397 -0.420Std. Dev. 0.025 0.052 0.077 0.099 0.155 0.182 0.194

AR(1) 0.754 0.942 0.977 0.985 0.933 0.995 0.995

Observations 1104 1098 1080 1058 992 924 855

Note:“St Dev”,“Skew”, and “Kurt” are the implied standard deviation, skewness, and

kurtosis of the risk-neutral distribution of ln(St+τSt

).

39

Table 3: Correlations between implied moments and corresponding short-cut indicators

Correlation (St Dev, ATM Straddle)

1W 1M 2M 3M 6M 9M 12MAUDUSD 0.99 0.99 0.98 0.97 0.95 0.94 0.94EURJPY 0.99 0.99 0.98 0.97 0.95 0.95 0.93EURUSD 0.99 0.99 0.98 0.98 0.96 0.96 0.94GBPUSD 0.99 0.99 0.98 0.97 0.95 0.95 0.94USDCAD 0.99 0.99 0.98 0.98 0.98 0.98 0.97USDJPY 0.99 0.99 0.99 0.98 0.97 0.88 0.87

Correlation (Skew, RR)1W 1 M 2M 3M 6M 9M 12M

AUDUSD 0.73 0.59 0.39 0.19 -0.2 -0.37 -0.06EURJPY 0.45 0.34 0.3 0.27 0.25 0.28 0.28EURUSD 0.91 0.52 0.37 0.29 0.11 0.01 -0.29GBPUSD 0.61 0.23 0.01 -0.1 -0.41 -0.26 -0.07USDCAD 0.88 0.84 0.78 0.78 0.72 0.64 0.58USDJPY 0.7 0.61 0.54 0.52 0.11 0.11 0.12

Correlation (Kurt, VWB)1W 1 M 2M 3M 6M 9M 12M

AUDUSD 0.4 0.29 0.18 0.05 -0.32 -0.13 -0.03EURJPY 0.27 -0.02 -0.05 0.01 -0.05 -0.12 -0.05EURUSD 0.28 0.24 0.16 0.11 -0.13 -0.39 -0.61GBPUSD 0.39 0.41 0.27 0.05 -0.38 -0.06 -0.05USDCAD 0.56 0.47 0.34 0.11 -0.41 -0.62 -0.66USDJPY 0.33 0.07 0.13 0.08 -0.03 0.01 -0.01

Note:“St Dev”,“Skew”, and “Kurt” are the implied standard de-viation, skewness, and kurtosis of the risk-neutral distribution of

ln(St+τSt

). See Section (3.3) for definitions of ATM straddle, Risk

Reversal, and Vega-Weighted Butterfly.

40

Tab

le4:

AU

DU

SD

mat

ched

freq

uen

cyO

LS

Reg

ress

ions

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C0.

052

0.06

10.

080.

122

0.81

40.

099

-0.0

66[0

.010

0]**

*[0

.021

3]**

*[0

.026

5]**

*[0

.048

7]**

[0.2

298]

***

[0.1

194]

[0.0

776]

D1

-0.0

450.

015

0.10

30.

186

-0.4

380.

464

0.36

5[0

.013

3]**

*[0

.036

0][0

.048

9]**

[0.0

696]

***

[0.2

396]

*[0

.135

1]**

*[0

.081

8]**

*

ST

DE

V0.

406

0.72

0.74

6-0

.202

-5.5

871.

509

4.03

4[0

.191

3]**

[0.2

235]

***

[0.3

128]

**[0

.304

8][1

.579

4]**

*[0

.959

6][0

.637

1]**

*

SK

EW

0.01

80.

074

0.12

80.

180.

329

0.23

70.

108

[0.0

086]

**[0

.028

3]**

*[0

.045

5]**

*[0

.062

4]**

*[0

.034

1]**

*[0

.016

5]**

*[0

.016

1]**

*

KU

RT

-0.0

14-0

.007

00.

019

0.01

60.

025

0.00

8[0

.002

4]**

*[0

.006

8][0

.008

0][0

.014

0][0

.008

1]**

[0.0

023]

***

[0.0

012]

***

D1*

ST

DE

V-1

.235

-2.5

47-4

.706

-3.3

252.

013

-6.0

2-6

.79

[0.3

348]

***

[0.4

595]

***

[0.6

834]

***

[0.5

257]

***

[1.6

640]

[1.0

112]

***

[0.6

594]

***

D1*

SK

EW

-0.0

2-0

.066

-0.0

89-0

.119

-0.4

72-0

.207

-0.1

05[0

.012

7][0

.032

1]**

[0.0

505]

*[0

.067

5]*

[0.0

549]

***

[0.0

263]

***

[0.0

168]

***

D1*

KU

RT

0.01

50.

004

-0.0

02-0

.024

-0.0

46-0

.026

-0.0

08[0

.003

0]**

*[0

.007

7][0

.009

2][0

.014

6][0

.011

1]**

*[0

.002

3]**

*[0

.001

2]**

*

Obse

rvat

ions:

1104

1098

1080

1058

992

924

855

Adj.

R-s

quar

ed:

0.13

80.

254

0.28

60.

340.

644

0.78

60.

829

F-s

tati

stic

:7.

558.

5211

.039

15.6

429

.15

160.

5124

4.6

Pro

b(F

-sta

t):

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

00B

reak

Dat

e2/

24/2

009

2/16

/200

91/

30/2

009

1/28

/200

98/

4/20

085/

15/2

008

5/2/

2008

Not

e:“X

R”

iscu

rren

cyex

cess

retu

rns

asd

efin

edin

equ

atio

n(2

.3).

New

ey-W

est

(NW

)H

AC

Sta

nd

ard

Err

ors

and

Cov

ari

an

cew

ith

auto

mat

icla

gtr

un

cati

on.

D1

=is

anin

dic

ator

vari

able

that

isze

rob

efor

eth

eb

reak

dat

ean

don

eot

her

wis

e.T

he

bre

akdate

for

each

regr

essi

onis

sele

cted

acco

rdin

gto

the

Bai

and

Per

ron

(200

3)m

eth

od

olog

y,al

low

ing

for

am

axim

um

of1

bre

ak.

F-s

tats

rep

ort

Wal

dte

stof

the

nu

llth

atth

esi

xn

on-i

nte

rcep

tco

effici

ents

are

all

zero

.N

WS

tan

dard

erro

rsar

ere

port

edin

bra

cket

s.A

ster

isks

ind

icat

esi

gnifi

can

ceat

1%(*

**),

5%(*

*),

and

10%

(*)

leve

l.

41

Tab

le5:

EU

RJP

YM

atch

edF

requen

cyO

LS

Reg

ress

ions

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C-0

.016

-0.0

19-0

.061

-0.1

2-0

.087

-0.0

64-0

.085

[0.0

098]

[0.0

123]

[0.0

163]

***

[0.0

304]

***

[0.0

207]

***

[0.0

243]

***

[0.0

366]

**

D1

0.03

30.

080.

192

0.31

80.

694

0.84

50.

7[0

.012

6]**

*[0

.020

1]**

*[0

.031

8]**

*[0

.049

6]**

*[0

.045

5]**

*[0

.042

0]**

*[0

.068

3]**

*

ST

DE

V0.

66-0

.839

-0.4

96-2

.301

-0.4

82-0

.972

0.06

4[0

.450

1][0

.317

4]**

*[0

.363

8][0

.426

0]**

*[0

.319

4][0

.337

8]**

*[0

.377

6]

SK

EW

0.00

3-0

.02

-0.0

1-0

.256

-0.0

73-0

.072

-0.0

3[0

.007

2][0

.016

0][0

.026

4][0

.044

0]**

*[0

.021

3]**

*[0

.020

4]**

*[0

.017

9]*

KU

RT

0.00

20.

005

0.01

2-0

.006

0.00

10

0[0

.001

3][0

.002

6]*

[0.0

035]

***

[0.0

041]

[0.0

011]

[0.0

004]

[0.0

002]

D1*

ST

DE

V-0

.894

0.22

5-1

.068

0.32

4-3

.128

-2.5

02-2

.462

[0.5

196]

*[0

.419

6][0

.502

0]**

[0.6

302]

[0.4

327]

***

[0.3

893]

***

[0.4

548]

***

D1*

SK

EW

-0.0

06-0

.082

-0.2

0.05

80.

219

0.24

90.

132

[0.0

121]

[0.0

319]

**[0

.058

1]**

*[0

.081

3][0

.038

2]**

*[0

.027

6]**

*[0

.030

3]**

*

D1*

KU

RT

-0.0

05-0

.025

-0.0

48-0

.03

0.00

70.

002

0[0

.002

0]**

[0.0

046]

***

[0.0

085]

***

[0.0

112]

***

[0.0

042]

[0.0

014]

*[0

.001

0]

Obse

rvat

ions:

1106

1100

1080

1058

992

926

861

Adj.

R-s

quar

ed:

0.03

50.

188

0.32

60.

462

0.74

80.

820.

636

F-s

tati

stic

:1.

611

.87

19.1

2714

.34

47.8

114.

5116

.8P

rob(F

-sta

t):

0.14

300.

0000

0.00

000.

0000

0.00

000.

0000

0.00

00B

reak

Dat

es10

/22/

2008

7/30

/200

88/

7/20

088/

7/20

084/

1/20

081/

3/20

0810

/5/2

007

42

Tab

le6:

EU

RU

SD

Mat

ched

Fre

quen

cyR

egre

ssio

ns

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C-0

.011

0.06

10.

116

0.13

40.

219

-0.1

03-0

.509

[0.0

051]

**[0

.026

8]**

[0.0

311]

***

[0.0

417]

***

[0.0

731]

***

[0.0

782]

[0.1

206]

***

D1

0.01

5-0

.033

0.08

60.

032

0.03

20.

575

0.89

7[0

.006

9]**

[0.0

345]

[0.0

587]

[0.0

844]

[0.0

844]

[0.0

952]

***

[0.1

369]

***

ST

DE

V0.

81-0

.167

-0.8

83-2

.048

-2.0

483.

643

7.96

6[0

.399

2]**

[0.3

606]

[0.2

345]

***

[0.8

967]

**[0

.896

7]**

[0.8

227]

***

[1.3

106]

***

SK

EW

0.01

10.

088

0.12

50.

150.

150.

102

0.01

9[0

.004

6]**

[0.0

197]

***

[0.0

204]

***

[0.0

184]

***

[0.0

184]

***

[0.0

131]

***

[0.0

100]

*

KU

RT

0.00

1-0

.002

0.00

50.

013

0.01

30.

009

0.00

4[0

.000

9][0

.002

5][0

.002

3]**

[0.0

020]

***

[0.0

020]

***

[0.0

008]

***

[0.0

007]

***

D1*

ST

DE

V-0

.473

0.05

6-1

.737

0.59

90.

599

-6.2

8-8

.955

[0.4

410]

[0.6

868]

[0.6

529]

***

[1.0

092]

[1.0

092]

[0.9

561]

***

[1.4

042]

***

D1*

SK

EW

-0.0

21-0

.123

-0.1

52-0

.197

-0.1

97-0

.072

0.04

1[0

.006

4]**

*[0

.022

4]**

*[0

.027

5]**

*[0

.033

7]**

*[0

.033

7]**

*[0

.035

0]**

[0.0

241]

*

D1*

KU

RT

-0.0

04-0

.008

-0.0

16-0

.032

-0.0

32-0

.025

-0.0

2[0

.001

1]**

*[0

.003

4]**

[0.0

046]

***

[0.0

057]

***

[0.0

057]

***

[0.0

062]

***

[0.0

022]

***

Obse

rvat

ions:

1096

1084

1075

1053

988

924

858

Adj.

R-s

quar

ed:

0.05

0.18

70.

294

0.37

20.

527

0.74

30.

737

F-s

tati

stic

:8.

58.

5818

.56

19.6

836

.19

110.

2815

7.97

Pro

b(F

-sta

t):

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

00B

reak

Dat

e10

/21/

2008

2/11

/200

91/

16/2

009

2/4/

2009

8/7/

2008

8/8/

2008

8/7/

2008

43

Tab

le7:

GB

PU

SD

Mat

ched

Fre

quen

cyR

egre

ssio

ns

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C-0

.023

-0.0

580.

064

-0.1

050.

061

-0.2

61-0

.085

[0.0

062]

***

[0.0

221]

***

[0.0

193]

***

[0.0

368]

***

[0.0

869]

[0.0

459]

***

[0.0

310]

***

D1

0.02

90.

048

0.11

3-0

.018

0.00

30.

653

0.59

3[0

.008

5]**

*[0

.023

7]**

[0.0

431]

***

[0.0

418]

[0.0

969]

[0.0

700]

***

[0.0

629]

***

ST

DE

V1.

091

2.72

91.

157

3.74

42.

385

6.91

24.

959

[0.4

012]

***

[0.4

609]

***

[0.2

792]

***

[0.4

595]

***

[0.9

436]

**[0

.776

8]**

*[0

.435

3]**

*

SK

EW

0.00

80.

041

0.07

70.

054

0.08

2-0

.006

0[0

.006

5][0

.016

8]**

[0.0

314]

**[0

.022

6]**

[0.0

233]

***

[0.0

027]

**[0

.000

2]*

KU

RT

0.00

30.

006

-0.0

040.

007

0.00

40

0[0

.000

8]**

*[0

.001

8]**

*[0

.003

0][0

.001

3]**

*[0

.001

0]**

*[0

.000

0]*

[0.0

000]

*

D1*

ST

DE

V-0

.883

-1.8

68-1

.327

-1.2

07-1

.77

-8.4

82-7

.932

[0.4

927]

*[0

.495

8]**

*[0

.926

9][0

.562

3]**

[1.0

567]

*[0

.870

1]**

*[0

.506

0]**

*

D1*

SK

EW

-0.0

07-0

.047

-0.0

78-0

.085

-0.0

580.

066

0.07

7[0

.010

0][0

.024

1]*

[0.0

372]

**[0

.029

0]**

*[0

.031

4]*

[0.0

153]

***

[0.0

199]

***

D1*

KU

RT

-0.0

06-0

.012

-0.0

34-0

.022

-0.0

2-0

.016

0.00

1[0

.001

4]**

*[0

.002

5]**

*[0

.011

4]**

*[0

.003

7]**

*[0

.003

8]**

*[0

.002

6]**

*[0

.000

2]**

*

Obse

rvat

ions:

1117

1100

1079

1058

992

920

795

Adj.

R-s

quar

ed:

0.07

90.

2828

0.34

10.

489

0.59

40.

752

0.69

7F

-sta

tist

ic:

5.09

615

.14

9.92

132

.379

26.6

466

.54

161.

7P

rob(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0B

reak

Dat

e11

/11/

2008

10/2

1/20

083/

17/2

009

10/2

3/20

0810

/20/

2008

8/12

/200

85/

7/20

08

44

Tab

le8:

USD

CA

DM

atch

edF

requen

cyO

LS

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C0.

024

-0.4

84-0

.075

-0.2

10.

417

0.45

20.

54[0

.011

1]**

[0.1

115]

***

[0.0

541]

[0.0

514]

***

[0.0

628]

***

[0.1

504]

***

[0.0

655]

***

D1

-0.0

340.

471

0.11

10.

113

-0.8

93-0

.806

-0.9

02[0

.013

4]**

[0.1

120]

***

[0.0

557]

**[0

.056

6]**

[0.0

699]

***

[0.1

578]

***

[0.0

843]

***

ST

DE

V-1

.462

5.21

11.

095

1.12

2-4

.802

-6.5

5-6

.713

[0.5

696]

**[0

.740

9]**

*[1

.354

5][0

.558

5]**

[0.6

222]

***

[1.3

511]

***

[0.6

135]

***

SK

EW

0.00

10.

094

-0.1

94-0

.151

0.03

8-0

.001

0.04

1[0

.010

0][0

.035

4]**

*[0

.051

2]**

*[0

.039

3]**

*[0

.024

9][0

.037

5][0

.018

1]**

KU

RT

-0.0

010.

103

-0.0

10.

005

-0.0

05-0

.003

0.00

1[0

.003

0][0

.027

1]**

*[0

.006

7][0

.009

2][0

.002

8]*

[0.0

036]

[0.0

022]

D1*

ST

DE

V1.

031

-5.3

55-2

.311

-0.4

878.

096

8.38

38.

259

[0.6

047]

*[0

.780

4]**

*[1

.393

9]*

[0.6

360]

[0.6

942]

***

[1.3

786]

***

[0.6

530]

***

D1*

SK

EW

0.00

1-0

.057

0.16

20.

133

0.10

50.

04-0

.028

[0.0

123]

[0.0

368]

[0.0

526]

***

[0.0

405]

***

[0.0

291]

***

[0.0

407]

[0.0

240]

D1*

KU

RT

0.00

6-0

.10.

018

0.01

20.

040.

036

0.03

3[0

.003

5]*

[0.0

272]

***

[0.0

069]

***

[0.0

094]

[0.0

073]

***

[0.0

068]

***

[0.0

078]

***

Obse

rvat

ions:

1105

1086

1074

1052

982

915

843

Adj.

R-s

quar

ed:

0.11

10.

172

0.30

60.

466

0.71

30.

770.

818

F-s

tati

stic

:1.

977

14.8

812

.667

33.5

672

.95

80.6

113

0.3

Pro

b(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0B

reak

Dat

e10

/21/

2008

10/1

2/20

0710

/16/

2008

2/4/

2009

4/3/

2008

8/21

/200

87/

30/2

008

45

Tab

le9:

USD

JP

YM

atch

edF

requen

cyR

egre

ssio

ns

Eq

Nam

e:1

WK

1M2M

3M6M

9M12

MD

ep.

Var

:X

RX

RX

RX

RX

RX

RX

R

C0.

001

-0.0

13-0

.043

-0.0

420.

057

0.12

70.

13[0

.006

1][0

.013

8][0

.013

2]**

*[0

.018

9]**

[0.0

255]

**[0

.020

3]**

*[0

.009

6]**

*

D1

0.02

60.

124

0.20

80.

239

0.24

30.

161

0.12

[0.0

079]

***

[0.0

300]

***

[0.0

314]

***

[0.0

340]

***

[0.0

395]

***

[0.0

274]

***

[0.0

280]

***

ST

DE

V0.

053

0.34

30.

720.

459

-0.9

6-1

.563

-1.0

17[0

.218

0][0

.209

8][0

.207

8]**

*[0

.248

6]*

[0.2

257]

***

[0.2

844]

***

[0.1

953]

***

SK

EW

0.00

5-0

.005

-0.0

11-0

.034

-0.0

26-0

.016

-0.0

01[0

.007

5][0

.016

8][0

.012

0][0

.014

3]**

[0.0

154]

*[0

.008

0]**

[0.0

003]

**

KU

RT

0.00

10.

001

0.00

30.

002

0.00

10

0[0

.000

7][0

.000

7]*

[0.0

008]

***

[0.0

004]

***

[0.0

005]

*[0

.000

2]**

[0.0

000]

**

D1*

ST

DE

V-0

.998

-2.2

38-3

.118

-2.5

97-0

.93

0.47

50.

151

[0.3

382]

***

[0.5

290]

***

[0.5

141]

***

[0.4

342]

***

[0.3

319]

***

[0.3

287]

[0.2

897]

D1*

SK

EW

-0.0

07-0

.018

-0.0

320.

008

0.00

90.

107

0.03

2[0

.009

0][0

.020

7][0

.019

1]*

[0.0

210]

[0.0

250]

[0.0

146]

***

[0.0

098]

***

D1*

KU

RT

-0.0

03-0

.011

-0.0

14-0

.012

-0.0

160.

004

-0.0

02[0

.001

1]**

*[0

.003

4]**

*[0

.002

0]**

*[0

.002

0]**

*[0

.004

0]**

*[0

.001

8]**

[0.0

018]

Obse

rvat

ions:

1109

1098

1079

1057

992

920

848

Adj.

R-s

quar

ed:

0.03

20.

110.

201

0.18

0.41

30.

551

0.35

1F

-sta

tist

ic:

5.46

93.

780

10.3

5411

.169

17.9

5064

.770

19.5

00P

rob(F

-sta

t):

0.00

00.

001

0.00

00.

000

0.00

00.

000

0.00

0B

reak

Dat

e1/

20/2

009

1/7/

2009

12/1

2/20

0811

/24/

2008

4/22

/200

81/

11/2

008

9/14

/200

7

46

Table 10: Add caption

Placeholder for Non-overlapping Observations

FX AUDUSD EURUSD GBPUSD USDCAD USDJPY

C 0.1595 0.087 -0.0676 -1.1367 -0.0208[0.0382]*** [0.0746] [0.0241]*** [0.1164]*** [0.0556]

St Dev 1.4816 0.264 2.7121 9.6125 0.6399[0.1883]*** [0.3438] [0.5195]*** [0.6839]*** [0.4725]

Skew 0.0266 0.0713 0.0523 0.2162 -0.0171[0.0554] [0.0420]* [0.0483] [0.0266]*** [0.0425]

Kurt -0.0436 -0.011 0.01 0.2502 -0.0017[0.0103]*** [0.0108] [0.0053]* [0.0283]*** [0.0102]

Break -0.0511 -0.0458 0.0638 1.1344 0.2033[0.0806] [0.0816] [0.0280]** [0.1169]*** [0.0708]***

Break*St Dev -3.6467 0.1621 -1.392 -10.105 -3.0348[0.8496]*** [1.4251] [0.5061]*** [0.7451]*** [1.1497]**

Break*Skew -0.0062 -0.128 -0.0627 -0.1728 0.0166[0.0595] [0.0527]** [0.0664] [0.0307]*** [0.0485]

Break*Kurt 0.0384 -0.0071 -0.0218 -0.2457 -0.0184[0.0132]*** [0.0124] [0.0076]*** [0.0288]*** [0.0111]

# Obs. 51 50 51 51 50Adj-R2 0.36 0.1 0.18 0.14 0.14F-stats 23.7197 4.2142 6.846 37.8234 5.6195P Value 0 0.0021 0 0 0.0002

47

Tab

le11

:A

UD

USD

QU

AN

TIL

ER

EG

RE

SSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

Q10

3MQ

253M

Q50

3MQ

753M

Q90

3MQ

95M

ethod:

LS

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C0.

122

-0.0

34-0

.032

-0.0

4-0

.033

0.17

80.

395

0.51

8[0

.048

7]**

[0.0

094]

***

[0.0

112]

***

[0.0

166]

**[0

.029

3][0

.033

5]**

*[0

.041

5]**

*[0

.040

0]**

*

D1

0.18

60.

387

0.46

40.

367

0.35

30.

116

-0.0

42-0

.117

[0.0

696]

***

[0.0

554]

***

[0.0

565]

***

[0.0

541]

***

[0.0

417]

***

[0.0

392]

***

[0.0

491]

[0.0

504]

**

ST

DE

V-0

.202

-0.5

24-0

.45

-0.1

320.

080.

329

0.29

40.

005

[0.3

048]

[0.1

524]

***

[0.1

541]

***

[0.1

445]

[0.1

659]

[0.1

129]

***

[0.2

029]

[0.5

259]

SK

EW

0.18

0.06

0.04

90.

053

0.13

0.24

0.41

60.

469

[0.0

624]

***

[0.0

240]

**[0

.017

8]**

*[0

.015

8]**

*[0

.044

3]**

*[0

.029

7]**

*[0

.032

0]**

*[0

.066

1]**

*

KU

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0.01

90.

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0.00

90.

026

0.02

30.

037

0.04

[0.0

140]

[0.0

059]

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.004

7][0

.004

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[0.0

102]

**[0

.006

1]**

*[0

.006

4]**

*[0

.010

1]**

*

D1*

ST

DE

V-3

.325

-3.7

06-4

.49

-4.1

55-3

.839

-3.5

11-3

.404

-3.4

7[0

.525

7]**

*[0

.586

6]**

*[0

.508

1]**

*[0

.457

5]**

*[0

.314

1]**

*[0

.214

2]**

*[0

.281

4]**

*[0

.599

4]**

*

D1*

SK

EW

-0.1

190.

035

0.07

80.

049

-0.0

51-0

.203

-0.3

58-0

.405

[0.0

675]

*[0

.037

5][0

.027

9]**

*[0

.024

6]**

[0.0

470]

[0.0

322]

***

[0.0

415]

***

[0.0

698]

***

D1*

KU

RT

-0.0

24-0

.021

-0.0

16-0

.006

-0.0

27-0

.029

-0.0

45-0

.047

[0.0

146]

[0.0

071]

***

[0.0

055]

***

[0.0

057]

[0.0

105]

**[0

.006

2]**

*[0

.007

7]**

*[0

.010

7]**

*

Obse

rvat

ions:

1058

1058

1058

1058

1058

1058

1058

1058

Adj.

R-s

quar

ed:

0.33

980.

401

0.34

180.

2317

0.16

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2138

0.35

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Pro

b(F

-sta

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Not

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edin

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atio

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ard

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able

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isze

rob

efor

eth

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reak

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and

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icat

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(***),

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%(*

)le

vel.

48

Tab

le12

:E

UR

JP

YQ

UA

NT

ILE

RE

GR

ESSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

Q10

3MQ

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LS

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Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C-0

.12

-0.1

52-0

.172

-0.1

35-0

.108

-0.1

37-0

.179

-0.2

23[0

.030

4]**

*[0

.015

0]**

*[0

.016

5]**

*[0

.026

3]**

*[0

.028

7]**

*[0

.032

2]**

*[0

.032

7]**

*[0

.057

9]**

*

D1

0.31

80.

329

0.36

90.

337

0.30

50.

348

0.35

40.

423

[0.0

496]

***

[0.0

236]

***

[0.0

186]

***

[0.0

327]

***

[0.0

460]

***

[0.0

482]

***

[0.0

408]

***

[0.0

606]

***

ST

DE

V-2

.301

-1.2

52-1

.465

-1.8

1-1

.735

-1.5

56-1

.445

-1.3

17[0

.426

0]**

*[0

.181

9]**

*[0

.193

2]**

*[0

.213

8]**

*[0

.366

6]**

*[0

.324

9]**

*[0

.335

9]**

*[0

.444

2]**

*

SK

EW

-0.2

56-0

.17

-0.2

12-0

.216

-0.1

92-0

.148

-0.1

31-0

.116

[0.0

440]

***

[0.0

264]

***

[0.0

269]

***

[0.0

326]

***

[0.0

463]

***

[0.0

351]

***

[0.0

283]

***

[0.0

387]

***

KU

RT

-0.0

06-0

.003

-0.0

04-0

.006

-0.0

030.

014

0.02

80.

04[0

.004

1][0

.001

2]**

*[0

.001

2]**

*[0

.003

2]*

[0.0

083]

[0.0

092]

[0.0

080]

***

[0.0

134]

***

D1*

ST

DE

V0.

324

-0.8

5-0

.68

-0.1

92-0

.014

-0.5

04-0

.862

-1.0

53[0

.630

2][0

.229

3]**

*[0

.214

8]**

*[0

.293

0][0

.573

9][0

.571

2][0

.398

3]**

[0.4

798]

**

D1*

SK

EW

0.05

80.

055

0.09

70.

128

0.10

6-0

.123

-0.4

53-0

.447

[0.0

813]

[0.0

340]

[0.0

372]

***

[0.0

445]

***

[0.0

763]

[0.0

877]

[0.0

542]

***

[0.0

660]

***

D1*

KU

RT

-0.0

3-0

.027

-0.0

26-0

.019

-0.0

2-0

.059

-0.1

08-0

.117

[0.0

112]

***

[0.0

035]

***

[0.0

045]

***

[0.0

065]

***

[0.0

132]

[0.0

154]

***

[0.0

102]

***

[0.0

158]

***

Obse

rvat

ions:

1058

1058

1058

1058

1058

1058

1058

1058

Adj.

R-s

quar

ed:

0.46

160.

3548

0.30

990.

2457

0.22

140.

2846

0.41

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4917

Pro

b(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

49

Tab

le13

:E

UR

USD

QU

AN

TIL

ER

EG

RE

SSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

Q10

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LS

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EG

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EG

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QR

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Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C0.

134

-0.0

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90.

082

0.13

20.

313

0.34

6[0

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7]**

*[0

.009

1]**

*[0

.013

3]**

*[0

.021

7][0

.012

3]**

*[0

.026

5]**

*[0

.036

3]**

*[0

.029

7]**

*

D1

0.08

60.

113

0.1

0.03

70.

137

0.13

1-0

.027

-0.0

61[0

.058

7][0

.021

6]**

*[0

.025

2]**

*[0

.042

1][0

.029

9]**

*[0

.034

9]**

*[0

.052

9][0

.046

2]

ST

DE

V-0

.883

-0.1

53-0

.158

-0.2

94-0

.586

-0.8

58-1

.598

-1.6

36[0

.234

5]**

*[0

.114

7][0

.125

6][0

.118

9]**

[0.1

074]

***

[0.1

407]

***

[0.1

995]

***

[0.2

176]

***

SK

EW

0.12

50.

050.

053

0.06

90.

102

0.12

20.

174

0.19

[0.0

204]

***

[0.0

058]

***

[0.0

078]

***

[0.0

121]

***

[0.0

087]

***

[0.0

131]

***

[0.0

132]

***

[0.0

088]

***

KU

RT

0.00

50.

006

0.00

60.

003

0.00

60.

007

0.00

10.

001

[0.0

023]

**[0

.000

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.000

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.001

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*[0

.001

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*[0

.002

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5]

D1*

ST

DE

V-1

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-1.0

93-1

.01

-0.8

36-1

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-1.6

27-0

.9-0

.654

[0.6

529]

***

[0.4

304]

**[0

.381

1]**

*[0

.421

7]**

[0.4

140]

***

[0.4

254]

***

[0.5

584]

[0.7

070]

D1*

SK

EW

-0.1

52-0

.032

-0.0

46-0

.076

-0.1

59-0

.197

-0.2

56-0

.276

[0.0

275]

***

[0.0

110]

***

[0.0

100]

***

[0.0

135]

***

[0.0

185]

***

[0.0

233]

***

[0.0

244]

***

[0.0

341]

***

D1*

KU

RT

-0.0

16-0

.015

-0.0

16-0

.012

-0.0

25-0

.024

-0.0

19-0

.018

[0.0

046]

***

[0.0

030]

***

[0.0

031]

***

[0.0

031]

***

[0.0

048]

***

[0.0

045]

***

[0.0

037]

***

[0.0

051]

***

Obse

rvat

ions:

1053

1053

1053

1053

1053

1053

1053

1053

Adj.

R-s

quar

ed:

0.37

250.

1687

0.14

210.

1267

0.18

860.

2895

0.32

790.

3777

Pro

b(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

50

Tab

le14

:G

BP

USD

3MQ

UA

NT

ILE

RE

GR

ESSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

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Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C-0

.105

-0.0

36-0

.07

-0.0

86-0

.131

-0.2

19-0

.223

0.00

8[0

.036

8]**

*[0

.027

4][0

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1]**

*[0

.014

0]**

*[0

.015

0]**

*[0

.026

7]**

*[0

.071

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.134

7]

D1

-0.0

18-0

.116

-0.1

07-0

.066

-0.0

130.

115

0.14

4-0

.105

[0.0

418]

[0.0

396]

***

[0.0

194]

***

[0.0

157]

***

[0.0

163]

[0.0

296]

***

[0.0

731]

**[0

.140

1]

ST

DE

V3.

744

0.30

50.

955

1.26

13.

233

6.33

87.

704

5.47

4[0

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9]**

*[0

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9]**

*[0

.364

8]**

*[0

.575

5]**

*[0

.972

8]**

*[1

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8]**

*

SK

EW

0.05

40.

030.

017

0.02

20.

039

0.02

80.

036

0.12

5[0

.022

6]**

[0.0

088]

***

[0.0

059]

***

[0.0

083]

***

[0.0

087]

***

[0.0

094]

***

[0.0

283]

[0.0

557]

**

KU

RT

0.00

70.

004

0.00

40.

006

0.00

90.

008

0.00

70.

007

[0.0

013]

***

[0.0

014]

***

[0.0

015]

***

[0.0

015]

***

[0.0

009]

***

[0.0

009]

***

[0.0

013]

***

[0.0

018]

***

D1*

ST

DE

V-1

.207

2.60

92.

569

2.00

8-0

.142

-3.5

24-5

.082

-2.5

35[0

.562

3]**

[0.6

457]

***

[0.2

557]

***

[0.2

850]

***

[0.3

827]

[0.5

929]

***

[0.9

929]

***

[1.8

208]

D1*

SK

EW

-0.0

85-0

.119

-0.1

09-0

.092

-0.0

67-0

.027

0.00

6-0

.071

[0.0

290]

***

[0.0

159]

***

[0.0

087]

***

[0.0

173]

***

[0.0

109]

***

[0.0

211]

[0.0

318]

[0.0

585]

D1*

KU

RT

-0.0

22-0

.035

-0.0

37-0

.035

-0.0

27-0

.021

-0.0

13-0

.011

[0.0

037]

***

[0.0

038]

***

[0.0

023]

***

[0.0

040]

***

[0.0

018]

***

[0.0

033]

***

[0.0

026]

***

[0.0

030]

***

Obse

rvat

ions:

1058

1058

1058

1058

1058

1058

1058

1058

Adj.

R-s

quar

ed:

0.48

910.

5041

0.43

490.

2849

0.26

20.

3354

0.44

090.

4773

Pro

b(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

51

Tab

le15

:U

SD

CA

DQ

UA

NT

ILE

RE

GR

ESSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

Q10

3MQ

253M

Q50

3MQ

753M

Q90

3MQ

95M

ethod:

LS

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C-0

.21

-0.4

32-0

.357

-0.2

93-0

.155

-0.1

04-0

.047

-0.0

22[0

.051

4]**

*[0

.033

8]**

*[0

.023

7]**

*[0

.036

7]**

*[0

.016

9]**

*[0

.021

6]**

*[0

.025

6]*

[0.0

376]

D1

0.11

30.

248

0.20

10.

175

0.07

80.

019

-0.0

7-0

.117

[0.0

566]

**[0

.042

7]**

*[0

.028

0]**

*[0

.040

0]**

*[0

.021

8]**

*[0

.025

7][0

.029

5]**

[0.0

475]

**

ST

DE

V1.

122

2.89

92.

544

1.35

30.

105

-0.1

08-0

.27

-0.6

28[0

.558

5]**

[0.3

289]

***

[0.3

884]

***

[0.4

482]

***

[0.1

704]

[0.1

902]

[0.3

994]

[0.4

078]

SK

EW

-0.1

51-0

.305

-0.2

62-0

.169

-0.0

95-0

.075

-0.0

67-0

.074

[0.0

393]

***

[0.0

292]

***

[0.0

408]

***

[0.0

374]

***

[0.0

122]

***

[0.0

127]

***

[0.0

243]

***

[0.0

242]

***

KU

RT

0.00

5-0

.011

-0.0

110.

010.

015

0.01

40.

010.

011

[0.0

092]

[0.0

068]

[0.0

105]

[0.0

089]

[0.0

036]

***

[0.0

032]

***

[0.0

034]

***

[0.0

035]

***

D1*

ST

DE

V-0

.487

-1.8

52-1

.595

-0.7

310.

233

0.92

41.

763

2.86

7[0

.636

0][0

.429

7]**

*[0

.460

5]**

*[0

.500

2][0

.232

1][0

.228

9]**

*[0

.492

0]**

*[0

.514

6]**

*

D1*

SK

EW

0.13

30.

322

0.26

40.

162

0.07

70.

052

0.04

30.

035

[0.0

405]

***

[0.0

313]

***

[0.0

416]

***

[0.0

374]

***

[0.0

131]

***

[0.0

167]

***

[0.0

272]

[0.0

266]

D1*

KU

RT

0.01

20.

031

0.02

90.

008

0.00

30.

002

0.00

70.

002

[0.0

094]

[0.0

072]

***

[0.0

105]

***

[0.0

091]

[0.0

036]

[0.0

036]

[0.0

040]

*[0

.007

6]

Obse

rvat

ions:

1052

1052

1052

1052

1052

1052

1052

1052

Adj.

R-s

quar

ed:

0.46

620.

4856

0.39

920.

2893

0.27

760.

3192

0.33

060.

3241

Pro

b(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

52

Tab

le16

:U

SD

JP

YQ

UA

NT

ILE

RE

GR

ESSIO

N

Eq

Nam

e:3M

LS

3MQ

053M

Q10

3MQ

253M

Q50

3MQ

753M

Q90

3MQ

95M

ethod:

LS

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

QR

EG

Dep

.V

ar:

XR

XR

XR

XR

XR

XR

XR

XR

C-0

.042

-0.0

8-0

.113

-0.0

97-0

.046

0.00

80.

025

0.11

3[0

.018

9]**

[0.0

232]

***

[0.0

277]

***

[0.0

336]

***

[0.0

067]

***

[0.0

146]

[0.0

383]

[0.0

605]

*

D1

0.23

90.

204

0.25

60.

269

0.25

50.

181

0.15

50.

055

[0.0

340]

***

[0.0

298]

***

[0.0

323]

***

[0.0

392]

***

[0.0

303]

***

[0.0

174]

***

[0.0

445]

***

[0.0

629]

ST

DE

V0.

459

-0.7

11-0

.413

0.14

30.

477

0.72

51.

625

1.15

6[0

.248

6]*

[0.1

855]

***

[0.2

310]

*[0

.328

3][0

.130

7]**

*[0

.123

3]**

*[0

.338

7]**

*[0

.376

4]**

*

SK

EW

-0.0

34-0

.05

-0.0

73-0

.064

-0.0

33-0

.01

0.00

80.

042

[0.0

143]

**[0

.017

8]**

*[0

.018

8]**

*[0

.022

0]**

*[0

.005

8]**

*[0

.010

2][0

.021

5][0

.041

1]

KU

RT

0.00

20.

003

0.00

30.

003

0.00

20.

001

0-0

.001

[0.0

004]

***

[0.0

004]

***

[0.0

005]

***

[0.0

008]

***

[0.0

003]

***

[0.0

003]

***

[0.0

007]

[0.0

011]

D1*

ST

DE

V-2

.597

-1.6

27-1

.96

-2.5

18-2

.805

-2.4

2-2

.902

-2.2

44[0

.434

2]**

*[0

.335

4]**

*[0

.350

9]**

*[0

.360

6]**

*[0

.352

7]**

*[0

.199

0]**

*[0

.444

0]**

*[0

.510

7]**

*

D1*

SK

EW

0.00

80.

031

0.06

10.

044

-0.0

12-0

.038

-0.0

28-0

.069

[0.0

210]

[0.0

197]

[0.0

219]

***

[0.0

244]

*[0

.014

0][0

.015

1]**

[0.0

245]

[0.0

445]

D1*

KU

RT

-0.0

12-0

.007

-0.0

08-0

.01

-0.0

14-0

.013

-0.0

09-0

.007

[0.0

020]

***

[0.0

011]

***

[0.0

012]

***

[0.0

015]

***

[0.0

019]

***

[0.0

011]

***

[0.0

019]

***

[0.0

018]

***

Obse

rvat

ions:

1057

1057

1057

1057

1057

1057

1057

1057

R-s

quar

ed:

0.18

030.

2613

0.21

030.

1129

0.10

240.

110.

1652

0.26

14P

rob(F

-sta

t):

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

53

Table 17: TERM STRUCTURE REGRESSIONS: MOMENTS

Eq Name: AUDUSD EURJPY EURUSD GBPUSD USDCAD USDJPYMethod: LS LS LS LS LS LS

Dep. Var: XR 3M XR 3M XR 3M XR 3M XR 3M XR 3M

C 0.098 -0.011 0.041 0.103 -0.162 -0.151[0.0569]* [0.0428] [0.0562] [0.0434]** [0.0410]*** [0.0680]**

1M-StDev -3.098 6.349 4.123 3.043 -8.626 9.882[2.0085] [1.4844]*** [2.5108] [1.6835]* [2.1233]*** [1.4123]***

1M-Skew -0.18 0.053 -0.326 -0.141 -0.022 -0.063[0.0568]*** [0.0705] [0.0585]*** [0.0377]*** [0.1001] [0.0299]**

1M-Kurt -0.008 0.026 -0.035 -0.002 -0.02 0.031[0.0137] [0.0082]*** [0.0137]** [0.0088] [0.0135] [0.0182]*

3M-StDev 8.391 -7.108 -5.095 -2.546 8.046 -10.169[2.6069]*** [1.7313]*** [2.5940]** [2.9914] [3.1372]** [1.4471]***

3M-Skew 0 -0.036 0.268 0.042 0.162 0.028[0.0574] [0.0959] [0.0798]*** [0.0225]* [0.0919]* [0.0248]

3M-Kurt -0.03 -0.052 0.025 -0.016 0.03 -0.002[0.0149]** [0.0177]*** [0.0127]* [0.0070]** [0.0163]* [0.0076]

12M-StDev -3.816 1.724 1.585 0.367 -0.945 2.951[0.7026]*** [0.6180]*** [0.8704]* [0.9458] [1.0742] [0.5100]***

12M-Skew 0.012 -0.09 -0.004 0 -0.073 0.001[0.0034]*** [0.0167]*** [0.0296] [0.0001] [0.0259]*** [0.0006]*

12M-Kurt 0 0.005 0.001 0 -0.003 0[0.0000]*** [0.0014]*** [0.0033] [0.0000] [0.0018]* [0.0000]*

Observations: 855 861 842 795 831 846Adj. R-squared: 0.3977 0.3261 0.332 0.2836 0.3015 0.23P(F-statistic) : 0.000 0.000 0.000 0.000 0.000 0.000

Note: Newey-West standard deviations are reported in brackets, with asterisks indicating significance at1% (***), 5% (**), and 10% (*) level. F-stats and P value below are based on the Wald test of the nullthat the coefficients on all risk-neutral moments are zero.

54

Table 18: TERM STRUCTURE REGRESSIONS: MOMENTS

Eq Name: AUDUSD EURJPY EURUSD GBPUSD USDCAD USDJPYMethod: LS LS LS LS LS LS

Dep. Var: XR 12M XR 12M XR 12M XR 12M XR 12M XR 12M

C 0.405 -0.229 0.197 0.555 -0.296 0.067[0.0750]*** [0.0537]*** [0.0570]*** [0.0600]*** [0.0506]*** [0.0469]

1M-StDev -6.122 7.171 6.986 8.832 8.738 4.47[2.7736]** [2.0070]*** [3.3438]** [1.9046]*** [2.6627]*** [1.0568]***

1M-Skew 0.13 -0.064 0.072 -0.225 -0.112 -0.031[0.1124] [0.0803] [0.0682] [0.1177]* [0.1134] [0.0190]*

1M-Kurt 0.146 -0.013 -0.037 -0.049 0.052 0.034[0.0303]*** [0.0096] [0.0113]*** [0.0158]*** [0.0184]*** [0.0139]**

3M-StDev 14.493 -16.694 -13.985 -21.092 -11.904 -5.183[3.2759]*** [2.3703]*** [3.6765]*** [4.0155]*** [4.0774]*** [0.9754]***

3M-Skew -0.286 -0.162 -0.022 0.22 0.412 0.039[0.1298]** [0.0800]** [0.0895] [0.1163]* [0.1307]*** [0.0142]***

3M-Kurt -0.196 -0.003 0.049 0.033 -0.044 -0.015[0.0298]*** [0.0144] [0.0123]*** [0.0129]** [0.0197]** [0.0061]**

12M-StDev -7.754 7.903 4.396 5.593 5.61 1.103[0.8878]*** [0.7708]*** [1.3538]*** [1.5940]*** [1.6695]*** [0.3079]***

12M-Skew 0.015 -0.069 0.028 0 -0.145 0[0.0099] [0.0185]*** [0.0297] [0.0002] [0.0431]*** [0.0003]

12M-Kurt 0 0.001 -0.005 0 0.001 0[0.0001] [0.0011] [0.0036] [0.0000] [0.0031] [0.0000]

Observations: 855 861 842 795 831 846Adj. R-squared: 0.6645 0.5196 0.5806 0.4335 0.57 0.3171P(F-statistic): 0.000 0.000 0.000 0.000 0.000 0.000

55

Table 19: Add caption

PLACEHOLDER FOR GLOBAL RISK REG 3M

FX AUDUSD EURUSD GBPUSD USDCAD USDJPY

C -0.0229 -0.0039 0.0084 0.0103 0.0187[0.0068]*** [0.0045] [0.0047]* [0.0043]** [0.0039]***

PC1 3M St Dev -0.0049 -0.001 0.0071 -0.002 -0.0017[0.0058] [0.0030] [0.0044] [0.0041] [0.0036]

PC1 3M Skew -0.0105 0.0029 -0.0139 0.006 -0.0061[0.0104] [0.0052] [0.0079]* [0.0076] [0.0058]

PC1 3M Kurt -0.0243 -0.0098 -0.0248 0.0157 -0.0137[0.0118]** [0.0055]* [0.0095]*** [0.0091]* [0.0056]**

PC2 3M St Dev -0.0172 -0.0198 0.0189 -0.0054 0.0139[0.0142] [0.0095]** [0.0111]* [0.0094] [0.0122]

PC2 3M Skew -0.0275 -0.0136 -0.0138 0.0117 0.0077[0.0074]*** [0.0046]*** [0.0043]*** [0.0043]*** [0.0037]**

PC2 3M Kurt -0.0132 -0.0099 0.005 0.0078 0.0032[0.0084] [0.0043]** [0.0049] [0.0051] [0.0052]

PC3 3M St Dev 0.0888 0.0448 0.0372 -0.0268 0.0082[0.0256]*** [0.0152]*** [0.0193]* [0.0175] [0.0138]

PC3 3M Skew 0.0271 0.0157 0.0173 -0.004 -0.0152[0.0098]*** [0.0060]*** [0.0069]** [0.0064] [0.0053]***

PC3 3M Kurt 0.035 0.0157 0.0117 -0.016 -0.0049[0.0127]*** [0.0059]*** [0.0081] [0.0081]** [0.0051]

# Obs. 1046 1046 1046 1046 1046Adj-R2 0.26 0.19 0.21 0.18 0.14F-stats 2.847 3.719 3.779 3.279 4.067

P Value 0.0026 0.0001 0.0001 0.0006 0

56

Table 20: Add caption

PLACEHOLDER FOR GLOBAL TERM STRUCTURE

FX AUDUSD EURUSD GBPUSD USDCAD USDJPY

C -0.0044 0.009 0.0206 0.0017 0.0153[0.0080] [0.0046]** [0.0060]*** [0.0054] [0.0048]***

PC1 all St Dev -0.0156 -0.007 -0.0035 0.0073 -0.0022[0.0040]*** [0.0023]*** [0.0029] [0.0024]*** [0.0022]

PC1 all Skew 0.0294 0.019 0.0104 -0.0177 0.0042[0.0066]*** [0.0040]*** [0.0055]* [0.0050]*** [0.0040]

PC1 all Kurt 0.0086 0.0082 -0.002 -0.0029 -0.0044[0.0068] [0.0037]** [0.0053] [0.0053] [0.0033]

PC2 all St Dev 0.0286 0.0026 0.0202 -0.0244 0.0187[0.0090]*** [0.0057] [0.0087]** [0.0061]*** [0.0061]***

PC2 all Skew -0.0208 -0.0147 -0.0137 0.0127 -0.0037[0.0033]*** [0.0021]*** [0.0027]*** [0.0025]*** [0.0025]

PC2 all Kurt 0.0276 0.015 0.0147 -0.015 0.0015[0.0040]*** [0.0024]*** [0.0027]*** [0.0026]*** [0.0028]

PC3 all St Dev -0.022 -0.0174 -0.0077 0.0074 0.0073[0.0112]** [0.0065]*** [0.0088] [0.0076] [0.0068]

PC3 all Skew -0.0232 -0.0102 -0.0083 0.0157 -0.0002[0.0057]*** [0.0033]*** [0.0037]** [0.0035]*** [0.0032]

PC3 all Kurt 0.0193 0.013 0.009 -0.0134 0.0083[0.0133] [0.0075]* [0.0097] [0.0098] [0.0054]

# Obs. 785 785 785 785 785Adj-R2 0.37 0.4 0.28 0.34 0.16F-stats 11.259 16.092 9.724 9.838 3.239

P Value 0 0 0 0 0.0007

57

Table 21: Add caption

DECOMPOSED XR CONDENSED OLS RESULTSDSPOT FP DSPOT FP DSPOT FP

AUDUSD#of obs. 1098 1098 1058 1058 855 855Adj. R2 0.2576 0.0852 0.3441 0.4346 0.8307 0.6409

p(F-stat) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000BreakDate 2/16/2009 1/11/2008 1/28/2009 1/21/2008 5/2/2008 1/11/2008EURJPY

#of obs. 1100 1100 1058 1058 861 861Adj. R2 0.1826 0.0692 0.4525 0.2144 0.6629 0.7181

p(F-stat) 0 0.0001 0 0 0 0BreakDate 8/7/2008 1/11/2008 8/7/2008 10/27/2008 8/29/2008 10/27/2008EURUSD

#of obs. 1084 1084 1053 1053 858 858Adj. R2 0.1948 0.0535 0.3895 0.2398 0.7577 0.7532

p(F-stat) 0 0 0 0 0 0BreakDate 2/12/2009 10/20/2008 2/4/2009 9/18/2008 8/7/2008 9/18/2008GBPUSD

#of obs. 1100 1100 1058 1058 795 795Adj. R2 0.2827 0.006 0.5131 0.3785 0.7391 0.8073

p(F-stat) 0 0.0947 0 0 0 0BreakDate 10/21/2008 10/21/2008 10/23/2008 9/22/2008 8/7/2008 1/17/2008USDCAD

#of obs. 1086 1086 1052 1052 843 843Adj. R2 0.174 0.0142 0.3887 0.0741 0.819 0.4391

p(F-stat) 0 0 0 0 0 0BreakDate 10/12/2007 10/12/2007 6/17/2008 9/19/2008 7/30/2008 10/1/2008USDJPY

#of obs. 1098 1098 1057 1057 848 848Adj. R2 0.1156 0.2423 0.1549 0.6336 0.4845 0.8914

p(F-stat) 0.0013 0 0 0 0 0BreakDate 1/7/2009 10/8/2008 12/12/2008 10/3/2008 10/3/2007 10/23/2008

58

Table 22: Add caption

Matched Frequency OLS Subsample Analysis

1M 3M 12MAUDUSD post pre post pre post pre

# of obs. 543 555 516 542 506 349Adjusted R2 0.186 0.21 0.485 0.151 0.695 0.85

F stat.(p. Value) 0.000 0.000 0.000 0.000 0.000 0.000

EURJPY post pre post pre post pre# of obs. 688 412 640 418 662 199

Adjusted R2 0.153 0.238 0.345 0.652 0.465 0.104F stat.(p. Value) 0.000 0.000 0.000 0.000 0.000 0.312

EURUSD post pre post pre post pre# of obs. 534 550 507 546 440 418

Adjusted R2 0.126 0.234 0.28 0.451 0.627 0.8F stat.(p. Value) 0.000 0.000 0.000 0.000 0.000 0.000

GBPUSD post pre post pre post pre# of obs. 629 471 585 473 509 286

Adjusted R2 0.133 0.416 0.489 0.415 0.53 0.649F stat.(p. Value) 0.000 0.000 0.000 0.000 0.000 0.000

USDCAD post pre post pre post pre# of obs. 883 203 507 545 432 411

Adjusted R2 0.091 0.466 0.487 0.387 0.526 0.793F stat.(p. Value) 0.000 0.000 0.000 0.000 0.000 0.000

USDJPY post pre post pre post pre# of obs. 571 527 562 495 677 171

Adjusted R2 0.196 0.022 0.328 0.044 0.354 0.343F stat.(p. Value) 0 0.123 0 0 0 0

59

Tab

le23

:A

dd

capti

on

MA

TC

HE

DF

RE

QU

EN

CY

OL

SW

ITH

10D

IMP

LIE

DM

OM

EN

TS

1WK

1M2M

3M6M

9M12

MA

UD

USD

#of

obs.

1108

1099

1080

1058

992

924

854

Bre

akD

ate

2/24

/200

92/

16/2

009

1/19

/200

91/

9/20

0911

/19/

2008

5/2/

2008

5/2/

2008

Adju

sted

R2

0.11

80.

248

0.30

90.

350.

668

0.79

30.

834

Fst

at.(

p.

Val

ue)

0.00

20.

000

0.00

00.

000

0.00

00.

000

0.00

0

EU

RJP

Y#

ofob

s.11

1710

9810

8010

5899

292

686

1B

reak

Dat

e10

/22/

2008

8/7/

2008

8/7/

2008

8/7/

2008

4/2/

2008

1/3/

2008

10/1

/200

8A

dju

sted

R2

0.04

10.

185

0.50

20.

502

0.72

90.

845

0.62

2F

stat

.(p.

Val

ue)

0.02

70.

000

0.00

00.

000

0.00

00.

000

0.00

0

EU

RU

SD

#of

obs.

1114

1099

1074

1058

990

924

858

Bre

akD

ate

10/2

1/20

082/

11/2

009

1/16

/200

92/

4/20

098/

7/20

088/

12/2

009

8/7/

2008

Adju

sted

R2

0.05

40.

174

0.28

40.

221

0.54

60.

763

0.73

7F

stat

.(p.

Val

ue)

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0

GB

PU

SD

#of

obs.

1115

1098

1079

1058

992

920

793

Bre

akD

ate

11/1

1/20

0810

/21/

2008

3/17

/200

910

/23/

2008

10/2

2/20

088/

12/2

008

5/5/

2008

Adju

sted

R2

0.08

30.

369

0.36

90.

474

0.61

60.

753

0.69

7F

stat

.(p.

Val

ue)

00

00

00

0

USD

CA

D#

ofob

s.11

1610

9810

8010

5899

292

686

1B

reak

Dat

e10

/21/

2008

10/1

0/20

0710

/20/

2008

10/2

0/20

084/

3/20

088/

21/2

008

7/31

/200

8A

dju

sted

R2

0.12

10.

157

0.36

90.

091

0.71

70.

772

0.82

4F

stat

.(p.

Val

ue)

00

00.

0387

00

0

US

DJP

Y#

ofob

s.11

1710

9910

7910

5799

292

084

8B

reak

Dat

e1/

20/2

009

1/7/

2009

12/1

1/20

087/

3/20

084/

22/2

008

1/11

/200

89/

14/2

007

Adju

sted

R2

0.03

0.1

0.20

90.

221

0.42

0.53

20.

418

Fst

at.(

p.

Val

ue)

00.

004

00

00

0

60

Figure 1: Time series Evolution of 1M AUDUSD option-implied moments

Figure 2: Time series Evolution of 1M EURJPY option-implied moments

61

Figure 3: Time series Evolution of 1M EURUSD option-implied moments

Figure 4: Time series Evolution of 1M GBPUSD option-implied moments

62

Figure 5: Time series Evolution of 1M USDCAD option-implied moments

Figure 6: Time series Evolution of 1M USDJPY option-implied moments

63