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. We would also like to thank Samir Bandaogo, Nick Basch, David Grad, Abe Robison, and Eric Zivot for their helpful comments and assistance. All remaining errors are our own. Chen’s and Gwati’s research are supported by Gary Waterman Scholarship and the Buechel Fellowship at the University of Washington. Chen: [email protected]; Gwati: [email protected]; Department of Economics, University of Washington, Box 353330, Seattle, WA 98195
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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.
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: [email protected]; Gwati: [email protected]; 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
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
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
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.
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:
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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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
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