du currency options equilibrium

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General Equilibrium Pricing of Currency Options

Du Du

Hong Kong University of Science and Technology

September 6, 2011

Abstract

This paper proposes a consumption-based model that explains currency option

pricing. The driving force is the imperfectly shared consumption disaster with vari-

able intensity which is investors concern under the recursive utility. With highly

but imperfectly shared disaster risks, exchange rate remains smooth and home dis-

aster intensity can either rise above or fall below its foreign counterpart. The former

generates the low option-implied volatilities, and the latter replicates the observed

stochastic skewness and kurtosis of currency returns that are suggested by the cross-

sectional option data. The model also matches the UIP anomaly for currency pricing

and salient features of various other assets.

JEL code: F31, G01, G11

Key words: currency options, stochastic skewness, variable disaster, recursive

preference, UIP anomaly

Email at [email protected]. I thank Redouane Elkamhi, Haitao Li, Nan Li, Tao Li, Tse-Chun Lin, MarakaniSrikant, Liuren Wu, Jin Zhang, and participants at various institutions and conferences for helpful com-ments. Fiancial support from the Hong Kong RGC Research Grant(HKUST 641010) is acknowledged.

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I Introduction

The foreign exchange market is the largest nancial market in the world, yet the pricing of

its derivatives is not well studied until recently. Using a large dataset of over-the-counter

option prices written on two most actively traded currency pairs, Carr and Wu (2007)

document several regularities about currency options. First, the implied volatility from at-

the-money (ATM)1 option contracts are low at round 811%. Second, the implied volatility

on average is a U-shaped function of moneyness, and the slope, which measures the risk-

neutral skewness of currency returns, exhibits substantial time variations. In particular,

the sign of the slope switch signs several times in their sample which is in sharp contrast

to equity index options in which the implied skewness stays highly negative. Third, the

curvature of the implied volatility smile, which measures the dierence between out-of-the-money (OTM) and ATM options is sizable and also varies signicantly over time.

This paper proposes a consumption-based general equilibrium model that explains cur-

rency option pricing. The central ingredients are the highly but imperfectly shared eco-

nomic disasters with variable intensities, and the recursive preference (e.g., Epstein and

Zin, 1989) which allows for a separation between elasticity of intertemporal substitution

(EIS) and risk aversion. Following Barro (2006) and Wachter (2011), disaster is modeled as

a peso component2 in the consumption process. Under the recursive preference, investors

care about not only the contemporaneous consumption shocks but also the prospects of

future consumption growths. When EIS is greater than one, they demand extra compensa-

tions for the increase of disaster rate out of the fear of the substantial drop of consumption

in the future. My results show that risks related to country-specic disaster components

with variable rates can quantitatively justify many observed features for currency option

pricing.

The mechanism of the model is as follows. First, since disaster risks are highly shared

across borders, the exchange rate needs to uctuate less in order to prevent international

arbitrage opportunities, which gives rise to the low ATM implied volatility. Second, with

imperfect risk sharing, disaster rate at home can either rise above or fall below its foreign1 Moneyness is dened as the ratio of the strike price of an option contract to the spot price of the

underlying on which the option is written. Therefore, an ATM option would have the moneyness of 1. Byconvention, out-of-the-money (OTM) and in-the-money (ITM) puts refer to put options with moneynessless than and greater than 1, respectively.

2 The name "peso problem" is attributed to Milton Friedmans comments about the eect of infrequentbut disastrous events on Mexico peso market in the early 1970s.

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counterpart generating, respectively, the negative and the positive skewness in currency

returns. Stochastic skewness emerges since the two country-specic disaster rates evolve

stochastically over time. Third, since OTM options provide hedge against jumps, they aretraded at higher prices than ATMs, and the dierences also vary signicantly driven by

the variable disaster rates.

In calibrating the model, I follow the literature (e.g., Backus, Foresi, and Telmer,

2001) by imposing the complete symmetry in that all model parameters are identical

across any two countries. Within a given country, I calibrate the variable disaster process

according to i) the international evidences about its intensity and magnitude provided by

Barro (2006), ii) the match of some key moments for stock pricing. Across countries, I

calibrate consumption correlation during normal times according to Brandt, Cochrane, and

Santa-Clara (2006), and I choose a predominant global disaster component to match the

observed high risk sharing. The preference parameters are at levels deemed reasonable in

the literature (e.g., Mehra and Prescott, 1985; Bansal and Yaron, 2004; Bansal, Gallant,

and Tauchen, 2007). Finally, I estimate an ination process to convert real variables into

nominal ones.

To compare the model with the data, I collect option quotes written on USDJPY and

USDGBP from bloomberg which covers the period from October 2003 to October 2010.

The calibrated model matches well the regularities in currency option markets. First, it

generates an average ATM implied volatility of of 8.5%, which is slightly below its empiricalrange of 1011%. Correspondingly, the implied exchange rate volatility is 8.4% which is

also close to the empirical level of 911%. This result is in contrast to the traditional

consumption-based model in which the implied volatility is usually more than an order of

magnitude higher than that in the data (e.g., Brandt, Cochrane, and Santa-Clara, 2006).

Second, the model replicates the magnitude of stochastic skewness measured as the

standard deviation of risk reversals (RR). Take the three-month near ATM contracts for

example.3 The model-implied standard deviation of RR is 7.67%, as compared to 9.34% and

6.62% for options written on USDJPY and USDGBP, respectively. In addition, the model

generates, as in the data, a rising term structure of the RR standard deviation for contract

horizon within one year. Intuitively, the longer time to maturity gives more chances for

3 More precisely, "near ATM" here refers to contracts at the 25 option delta. Delta is usually used forcurrency options as an equivalent way to quote strike/ moneyness, and more detailed discussions are inSection III.C.

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the greater uctuations of the variable disaster rates, which enhances the variations of the

risk-neutral skewness. For long enough contract horizon, this eect is oset by the mean

reversion nature of the disaste rates, hence the lower RR standard deviation. The latterprediction is also supported by the data.

Third, the model captures the large fraction of the time series variations of risk rever-

sals, and it generates the observed positive correlation between changes in risk reversal

and currency returns, and the positive correlation between risk reversals and interest rate

dierentials. In addition, the model matches well both the average and the standard de-

viation of the volatility dierences between OTMs and ATMs measured by the buttery

spreads (BF). Simultaneously, it generates the observed rising term structure for both the

average and the standard deviation of the BF.

In my model, the key to simultaneously match the low level of ATM volatility and

the substantial cross sectional variations for currency options lies in the highly but im-

perfectly shared economic disasters. Indeed, I show in the sensitivity analysis that there

are essentially no risk reversal variations when we shut down the un-shared disasters. To

understand the source for the un-shared component to which exchange rates are subject,

it is helpful to interprete country-specic disaster as a broader concept than "a disaster

that strikes one country but not the other". For example, Table I in Barro (2006) reveals

that the disaster associated World War II struck major economies at dierent timing. It

is arguable, at least qualitatively, that such a disaster is not perfect shared across borders.Since option prices are very sensitive to jumps, even a small fraction of the peso compo-

nent that is un-shared can produce substantial cross sectional variations as in the data.

Therefore, while it is hard to distinguish econometrically my model from a model featuring

100% global disasters, the two setups have very dierent pricing implications.

In the literature, Carr and Wu (2007) provide the rst theoretical explanation for

the stochastic skewness suggested by currency option data. Built within the reduced-form

setting, their model starts with the exchange rate process which is exogenously imposed two

Levy type jumps exhibiting positive and negative skewness, respectively. By randomzing

the time clock on both jumps, stochastic skewness emerges from the stochastic variation

in the relative proportion of positive to negative jumps. Despite the success, Carr and

Wu (2007) point out that "for future research, it is important to understand the economic

underpinnings of the stochastic skewness suggested by currency option prices". My paper,

which is motivated by their research, represents the rst step in that direction.

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The present model has rich implications beyond the pricing of currency options. First,

it quantitatively replicates the UIP anomaly, i.e., high interest rate currencies tend to

appreciate. The key lies in the time-varying foreign currency risk premium attributed to thevariable disaster rates. Building on the work of Campbell and Cochrane (1999) and Case II

of Bansal and Yaron (2004), respectively, Verdelhan (2010) and Bansal and Shaliastovich

(2010) show that habit formation and long-run risk models are also able to generate the

UIP anomaly. In their papers, time-varying currency risk premiums arise from variable

risk aversions induced by habit formation as in Verdelhan (2010), and from the short-

term uctuations of consumption volatility as in Bansal and Shaliastovich (2010). Farhi

and Gabaix (2009) provides another disaster story for the UIP anomaly. Unlike my setup,

they model disasters through the time-varying "expected resilience" that incorporates both

disaster intensity and disaster magnitude. None of the above papers considers derivative

pricing.

Second, the model captures the aggregate stock market behaviors such as the high

equity premium and the excess return volatility. These results essentially replicate Wachter

(2011) who is the rst to show that variability in disaster intensity combined with the

recursive preference provides another way to understand stock pricing. My analysis extends

her model to the two-country setup, and focuses on the pricing of currency options using

the variable disaster framework which is not touched previously.

Third, the model explains the salient pricing feature for equity index options in that itcaptures well both the unconditional and the time series variations of the volatility smirk.

The match of the averagel level is consistent with the well documented fact that a small

peso component is crucial to the pricing of index options (e.g., Bates, 1996; Pan, 2002;

Eraker, 2004). The time series t is attributed to the variations of disaster rates that can

be interpreted as the "investor fear gauge". Simultaneously, the model predicts a rising

smirk premium, measured as volatility dierence between deep OTM and ATM options,

as ATM implied volatility decreases, which is also supported by the data (e.g., Du (2011)).

Fourth, the model matches the average and the standard deviation of short-term rates,

and generates a slightly downward sloping term structure for real bond yields which is

associated with the procyclical real interest rates. Using both the U.S. and the international

data, Verdelhan (2010) provides evidences that real rates indeed tend to be lower during

bad times. Simultaneously, the model explains 32% time series variations of the observed

interest rates using disaster intensities backed out from the index option data. This result

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is in contrast to the reduced form model of Carr and Wu (2007) who assume constant

interest rates.

To provide additional support to the main nding of the paper, I also investigate theimplications of a two-country stochastic volatility model under the long-run risk framework

of Bansal and Yaron (2004, BY), which is previously studied by Bansal and Shaliastovich

(2010). I replicate Bansal and Shaliastovich (2010)s nding that the model can resolve

the UIP anomaly. However, the model performs poorly in capturing currency opion prices:

the implied risk reversal standard deviations are no more than 0.06% which is far below

their empirical levels. This nding rearms that modeling disaster components is essential

to understanding currency option pricing.

The remainder of the paper is organized as follows. Section II&III presents the setup

of the model and summarizes the theoretical results. Section IV describes the data and

calibrate the model. Section V discusses model implications, and Section VI studies impli-

cations of a stochastic volatility model. Section VII concludes the paper.

II The setup

This section discusses the model setup. I start with the variable disaster components in the

consumption process that are imperfectly shared across borders. I then derive the implied

pricing kernel under which variations of disaster rate are directly priced.

A. Imperfectly shared disasters

Following the peso problem literature (e.g., Barro, 2006), I assume that the aggregate home

consumption evolves according to

dCtCt

= dt + dBct +

eZ 1 dNt; (2.1)where dBct is a standard Brownian; dNt denotes the Poisson jumps with the arrival intensitytdt; which models the rare economic disasters; Z < 0 denotes the log consumption jump

size upon the occurrence of the disaster, which is assumed to be constant for simplicity

without aecting model intuitions. Following Wachter (2011), I introduce variability in

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disaster intensity t as follows:

dt =

t dt + ptdBt; (2.2)where dBt is another standard Brownian independent ofdBct: As plotted in Wachter (2011,

Figure 1), the square root term makes the stationary distribution of t highly skewed. In

particular, there are times when "rare" disasters can occur with high probability, but these

times are themselves unlikely. On average, disasters strike at the intensity .

Following the international nance literature (e.g., Backus, Foresi, and Telmer, 2001;

Verdelhan, 2011), I impose the perfect symmetry so that parameters are identical across

any two countries. The aggregate consumption and disaster intensity in the foreign country,

super-indexed by "*", thus follow

dCtCt

= dt + dBct +

eZ 1 dNt ; (2.3)dt =

t

dt +

ptdB

t; (2.4)

where Nt with intensity t models the foreign economic disaster; B

ct and B

t are mutually

independent. Across borders, Bct is correlated with Bct by a constant hfC ; dBt and dB

t

are also correlated which will become clear in the following.

In order to capture the imperfect correlation of variable disasters in dierent countries,I decompose each of the two Poisson processes into two components: the global component

and the country-specic component. In particular,

f dNt = dNht + dN

gt

dNt = dNft + dN

gt

;

where Nht and Nft model disasters specic to the home and the foreign country with

intensities denoted by ht and ft ; N

gt models the global disasters shared across borders

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with the intensity gt ; Nht ; N

ft ; and N

gt are mutually independent.

4 I further specify that

dht =

h

ht

dt + q

ht dBht; (2.5)

dft =

f ft

dt +

qft dBft; (2.6)

dgt =

g gt

dt +

pgtdBgt; (2.7)

where Bht; Bft; and Bgt are mutually independent which are also independent of Bct and

Bct: To be consistent with perfect symmetry, I impose that

ht ; ft ;

gt

share and and

that h

= f

; but country-specic rates and the global rate are allowed to have dierent

long-run averages. Under (2.5)(2.7), t

= h

t + g

t

and

t

= f

t + g

t

indeed follow theprocesses of (2.2) and (2.4) if we dene

dBt 1pt

qht dBht +

pgtdBgt

; (2.8)

dBt 1pt

qft dBft +

pgtdBgt

: (2.9)

It can be shown that5

(dNt; dNt ) = (dBt; dBt) =

gtr

ht + gt

ft +

gt

;

where (:) denotes the correlation coecient. Apparently, (dNt; dNt ) is close to one

when disasters are highly shared, i.e., gt >> ht ;

ft :

Decompositions of Poisson processes are not uncommon in previous studies (e.g., Due

and Garleanu, 2001). Such researches tend to focus on processes with constant intensities.

4 To clarify the potential confusions about names, I reiterate the dierence between home-specic dis-

asters and home disasters: the former is modeled by dNh

t which only strike at home; the latter is modeledby dNt which comprises both the home-specic disasters and the global disasters. Similar dierentiationapplies to foreign-specic disasters and foreign disasters.

5 Take the correlation between dNt and dNt for example. (dNt; dN

t ) =

Cov(dNt;dN

t )pV ar(dNt)

pV ar(dNt )

=

Cov(dNht +dNgt ;dN

ft +dN

gt )p

tp

t dt=

Cov(dNgt ;dNgt )p

tp

t dt=

gtq

(ht+gt )(

ft+

gt )

, where Ive used the mutual independences of

Nht ; Nft ; and N

gt for the third equality.

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To my best knowledge, the above decomposition of Poisson process with stochastic inten-

sity, also known as the Cox process, is new to the literature. It is built on the result, which

I formally show in Appendix A.1, that the sum of two independent Cox processes is still aCox process if their intensities are also independent of each other.

B. Pricing kernels

Assume the existence of a representative agent in the home country whose preference

is described by the stochastic dierential utility (SDU) developed by Due and Epstein

(1992), which is the continuous-time version of the recursive utility considered by Kreps and

Porteus (1978) and Epstein and Zin (1989). Given the consumption process fCs : s 0g,the period-t utility of the agent, denoted by Jt; is dened recursively by

Jt = Et

Z1t

f(Cu; Ju) du

; (2.10)

where

f(Ct; Jt) =

1 1=C11=t [(1 ) Jt]

11=1

[(1 ) Jt]11=1

1(2.11)

which is the normalized aggregator. In the above formula, ; ; and denote the subjective

discount rate, the degree of risk aversion, and the elasticity of intertemporal substitution

(EIS), respectively. As discussed in Introduction, recursive utility ensures that stochastic

disasters are directly priced, which is essential for pricing.

By homogeneity, Jt is separable in Ct and t; and can be written as:

Jt =C1t1 [I(t)]

; (2.12)

where 111=

: By Proposition 2 in Benzoni, Collin-Dufresne, and Goldstein (2010), I(t)

in (2.12) denotes the home wealth-consumption ratio implied from this model. Using an

accurate log linear approximation, Appendix A.2 shows that I(t) can be written as

I(t) = ea+bt ; (2.13)

where a and b are two constants: In particular, b < 0 (b > 0) if > 1 ( < 1) : This is

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because a positive t shock implies a higher intensity of economic disasters which tends to

substantially depress consumption. On the one hand, the income eect makes the agent

consume less today which raises the wealth-consumption ratio. On the other hand, theintertemporal substitution eect encourages the agent to borrow from the future which de-

presses the wealth-consumption ratio. If > 1; EIS is large, and therefore the substitution

eect dominates. As a result, the wealth-consumption ratio is decreasing in t: On the

contrary, the income eect dominates when < 1 leading to the positive relation between

the wealth-consumption ratio and t: The approximate analytical solution (2.13) greatly

facilitates the exposition of model intuitions as well as the computation of option prices.

Under the recursive preference, Due and Epstein (1992) show that the implied pricing

kernel is given by

t = expZ

t

0

fJds

fC;

where fJ and fC denote the derivatives of the normalized aggregator with respect to its

second and rst argument, respectively. By evaluating derivatives in (2.11) and substituting

for Jt using (2.12), I obtain

t = exp

Zt0

+

1 I(s)

ds

Ct I(t)

1 : (2.14)

When implementing the model, I set both and to be greater than one, which isconsistent with previous estimations (e.g., Mehra and Prescott, 1985; Bansal, Gallant, and

Tauchen, 2006). Under such preferences, < 0 and b < 0; hence t loads positively on

t. Intuitively, a higher disaster intensity at home raises the home agents marginal utility

which is captured by the higher t:

Applying Itos lemma with jumps (e.g., Appendix F of Due, 2001) to (2.14),

dtt

= rtdt + ( 1) bp

gtdBgt +

eZ 1 (dNgt gtdt)dBct + ( 1) bqht dBht + eZ 1 dNht ht dt ; (2.15)

where rt denotes the short-term real interest rate at home that equals

rt = +12

(1 + ) 2+1

2( 1) b22tt

eZ 11

t

e(1)Z 1 : (2.16)

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In (2.15), the rst (excluding rtdt) and the second line captures the impacts of the globaland the home-specic risk components, respectively. Bakshi, Carr and Wu (2008) also

emphasize the dierentation between global and country-specic risks, which they ndis the key to capture the multi-dimensional structure of pricing kernels in international

economies. The dierence lies in that they adopt the reduced-form approach by exoge-

nously partitioning the pricing kernel into the global diusion risk component and the

country-specic component involving both diusions and Levy type jumps. Finally, pricing

kernel process and interest rate at the foreign country can be obtained from (2.15)(2.16)

via perfect symmetry. In particular,

rt = +

1

2

(1 + ) 2 +1

2

(

1) b22

t

t eZ 1

1

t e(1)Z 1 :(2.17)III Asset valuations

This section presents the models theoretical pricing implications. The rst two subsections

derives the implied real and nominal exchange rates. Section III.C discusses the implied

currency option pricing which is the focus of this paper. The last subsection summarizes

model implications about the aggregate equity, equity index options, and (default-free)

bonds.

A. Real exchange rate

Throughout this paper, I treat USD as the home currency. The exchange rate, denoted by

e; is dened as the home currency price of the foreign currency: Under this denition, an

appreciation of the foreign currency, or equivalently a depreciation in the home currency,

results in an increase in e:

When the market is complete, it is well known that exchange rate is linked to the

pricing kernels byett =

t ; (3.1)

where t and t denote the home and the foreign pricing kernel, respectively.

6 From (3.1),

6 When the market is incomplete, t (t ) in (3.1) is interpreted as the minimum-variance pricing kernel

derived from the projection of all candidate home (foreign) pricing kernels onto the space of all home

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a rise in t relative to t leads to the decrease in et. This observation, combined with

the result that t loads positively on t (see Section II.B), implies that a higher disaster

rate at home drives down the exchange rate. Intuitively, t can be interpreted as the "feargauge" for the home investors who reduce their investment in risky assets, including the

foreign currency, when they become more fearful, hence the depression of asset valuations.

By (2.15) and the perfect symmetry for t ; the log real exchange rate follows:

d ln (et) =

ht ft 1

e(1)Z 1 + 1

2( 1) b22

dt + (dBct dBct)

( 1) bq

ht dBht q

ft dBft

+ Z

dNht dNft

: (3.2)

As expected, the dynamics of exchange rates are only driven by shocks that are not shared,

which are captured by (dBct dBct) for consumption shocks, by ( 1) bq

ht dBht q

ft

for disaster intensity shocks, and by Z

dNht dNft

for the arrival of disasters.

When risk aversion is not too low, the model implies that real interest rate dierential,

rt rt ; and the expected change in log real exchange rate move in opposite direction. Tosee it, rst obtain from (2.16), (2.17), and (3.2) that:

rt

rt = ht

ft 1

2

(

1) b22

eZ

1

1

e(1)Z 1 ; (3.3)Et (d ln et) =dt =

ht ft

12

( 1) (b)2 1

e(1)Z 1 + Z : (3.4)

Next, denote by ret the expected foreign currency premium, i.e., the expected excess return

of a domestic investor who borrows fund at home, converts them into a foreign currency,

lends at the foreign risk-free rate, and then converts his earnings back an instant later to

the home currency. In logs, ret is computed by (e.g., Backus, Foresi, and Telmer, 2001):

r

e

t Et (d ln et) =dt (rt r

t ) : (3.5)(foreign) asset returns.

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Substituting for rt rt and Et (d ln et) =dt in the above equation gives

ret =

ht

ft1

2 ( 1)2

(b)2

+ eZ

1 + Z : (3.6)Under reasonable parameterizations, rtrt as in (3.3) loads negatively on ht ft : The

intuition is straightforward. When the home-specic disaster intensity is higher than its

foreign counterpart, the home agent saves more than the foreign agent out of the stronger

precautionary saving motivations, which drives down the home interest rate relative to its

foreign counterpart. Since ex 1 x 0 for all x, (3.6) implies that ret loads positively onht ft :7 When the loading of ret on ht ft dominates that of rt rt ; which is the casewhen the risk aversion is not too low, (3.5) implies that Et (d ln et) =dt also loads positively

on ht ft ; and hence moves in opposite direction to rt rt : This mechanism enables thepresent setup to resolve the UIP anomaly for currency pricing, and I relegate the further

discussions to Section V.B.1.

B. Nominal exchange rate

So far all variables are in real terms. To introduce ination, I take the home country as

the example and assume the following process for the consumption price index Pt:8

dPtPt

= tdt + PdBpt; (3.7)

7 To see the economics, I focus on the usual case where and are both greater than one. First notefrom Section II.B that a positive shock in ht raises the home pricing kernel. Next, let

ht >

ft ; under

which the positive loading translates into a positive premium. Since economic disasters are more probableat home than abroad in this case, the home-specic disaster intensity dominates the eect of its foreigncounterpart on the exchange rate determination. In particular, the rise in t due to the positive shockin ht dominates the impact of changes in

t ; which, acccording to (3.1), leads to the depression of the

exchange rate. Put together, conditional on ht > ft ; the home agents marginal utility rises exactly when

the value of foreign currency drops, hence, the positive foreign currency premium. Note that the analysisis perfectly symmetric which takes into account that a positive currency risk premium for home investors

means a negative currency risk premium for foreign investors. Unconditionally, h

= f

implies a zeropremium, which is consistent with the empirical observations that average currency excess returns are notsigncantly dierent from zero.

8 One may argue that ination bursts during disaster time. However, we might as well see deation, ora ight to quality, or counter-cyclical monetary policy during a disaster, in which case the opposite resultswould obtain. Empirically, while some countries do experience hyper-inations during disasters, the GreatDepression in the U.S. is clearly a counter-example. Given these mixed evidences, I opt not to model a

jump component in the ination process.

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where the expected ination t is mean reverting according to:

dt = ( t) dt + dBpt: (3.8)The above specication implies that expected ination and realized ination are perfectly

correlated, which allows the former to be identied from ination data alone (e.g., Wachter,

2006). I further assume that dBpt is independent of all real shocks.9

The nominal (home) pricing kernel $t is formulated in the standard way as

$t = t=Pt:

Denote by

$;

t the nominal pricing kernel in the foreign country. In nominal terms, (3.1)becomes:

e$t$t =

$;t ; (3.9)

Again, by perfect symmetry and Itos lemma with jumps,

d ln

e$t

=

ht ft

1

e(1)Z 1 + 1

2( 1) b22

+ t t

dt + (dBct dBct

( 1) bq

ht dBht q

ft dBft

+ Z

dNht dNft

+ P

dBpt dBpt

:(3.10

Compared to (3.2), in addition to all the real shocks, the dynamics of nominal exchange

rates are also driven by nominal shocks that are not shared, which is captured by P

dBpt dBpt

:

Denote by r$t Etd$t$t

the nominal short-term rate at home. It is easy to show

that

r$t = rt + t 2P: (3.11)

In nominal terms, the interest rate dierential and the expected change in log exchange

rate become

r$t

r$;t = rt

rt + t

t ; (3.12)

Et

d ln e$t

=dt = Et (d ln et) =dt + t t ; (3.13)

where rt rt and Et (d ln et) =dt are given by (3.3)(3.4); t denotes the expected ination9 The pricing of currency options is insensitive to nominal shocks, and hence insensitive to the more

general specication allowing correlation between Bpt and consumption innovations.

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in the foreign country that follows

d

t = (

t ) dt + dB

pt; (3.14)

where dBpt is another Brownian independent of dBpt and all real shocks. While rt rt andEt (d ln et) =dt move in opposite direction when risk aversion is not too low, as discussed

in Section III.A, nominal shocks as captured by t t generate the comovement inEt

d ln e$t

=dt and r$t r$;t . Intuitively, a positive innovation of foreign ination raises

r$t r$;t by depressing the valuation of the foreign currency, due to the purchasing powerparity, and also raises the nominal interest rate prevailing in the foreign country. The

latter eect, however, is dominated in the model by the impacts of currency risk premium

driven by the disaster intensity shocks. This result is consistent with Holield and Yaron(2003)s conclusion that risks from the real side of the economy are vital to capturing the

UIP anomaly.

C. Currency options

Like exchange rates, prices of currency options are denominated in the home currency.

The following proposition summarizes the semi-closed form valuations for both call and

put options written on the nominal exchange rates.

Proposition 1 Conditional on the approximate analytical solution (2.13) for the wealth-

consumption ratio, the period-t prices of a call and a put currency option with strike price

K and time to maturity are

Callt (K; ) = Gd;d( ln K; t; T) KG0;d( ln K; t; T); (3.15)

P utt (K; ) = KG0;d(ln K; t; T) Gd;d(ln K; t; T); (3.16)

respectively, where T = t + ; d = [1; 0; 0; 0]0 ;

Ga;b(y; t; T) = (a; t; T)

2 1

Z10

Im

(a + ib; t; T) eiy

d (3.17)

for any a; b 2 C6; the set of six-tuples of complex numbers. In (3.17), i is the square root

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of -1; Im (c) denotes the imaginary part of c 2 C; (:) is in closed form of

(:; t; T) = exp"

aC () + bC;Y () ln et + bC;h ()

h

t + bC;f ()

f

t

+bC;g () gt + bC; () t + bC; ()

t

#; (3.18)

where aC (:) and bC (:) satisfy a series of ordinary dierential equations (ODEs) presented

in Appendix B.2:

From (3.15)(3.18), currency option prices vary at the dierent home- and foreign-

specic disaster rates, which is clear. Through the home pricing kernel, the global com-

ponent gt also aects the implied prices. Since they enter into the nominal exchange rate

and the pricing kernels in dierent manners, option prices respond separately to variations

in the real exchange rate, et; and variations in the expected inations, t and t . Quanti-

tatively, however, the implied option prices are very insensitive to shocks from the nominal

side.

Given the option price Ot (O = Call;Put) ; the option-implied volatility is computed

by

G/K-volt = GK1( ; K ; Ot; rt;t+; r

t;t+); (3.19)

where GK1 is the inverse of Garman-Kohlhagen formula, the currency option version of

the Black-Scholes (B/S) formula, which is inverted over the exchange rate volatility as one

of its arguments; rt;t+ and rt;t+ are the home and the foreign interest rate over the period

[t; t + ] : In the actual implementation, quotes of G/K-volt are directly available in the

data, and I calculate the model-implied G/K-volt using the general equilibrium values of

rt;t+ and rt;t+.

Following industry convention, the G/K-vol are quoted against option deltas computed

by

f c = er

N(d1) for call options

p = erN(d1) for put options; (3.20)

where the super-indices, "c" and "p", denote the call and the put options; N(:) denotes

the cumulative normal function;

d1 =ln (et=K) + (r r)

IVp

+

1

2IV

p ; (3.21)

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where K; and IV denote, respectively, the strike, the time to maturity, and the volatility

input quoted in the data or implied from a model.10 In the data, G/K-vol are directly

available only for the delta-neutral straddle dened as the sum of a call option and a putoption with the same strike satisfying c + p = 0: Since the delta-neutral restriction

implies d1 = 0; the implicit strike is close to the spot price. For this reason, we refer to the

implied volatility as at-the-money implied volatility (ATMV) thereafter.

Besides ATMV, currency option data provide quotes for risk reversals (RR) and but-

tery spreads (BF), which record variations of G/K-vol across dierent option deltas. RR

is dened as the volatility dierence between an out-of-the-money (OTM) call option and

that of an OTM put option at the same time to maturity, which captures the skewness of

the risk-neutral currency return distribution. Specically, risk reversal at the given option

delta, ; is computed by

RR [] IVc [] IVp [] ; (3.22)

On the other hand, option traders use BF, dened as the dierence between the average

OTM implied volatilities and ATMV, to quantity the kurtosis of risk-neutral currency

return distribution. At the given option delta, ; the implied BF is computed by

BF [] (IVc [] + IVp []) =2 ATMV: (3.23)

Quantitatively, valuations of currency options are sensitive to variations in ht and ft

which capture the un-shared disaster risks. Since the pricing kernel loads positively on

the disaster intensity (Section II.B), a positive shock on ht raises the home pricing kernel

relative to its foreign counterpart, which, from (3.1), adds to the depreciation and hence

the negative skewness of the foreign exchange rate. Consequently, the home agent trades

up prices of call options to hedge the upside risks which leads to the positive risk reversal.

Conversely, a positive shock on ft adds to the positive risk reversal. Empirically, Carr

and Wu (2007) document that the skewness of two major currency pairs, USDJPY and

USDGBP, varies so much that the signs risk reversal switches several times in their sample.

The present model thus provides an economic interpretation for this observation with the

details presented in Section V.A.2.

10 Deltas are normally quoted in absolute percentage terms. For example, ten-delta options correspondto c = 0:1 for calls and p = 0:1 for puts. Hence, deltas for both calls and puts decrease as options goout of the money. In particular, both deltas approach to zeros as K goes to innity for calls and to 0 forputs.

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D. Aggregate equity, index option, and bonds

Following Abel (1999) and Campbell (2003), I model the aggregate dividend as the levered

consumption, i.e., D = C: Denote by St the period-t price of the aggregate equity as the

claim to all future dividends. Appendix B.3. shows that the log-linear approximation to

the price-dividend ratio is

IS (t) StDt

= eaD+bDt ; (3.24)

where aD and bD are constants. Under (3.24), the equity premium and the stock return

volatility are

EPt = 2

(

1) bbD

2t + t 1 e

Z eZ 1 ; (3.25)volRt =

q22 + b2D

2t + t (e

Z 1)2; (3.26)

respectively. In (3.25), the rst and the third term are the usual compensations for con-

sumption shocks and the potential jumps; the second term is unique to variable disaster

models which compensates the agent for bearing the risks of disaster intensity shocks.

Similar interpretations apply to (3.26).

Like currency options, index options written on the aggregate equity can also be valued

in semi-closed form, which is summarized by the following proposition.

Proposition 2 Conditional on the approximate analytical solution (3.24) for the price

dividend ratio, the period-t prices of a call and a put option written on the aggregate stock

with strike price K and time to maturity are

CallSt (K; ) = Gd;d( ln K; t; T) KG0;d( ln K; t; T); (3.27)

P utSt (K; ) = KG0;d(ln K; t; T) Gd;d(ln K; t; T); (3.28)

respectively, where T = t + ; d = [1; 0]0 ; Ga;b(:) is dened in (3.17), where the (:)

function is given by:

(:; t; T) = exp [aC () + bC;Y () Yt + bC;h () t] ; (3.29)

where aC (:) and bC (:) satisfy a series of ODEs presented in Appendix B.3:

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Following the industry convention, I quote index option prices in terms of the implied

Black-Scholes volatility (B/S-vol). When calculating B/S-vol from both the model and the

data, I x the interest rate and the payout ratio as the two arguments for the inversion ofB/S formula at 5% and 3%, respectively. Finally, the present setup permits closed form

valuation for (default-free) nominal bonds. Specically, the periodt price of a periodnominal bond at home equals

PB;$t;t+ = eaB()+bB()t+cB()t; (3.30)

where aB (:), bB (:) ; and cB (:) satisfy a series of ODEs given by (A.22)(A.24).

IV Data and calibration

A. Data

I obtain over-the-counter quotes on currency options from Bloomberg for two currency

pairs: USDJPY and USDGBP, where the sample period is from October 1, 2003 to May

20, 2011. The underlying exchange rates are the dollar price of JPY and the dollar price

of GBP which are also taken from Bloomberg. Options on each pair have seven xed

time to maturities at one, two, three, six, nine, 12, and 18 months, and I ignore the one-

week quotes out of the concern that short-dated contracts are more subject to liquidity

or supply/demand premium (e.g., Pan and Singleton, 2008). At each maturity, quotes

are available at ve deltas in the form of delta-neutral straddle implied volatilities, 10-

and 25-delta risk reversals, and 10- and 25-delta buttery spreads, which are described in

Section III.C There are a total of 60,685 option quotes.

From the above quotes, we can derive the implied volatilities at the ve levels of delta

using (3.22)(3.23) as follows: i) IVc [25] = BF [25]+ IV AT M+ RR [25] =2; ii) IVp [25] =

BF [25] + IV AT M RR [25] =2; iii) IVc [10] = BF [10] + IV AT M + RR [10] =2; andiv) IV

p

[10] = BF [25] + IV AT M RR [10] =2: To back out the time series variations ofdisaster rates implied from the model, we need to convert the implied volatilities into option

prices and the deltas into strike prices using the domestic and foreign interest rates. The

maturity-matched interest rates are stripped using LIBOR and swap rates from Bloomberg

for all the three currencies involved, assuming piece-wise constant forward rates.

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For equity index option data, I obtain the daily S&P 500 index option quotes from

OptionMetrics at four moneyness (0.9, 0.95, 1, 1.05) and three time to maturities (1m,

3m, 6m). The period is from January 4, 1996 to October 29, 2010, and I apply the usualexclusion lters to choose option observations used in this paper. All other data (e.g.,

consumption, CPI, and population) are from the FRED database at the St. Louis.

B. Calibration

Table I reports the base case calibration of the model at the annual frequency. Panel A

describes the preference parameters. Consistent with the macro nance literature, I set the

subjective time discount rate to 0.02. The degree of risk aversion is xed at 6 which is

well within the range considered to be reasonable (e.g., Mehra and Prescott, 1985). I use = 2 which is close to the estimation by Attanasio and Weber (1989), Bansal, Gallant,

and Tauchen (2007), and Bansal, Tallarini, and Yaron (2006).

In the model, and denote the average and the volatility of the aggregate consump-

tion growth conditional on no disasters occuring. I thus calibrate their annualized values

to the quarterly real consumption divided by the total population data in the U.S. for

the period between 1952Q2 and 2006Q4, during which no economic disasters are docu-

mented. The correlation of the home and the foreign consumption growths, hfC ; is set

to 0.3, which is roughly the midpoint of its empirical range from 0.17 to 0.42 reported in

Brandt, Cochrane, and Santa-Clara (2006, Table 4). The leverage parameter is set at

1.8, which is conservative compared to the high ratio of dividend volatility to consumption

volatility. These numbers are reported in Panel B of Table I.

Parameters related to disasters are presented in Panel C. I set the mean-reversion para-

meter and the volatility parameter in the tprocess at 0.142 and 0.09, respectively.11Using the postwar data in G7 countries, Barro (2006) reports an average disaster probabil-

ity of 0.017 per annum, to which the long-run disaster intensity is calibrated. From the

U.S. experience, the largest consumption drop (1932) is about 10%. Using the S&P 500

index option data for 19882008, Du (2011) estimates a 15.8% consumption jump in the11 Since t is the only state in the one-country version of the model, the autocorrelation of price-dividend

for aggregate equity is approximately the autocorrelation of t: Setting =0.142 generates an autocorre-lation for the price-dividend ratio at its data value of 0.87 (e.g., Lettau and Wachter, 2011). The choiceof enables the model to generate a reasonable level of volatility in stock returns. Section V.C providesrobustness check regarding the sensitivity of model implications to both and :

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future that is factored into option pricing. Consumption jump sizes tend to be much higher

from international experiences12 which are more relevant to the analysis in this paper. In

view of these evidences, I choose, on balance, a calibration of 20% consumption jump size,i.e., eZ1 = 0:2: The last disaster parameter, g=; is set at 0.98, which implies that themajority of economic disasters are shared across borders. This calibration ensures that the

model simultaneously matches the magnitudes of ATM-implied volatility and stochastic

skewness for currency options with the detailed discussions in Section V.A.1A.2.

Finally, Panel D of Table I reports parameterization on the nominal side which are

estimated using Kalman lter based on the US ination data from 1952Q1 to 2008Q4. 13

These numbers are close to that used by Lettau and Wachter (2010), and Bansal and

Shaliastovich (2010). It is worth noting that none of the model parameters, except for

g=; are chosen to match currency option data, whose pricing is largely out of the sample.

V Results

Section V.A discusses the pricing of currency options which is the focus of the paper.

Section V.B presents model implications about various other assets, including the UIP

anomaly for currency pricing, the aggregate equity behavior, volatility smirk for index

options, and the moments and the times series for bond yields. Section V.C provides

sensitivity analysis for the key matches of the model.

A. Pricing of currency options

I rst report the low levels of ATM implied volatility denoted by ATMV. I then study the

model implications about cross sectional variations in currency option prices.

12 For example, Barro (2006) reports that both Germany and Japan experienced delines of more than50% in real GDP per capita over the two-year period towards the end of World War II. Using a noveldataset, Barro and Ursua (2009) show that the results on consumption are similar. Heston and Summers(1991) report that several countries have experienced a one-year decline in real GDP or consumption of

more than 20% since 1950, which include Algeria, Angola, Chad, Iran, Iraq, Namibia, Nicaragua, Niger,Nigeria, Sierra Leone, and Uganda.

13 To apply the Kalman lter, I need to discretize (3.7)(3.8). The implied state and observation equa-

tions are

pt+1t+1

=

12

2P

+

0 10 1

ptt

+

P"t+1"t+1

and pt = (1; 0)

ptt

;

respectively, where pt log(Pt) ; f"tg are i.i.d. standard normal series. At the quarterly fre-quency, the estimates of (; ; ; P) are(0:886%; 0:124; 0:272%; 0:652%) with standard errors being(0:180%; 0:0423; 0:0382%; 0:0288%) : Parameter values reported in Panel D of Table I are annualized.

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A.1. ATM implied volatility

Column 25 of Panel A&B in Table II report the model- and the data-implied average

ATMV at the four time to maturities: 1m, 3m, 6m, and 12m. To obtain the model values,

I compute ATMV using the maturity-matched home and foreign interest rates implied

from the model, and then take the averages of ATMV over the stationary distributions ofht ;

ft ;

gt

: Using the equilibrium values of interest rates recognizes the result that bond

yields implied from the model match well their data counterparts with details presented in

Section V.B.3. Since currency option prices are very insensitive to uctuations of expected

ination, to relieve the computation burden I x both and at their long run average

in the following calculations unless indicated otherwise.

The model-implied ATMV is low with a at term structure which is consistent with thedata. The low ATMV is attributed to the smooth exchange rate process, as can be seen

from the last columns of Panel A&B which report the exchange rate volatilities implied

from both the model and the data. In contrast, previous general equilibrium models tend

to generete a highly volatile exchange rate process. Indeed, Brandt, Cochrane, and Santa-

Clara (2006) report the the implied exchange rate volatility from traditional consumption-

based asset pricing model (e.g., chapter 1 of Cochrane, 2005) is at least 50% per year. 14

The low volatility for exchange rates and hence for ATM options generated in the present

setup thus lays the foundation for explaining the cross sectional variations in currency

option prices.

To understand the model mechanism for the low volatility, I calculate from (3.10) the

instantaneous variance of changes in log nominal exchange rate as follows:

V art

d ln

e$t

=dt = 2 ()2

1 hfC

+

( 1)2 (b)2 + (Z)2

ht + ft

+ 22P;

(5.1)

where Ive used the independences between dBht and dBft; and between dNht and dN

ft :

14 Using the two-country version of the traditional consumption-based asset pricing model, real exchangerate determined by (3.1) evolves according to d ln (et) = d ln

t

d ln t = (dBct

dBct) ;where ; ;

Bct; and Bct denote, respectively, the risk aversion, the standard deviation of the consumption growth rate,

and the two standard Brownians that drive the home and the foreign consumption shocks. The instanta-

neous variance of the change in the log exchange rate is thus V ar (d ln (et)) =dt = 2 ()2

1 hfC

;where

hfC is the correlation between the two consumption shocks. To see the implied high exchange rate volatil-ity, rst notice that the cross-border risk sharing of consumption shocks is low (around 0.3). Second, since

is low, a high is needed to replicate the high equity premium, which makes ()2

high.

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(5.1) consists of three terms which are due to the imperfectly shared consumption shocks,

the imperfectly shared economic disasters, and nominal shocks, respectively. Empirically,

volatilities of real and nominal exchange rates are very close implying that nominal shocksare small. Due to risk compensations for variable disaster rates, the present setup can

generate the observed high equity premium under low risk aversion. This result, combined

with the low ; implies the low volatility attributed to consumption shocks. If arrivals of

economic disasters are highly shared across borders, i.e., ht + ft is close to 0, the second

terms contribution is also small, hence the low exchange rate volatility.

The above mechanism can be further illustrated with the risk sharing index (RSI)

dened by

RSI 1 V ar (d ln t d ln t )V ar (d ln t) + V ar (d ln t )

= 1 V ar (d ln et)V ar (d ln t) + V ar (d ln t )

; (5.2)

(5.2) is introduced by Brandt, Cochrane, and Santa-Clara (2005), where the numerator

measures the dierence in marginal utility growth between the two countries, i.e., the

amount of risk that is not shared; the denominator measures the volatility of the marginal

utility growths in the two countries, i.e., the amount of risk to be shared. At the long-run

averages of the disaster rates, the implied RSI in my model equals:

1 ()2

1 hf

C

+

( 1)2

(b)

2

+ (Z)

2

h

+

2

P

()2 +

( 1)2 (b)2 + (Z)2

+ 2P; (5.3)

where Ive set h

= f

: In the standard consumption-based model without the disaster

components and ination, this index degenerates to hfC ; the cross-border correlation of

consumption growth rates, which tends to be much lower than that suggested by asset

market data (e.g., Brandt, Cochrane, and Santa-Clara, 2005). If disaster risks are highly

shared, i.e., >> h

; and since the present setup permits a low for equity pricing, the

risk sharing index computed by (5.3) can be substantially higher than hfC : In particular,

when there are no consumption shocks; (5.3) becomes

1 h

+ 2P + 2P

: (5.4)

Given that 2P is small, (5.4) is close to one when disaster risks are largely shared across

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borders. Quantitatively, Panel C of Table II reports both the model- and the data-implied

RSI which shows a good match.

A.2. Stochastic skewness

Starting from ATMV, option implied volatilities vary as we change moneyness, or equiv-

alently, option deltas. In particular, variations in pricing between OTM puts and OTM

calls are captured by changes in risk reversals (RR) dened in (3.22), which reects the

skewness of the risk-neutral exchange rate distribution. Since the positive (negative) risk

reversals imply the positive (negative) skewness,15 I use interchangeably variations of risk

reversals and stochastic skewness thereafter.

Panel A of Table III reports the model- and the data-implied standard deviation for the10- and the 25-delta risk reversals, denoted by RR10 and RR25, respectively, over various

time to maturities, where RR are normalized as percentages of the maturity-matched

ATMV. To obtain the model values, I compute standard deviations over the stationary

distributions of

ht ; ft ;

gt

: Conditional on the given disaster rates, I rst search for the

strike, using (3.20)(3.21), to match the option deltas, where the implied volatility, IV; is

backed out of the model-implied prices given by (3.15)(3.16). I then use (3.15)(3.19) and

(3.22) to compute RR at the given deltas, where the home and the foreign interest rates

are at their maturity-matched equilibrium values.

The model does a good job in capturing the substantial variations of risk-neutral skew-

ness observed in the data. Take the three-month risk reversals for example. The model-

implied standard deviation of RR10 is 19.6% which is very close to the observed 19.1% for

options written on USDJPY. Turning to RR25, the implied standard deviation is 7.67% in

the model as compared to 9.34% and 6.62% for options written on USDJPY and USDGBP,

respectively. Except for the 18m contracts, the model generates a rising term structure for

the RR standard deviations which is strongly supported in the data. Intuitively, within

certain range, the longer time to maturity gives more chances for the greater uctuations of

the variable disaster rates, which enhances the variations of the risk-neutral skewness. Forlong enough contract horizon, however, this eect is oset by the mean reversion nature of

the disaste rates which produces the lower RR standard deviation. The latter prediction

15 Contrary to this paper, Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2010) dene risk reversalas the price dierence between OTM puts and OTM calls. Under their denition, risk reversals move inopposite directions to the skewness of currency returns.

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is also observed in the data as with options written on USDGBP.

Panel B of the same table reports the ranges of RR variations. The implied variations

are largely symmetric in the model which is expected. In the data, on the other hand, weobserve more asymmetry towards the negative skewness implying, on average, higher ht

than ft : as explained in Section III.C, a positive shock on home disaster rate adds to the

negative risk reversal. The reason is that our sample period is over-represented, relative

to that implied from the stationary distribution of

ht ; ft ;

gt

, by the high realizations of

ht which is due to the recent nancial crisis originated from the U.S. Indeed, using data

from the earlier sample period of 19972004 before the nancial crisis, Carr and Wu (2007)

report much more symmetric ranges for RR variations (based on the three-month ten-delta

risk reversals, they report -30%60% for USDJPY and -22%22% for USDGBP). Focusing

on the size of the range measured as the maximum minus the minimum, numbers implied

from the model are still close to their data counterparts.

To study the implied stochastic skewness in more details, I rst back out the time series

of t

= ht + gt

using the index option data which is described in Section V.B.2. The

reason is that currency option pricing is not sensitive to gt ; while t is the only state for

index option pricing in the present setup. Given the backed outn

to

and the currency

pair, I then use the daily currency option prices to back out ht and ft on each of the trading

days within the common sample period for the index and the currency options, which is

from October 1, 2003 to October 29, 2010. More specically, let Nt be the number of optionprices on day t,16 and O

(d)n (t; n; Kn) and O

(m)n (t; n; Kn; ;

ht ;

ft ;

gt ) be, respectively, the

observed and the model price of the nth option (n = 1; 2;:::Nt), where n and Kn denote

the time to maturity and strike price; denote the calibrated parameters reported in Table

I. Day ts disaster rates are identied by minimizing the sum of the squared pricing errors

as follows:

h

t ; f

t ; g

t= arg min

ht ;ft ;

gt

Nt

Xn=1 hO(d)n (t; n; Mn) O(m)n (t; n; Mn; ; ht ; ft ; gt )i

2

; (5.5)

where I use h

t + g

t = t. To avoid over-weighting for high price quotes, I scale all op-

16 For most of the trading days, I have option quotes for seven xed time to maturities and ve strikes(converted from the ve deltas), hence Nt = 35: There are trading days when some quotes are missing. Inthose cases, Nt < 35:

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tion prices by their Garman-Kohlhagen vega. The estimation results are summarized by

fh;JPYt ; f;JPY

t ; g;JPY

t g and fh;GBP

t ; f;GBP

t ; g;GBP

t g; where "JPY" and "GBP" in the su-perscripts denote estimations based on option quotes for the two currency pairs: USDJPYand USDGBP.

With the estimated fht ; f

t ; g

tg as inputs, I compute the model-implied RR series,which are plotted in the top two panels of Figure 1 together with the corresponding data

values, for USDJPY and USDGBP, respectively. Specically, I use the three-month ten-

delta risk reversal as the example and choose the weekly frequency which is more clear

for comparison. In general, the model- and the data-implied RR match each other well

with correlations being 0.93 and 0.95 for the two currency pairs. Consistent with empirical

evidences, risk reversals from the model vary greatly over time for both USDJPY and

USDGBP, with standard deviations being 14.9% and 13.6% as compared to 19.5% and

12.9% in the data, which change signs several times over the sample period.17 While

the pricing errors appear large during certain periods, it is worth noting, from the above

estimation procedure, that the model is also required to capture the time series variations

for index options.

Whats the model mechanism that generates the stochastic skewness? First, investors

are concerned about the prospects of future consumption growths under the recursive

utility. Variable disaster intensities are therefore directly priced through which they have

impacts on exchange rate dynamics. Second, under the imperfectly shared disaster risks,home disaster rate can either rise above or fall below its foreign counterpart generating,

respectively, the negative and the positive skewness in exchange rates. Third, since the

two country-specic components evolve independently, their relative weights change over

time leading to the substantial variations in option risk reversals. To provide the graphical

illustration of the mechanism, I calculate the implied time series in a degenerated model

assuming perfect sharing of disaster risks. While the implied risk reversals still vary over

time due to changes in gt ; the bottom two panels of Figure 1 show that the degenerated

model fails miserably in capturing the observed uctuations.

The stochastic skewness suggested by currency option quotes is rst addressed by Carr

and Wu (2007) using a class of reduced-form models built on the general framework of time-

changed Levy processes developed in their earlier work, i.e., Carr and Wu (2004). Unlike

17 Changes in sign are mostly observed before 2007. Following the outbreak of the recent nancial crisis,risk reversals are mostly negative which is commented previously.

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the present setup where stochastic skewnewss arises endogenously from imperfectly shared

disaster risks, Carr and Wu (2007) exogenously impose two Levy-jump components in

the exchange rate process which exhibit positive (upward jumps) and negative (downwardjumps) skewness, respectively. Stochastic skewness emerges when the two jump compo-

nents, and hence their relative weights, vary stochastically over time. Despite the success,

Carr and Wu (2007) point out that "for future research, it is important to understand the

economic underpinnings of the stochastic skewness suggested by currency option prices".

My model represents the rst step in that direction.

As another important dierence between the present model and Carr and Wu (2007),

jumps in their setup occur innite number of times within any nite time interval.18 As

a result, stochastic skewness in their model arises from the actual strike of small jumps.

In contrast, jumps are rare in my model, and variations in risk reversal are mainly driven

by variable disaster rate at home relative to that abroad. In other words, the present

model advocates that it is the concern about its potential arrival, not the jump itself,

that matters the most for the observed pricing. The same mechansim applies to index

options for which the variations in t captures well the time series variations of the smirk

premium (see discussions in Section V.B.2). My model thus unies the pricing in various

derivative markets by formalizing the intuition that the changing sentiment toward rare

but disastrous events, interpreted as investors fear gauge, has profound impacts on the

derivative prices.In a related paper, Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2010, FFGRV)

consider a dierent disaster framework to study the disaster risk premium implied in the

carry trade excess return. Still built within the reduced-form setting, their model starts

with the home and the foreign pricing kernel processes, both of which are subject to small-

probability jumps. Unlike my setup in which disasters at dierent countries have symmetric

magnitudes but imperfectly shared arrival rates, FFGRV model disasters as world events,

but their magnitudes are allowed to vary across countries. Instead of explaining currency

option pricing, the main nding in their paper is to identify, with the help of currency

option data, that 25% of excess currency returns are due to the potential disasters.

18 Such jumps are referred to as innite activity jumps. In contrast, the usual Possion jump exhibit niteactivity, i.e., nite number of jumps during any time interval. For detailed dierentiations, see Section 4of Carr and Wu (2004).

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A.3. Other implications

Before addressing other implications related to currency options, it is worth pointing out

that risk reversals in my setup can be proxied by ft ht : To see it, I x the global intensitygt at its long-run average

gand plot in Figure 2 the implied three-month ten-delta RR as a

function of the home-

ht

and the foreign-specic

ft

disaster intensities, where ht and

ft are both quoted in terms of h

and vary between zero to 5h

: Risk reversal is virtually

zero when there are no jumps in the exchange rate process at ht = ft = 0 (the rightmost

point). Starting from ht = ft = 0; risk reversal drops monotonically with the increase

in ht ; since higher ht adds to the negative skewness of currency returns by raising the

likelihood of the potential downward jumps. Conversely, risk reversal rises monotonically

with the increase of ft for the opposite reason. By comparison, RR [10] changes littlealong the line ht =

ft and is still close to zero at

ht =

ft = 5

h(the leftmost point): This

exercise rearms the importance of country-specic disaster component for generating risk

reversal variations.

Empirically, both Carr and Wu (2007) and FFGRV document a strong positive cor-

relation between changes in risk reversals (after adjustment for dierent denitions) and

the contemporaneous nominal currency returns, and FFGRV show that this relation also

holds for real exchange rates. To see this result analytically from the lens of my model,

note that the change of risk reversal, which is proxied by the change in ft

ht , equals

d

ft ht

=

ft ht

dt +

qft dBft

qht dBht

; (5.6)

where Ive used (2.5)(2.6). Comparing (5.6) with (3.10), the positive correlation between

risk reversal innovation and innovation in log exchange rate follows from < 0 and b < 0

under the preference that and are both greater than one. Intuitively, the decrease of

ft and the increase of ht both add to the potential downward jumps of the underlying

foreign curreny, which leads to its contemporaneous devaluation.

Another observation reported in Carr and Wu (2007) and FFGRV is the positive corre-lation between risk reversals and (nominal) interest rate dierentials, which is also apparent

from the lens of the present setup. Other things equal, a positive risk reversal, which is

associated with ht > ft ; implies that precautionary saving is stronger abroad than at

home. This eect leads to the positive r$t r$;t provided that ination dierential is not

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too large.

While risk reversals capture cross sectional variations between OTM puts and OTM

calls, buttery spreads, as dened in (3.23), capture dierences between OTM and ATMoptions. Panel A&B of Table IV report the model- and the data-implied average and

standard deviations for the 10- and the 25-delta buttery spreads, denoted by BF[10]

and BF[25], respectively, over various time to maturities, where the BFs are normalized

as percentages of the maturity-matched ATMV. Overall, the matches are good. Take

the three-month ten-delta buttery spreads for example. The model-implied average and

standard deviation of RR[10] are (6.60%, 3.27%), as compared to (11.9%, 3.27%) for US-

DJPY and (9.18%, 3.17%) for GBPUSD. Simultaneously, the model generates the rising

term structure for both the average and the standard deviations of the BF which is also

supported by the data.

B. Pricing of other assets

B.1. UIP anomaly for currency pricing

A model for derivatives is not a successful one unless it can also capture salient features

of the underlying asset. In this subsection, I show that the present setup explains the key

empirical feature for currency pricing: the UIP anomaly. To see the details, denote by

ret the expected foreign currency premium which is dened in (3.5). If ret is a constant,(3.5) implies a one-to-one relation between rt rt and Et (d ln et) =dt which is sometimesreferred to as the uncovered interest rate parity (UIP). If UIP holds, the slope coecient

in the following regression

ln et+1 ln et = a1 + a2 (rt rt ) + residual

should be close to one. Since Fama (1984), however, people have found that a2; referred

to as the UIP coecient, is consistently less than one and usually negative: high interest-

paying currencies tend to appreciate instead of depreciating, which indicates that thecurrency risk premium, ret ; is time-varying.

The model mechanism to resolve the above UIP anomaly is discussed in Section III.B.

Panel A of Table V reports the quantitative implications, where the UIP regressions are

run at the monthly frequency based on the simulated data of nominal exchange rates and

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nominal yields of bonds with one month to expiration.19 The implied UIP coecient is

-2.0 with the standard deviation being 1.2. These numbers are in line with their empirical

values documented in previous studies (e.g., Bansal and Shaliastovich 2009; Verdelhan,2011). The basic intuition is that the home agent requires a positive premium to hold

the foreign currency when disaster is more likely at home than abroad. When the risk

aversion is not too low, the required premium will be able to oset the eect of interest

rate dierentials and in addition drives up the appreciation of the foreign currency.20

Building on the work of Campbell and Cochrane (1999) and Case II of Bansal and

Yaron (2004), respectively, Verdelhan (2010) and Bansal and Shaliastovich (2010) show

that habit formation and long-run risk models are also able to resolve the forward premium

anomaly. Time-varying currency risk premiums arise from variable risk aversions induced

by habit formation according to Verdelhan (2010), and from the short-term uctuations of

consumption volatility according to Bansal and Shaliastovich (2010). In this paper, I resort

to the disaster story instead because I aim at pricing currency derivatives. As suggested by

Benzoni, Collin-Dufresne, and Goldsterin (2011) and Du (2011), neither habit formation

nor long-run risk models price derivatives well if the underlying asset returns are not subject

to jumps. More evidences about the importance of a disaster component to currency option

pricing are provided in Section VI.

B.2. Aggregate equity, index options, and bonds

Panel BD of Table V present the key moments characterizing behaviors of the aggregate

equity, the equity index options, and the short term (default-free) bonds. Both the model

values and their usual empirical value ranges are reported. The model-implied average

price-dividend ratio, average equity premium, and volatility of the aggregate equity returns

are 35, 6.5%, and 14.2%, respectively, which are all line with their usual data estimates.

Regarding the pricing of equity index options which is presented in Panel C, B/S-vol from

ATM options, referred to as ATMV thereafter, are close to but higher than the stock return

19 In particular, I rst compute the yield dierentials of nominal bonds with period to expiration,which are given by y$t;t+ y$;t;t+ = 1

hbB ()

ht ft

+ cB () (t t )

i; where Ive used (A.21)

and the perfect symmetry; bB (:) and cB (:) are given by (A.23)(A.24). As expected, y$t;t+ and y

$;t;t+

approaches to the short term nominal rates r$t and r$;t when ! 0. I then regress the monthly changes

in log exchange rates onto last months bond yield dierentials.20 Quantitatively, I nd that, with all other parameters xed at their base case levels, UIP coecient

turns negative once the degree of risk aversion exceeds 2.2.

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volatility in both the model and the data. The smirk premium, measured as the dierence

in B/S-vol between 10% OTM puts and ATMs, are on average 9.7% in the model. This is

consistent with the 810% smirk premium implied from the data that has been documentedsince the 1987 market crash (e.g., Benzoni, Collin-Dufresne, and Goldstein (2010)).

Within the one-country setup, variations of index option prices in the present model

are exclusively driven by home disaster rate t for the given stock price. To test this

prediction, I back out the model-implied time series of t using the index option data over

the sample period from January 4, 1996 to October 29, 2010. More specically, let Nt

be the number of option prices on day t, and O(d)n (t; n; Kn) and O

(m)n (t; n; Kn; ; t) be,

respectively, the observed and the model price of the nth option (n = 1; 2;:::Nt), where n

and Kn denote the time to maturity and strike price; denote the calibrated parameters.

Day ts disaster rate is identied by minimizing the sum of the squared pricing errors as

follows:

t = arg mint

NtXn=1

hO(d)n (t; n; Mn) O(m)n (t; n; Mn; ; t)

i2: (5.7)

The top panels of Figure 3 plot together the model- and the data-implied time series of

ATMV and smirk premium for 1m options, where the model values are calculated with the

backed-out ftg as the inputs. To facilitate the comparison with Figure 1, I focus on thesample period from October 1, 2003 to October 29, 2010.

The model captures a large fraction of variations in both ATMV and smirk premiumwith correlations between the model- and the data-implied series being 0.96 and 0.42,

respectively. In the model, as in the data, ATMV rises monotonically in t: For example,

during the peak of nancial crisis in October 2008, the implied t rises dramatically which is

associated with very high levels of ATMV from both the model and the data. The monotone

relation between ATMV and t is intuitive, since empirically ATMV for index options has

been called the "investor fear gauge". From the lens of the present model, the degree of fear

is captured by the assessment about the arrival rate of economic disasters. In contrast,

raising disaster rate has two osetting eects on the implied smirk premium. First, it

increases the likelihood of potential disasters which gives the agent stronger incentives

to hedge, hence the higher option premium for the given diusive stock volatility sPt.

Second, it raises sPt which depresses the relative importance of jumps in the total stock

return variations. As a result, the agent has less incentives to buy insurance from OTM

puts which drives down the smirk premium for the given t. The second eect dominates

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leading to the decreasing smirk premium in t. Taken together, the model predicts a rising

smirk premium as ATMV decreases. The data largely supports this prediction (e.g., Du

(2011)).Moments about short-term real interest rates are reported in Panel D of Table V.

Consistent with the data, the implied mean and the standard deviation are both low, and

the rst autocorrelation is high. Simultaneously, the model generates a fairly at and

slightly downward-sloping yield curve for real bonds plotted in the bottom left panel of

the Figure 3, which is also consistent with empirical ndings (e.g., Ang, Bekaert, and Wei,

2008; Verdelhan, 2010). The bottom right panel of the same gure plots the time series,

from January 1996 to October 2010, for the short-term interest rates, where the model

values are obtained from (2.16)(3.11) with

ft

gas the inputs and the expected ination

t xed at its long-run average; the data values are proxied by the 3m LIBOR rates for

USD. Driven only by disaster rate shocks, the model-implied series capture 32% of the

variations in the data. For example, during 20052006, the estimated t decreases in a

persistent way, as evidenced by the decreasing ATMV for index options. The model thus

implies the gradual increases of interest rates due to the less precautionary saving, which

is supported by the data.

C. Sensitive analysis and further discussions

So far, all reported matches are based on the base case calibration presented in Table

I. Since the key for the models success lies in the time variations of disaster intensities,

an important question remains: how robust are the models pricing implications to the

dierent calibration of the variable disaster rates? To answer it, I focus on the risk reversal

variations which is the key match of this paper compared to the previous literature.21

Specically, I vary the three parameters, ; ; and ; which control the evolution of

disaster rates at both home and abroad, by 30% relative to its base case level in both

directions. In addition, I consider the impact of variations in g= which governs the

degree of disaster risk sharing: Figure 4 plots the implied standard deviations of the ten-delta risk reversals across various time to maturities when we change each of the four

parameters separately. For comparison, I plot together the base case implications.

21 While not reported, I nd that the models other implications, such as the resolution of UIP anomaly,the high equity premium and the high smirk premium, all survive moderate variations of disaster-relatedparameters.

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The top two panels plot the impacts of and controlling, respectively, the volatility

and the mean reversion of the disaster rate. As expected, increasing (decreasing) raises

(reduces) the standard deviation of risk reversals relative to their base case levels. Therelation is nonlinear in that lower reduces the RR variations more eectively, but the

implied stochastic skewness remains sizable even at the 30% reduction in : In contrast,

RR variations rise for both the higher and the lower : The rst result is intuitive since

the high mean reversion depresses extreme realizations of t: On the other hand, the high

persistence associated with very low tends to generate concentrated realizations of t;

which also depresses tvariations.Compared to the literature, Wachter (2011) uses a dierent calibration by setting

(; ) = (0:080; 0:067) ; both of which are lower than their base case levels in the present

paper.22 Using her calibration, I nd that the models implications about RR variations is

almost unchanged, and the above sensitivity analysis justies this result. In summary, the

implied stochastic skewness is robust to a wide range of choices about and .

The bottom left panel plots the impact of : As expected, RR standard deviations

increase (decrease) when the economic disasters become more (less) likely, and the mag-

nitudes of the impact are largely symmetric. To study the impact of g=; I consider

two values: 0.999 and 0.96 which are higher and lower, respectively, than the implied risk

sharing at the base case level (=0.98). The rst choice essentially shuts down the country-

specic disaster components. As a result, the implied currency return skewness becomesrather stable which is counterfactual. In contrast, lowering the degree of risk sharing sub-

stantially raises the implied RR variations. The bottom right panel of Figure 4 plots the

results.

The above exercise rearms that the key for matching the observed stochastic skewness

lies in the imperfectly shared disaster components to which exchange rates are subject.

Given the rare nature of economic disaster, which happens only once every 59 years in my

calibration, detecting a small fraction of its occurrence is dicult. Since option prices are

very sensitive to jumps, however, even a small fraction of the peso problem component

is able to produce substantial variations in risk reversals. Indeed, as will be shown in

next section, a stochastic volatility model generates little RR variations even though the

diusive risks at home and abroad are not shared at all. Therefore, while it is hard to

22 In the earlier version of her paper, Wachter (2011) uses the same calibration of (; ) as reported inmy Table I.

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distinguish econometrically my model from a model featuring 100% global disasters, the

two setups have very dierent pricing implications about currency options.

Overall, the 98% risk sharing at the base case implies that economic disasters are verylikely to occur simultaneously in dierent countries, which is consistent with the historical

evidence of the Great Depression. Given the increasing economic ties among countries,

this implication seems more valid nowdays than ever before, at least for major economies.

To understand the source of the un-shared components, it is helpful to interprete country-

specic disaster as a broader concept than "a disaster that strikes one country but not the

other". For example, Table I in Barro (2006) reveals that the disaster associated World War

II struck major economies at dierent timing. It is thus arguable, at least qualitatively,

that this disaster is not perfect shared across borders. My work demonstrates that the

imperfect risk sharing is vital to the understanding of currency option pricing.

VI Stochastic volatility and currency option pricing

To further emphasize the importance of disaster components to currency option pricing,

I investigate in this section the implications of the a stochastic volatility model under

the long-run risk framework of BY. Similar model is considered previously by Bansal and

Shaliastovich (2010) to study the UIP anomaly.

A. The setup

I use the home country as the example to present the model. Aggregate consumption

follows:dCtCt

= ( + xt) dt +p

tdBct;

where

dxt = xxtdt + xdBxt;

dt =

t dt + ptdBt;Bct; Bt; and Bxt are mutually independent. Under the recursive utility of (2.10)(2.11), xt

and t are both directly priced which capture the long-run risk and the stochastic volatility,

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respectively. As a result, the home pricing kernel t takes the form of

t = expZ

t

0

+

1

I(s; xs)

ds

Ct I(t; xt)

1

: (6.1)

Using the approximation

I(t; xt) = ea+bt+cxt ;

t follows

dtt

= rtdt p

tdBct + ( 1) bp

tdBt + ( 1) cxdBxt;

where the short-term rate is

rt = + ( + xt) 12

(1 + ) t +1

2( 1) b22t + c22x :

On the nominal side, again I specify the process for consumption price index Pt by (3.7)

(3.8).

B. Pricing implications

As usual, I impose perfect symmetry between the home and the foreign country. Following

Bansal and Shaliastovich (2010), I further assume that i) the long-run growth prospects

across two countries are identical in that xt is perfectly shared; ii) there are short-run dier-

ences captured by the independent evolutions of consumption volatility between countries;

iii) Bct and Bct are independent. By Itos lemma with jumps, the implied log nominal

exchange rate follows:

d ln

e$t

=

(t t )

1

2( 1) (b)2 + 1

2( 1 )

+ t t

dt

+ptdBct ptdB

ct ( 1) b ptdBt p

tdB

t + P dBpt dB

where innovations due to imperfectly shared risks are driven by p

tdBct p

tdBct

for consumption shocks, by ( 1) b

ptdBt

ptdB

t

for stochastic volatility,

and by P

dBpt dBpt

for nominal shocks. By applying similar procedures as those

for the variable disaster model, I obtain the semi-closed form valuations of the currency

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options given by (3.15)(3.17) with the closed form of the function changed to

(u; t; T) = exp"

aC () + bC;Y () ln et + bC; () t + bC;

()

t

+bC;x () xt + bC; () t + bC; () t

#; (6.2)

where et denotes the real exchange rate; aC (:) and bC (:) satisfy a series of ODEs. Note xt

shows up in (6.2) through the pricing kernel.

Again, the above long-run risk model is calibrated at the annual frequency. To facilitate

comparison, I keep calibration as close to that of BY and Bansal and Shaliastovich (2010)

as possible. Except for hfC , I choose parameters describing the real side of the economy

according to BY after the adjustment for continuous-time specication and dierences in

scaling. Since BYs model is built within the one-country setup and does not considernominal shocks, I calibrate hfC and parameters related to ination according to Bansal

and Shaliastovich (2010).23 Panel AD of Table VI summarize the calibration.

The model replicates Bansal and Shaliastovich (2010)s ndings that long-run risk

model is able to resolve the UIP anomaly (with the UIP coecient equaling -3.7).24 The

diculty comes from currency option pricing which is reported in Panel E of Table VI.

Column 2 shows that ATMV, which is very close to the exchange rate volatility, appears

too high compared to the data. Column 34 reports standard deviations of the 10- and

25-delta risk reversals over various time to maturities. Even though the diusive risks

as in t and t are not shared at all, the implied RR standard deviations are no more

than 0.06% which falls far b

du currency options equilibrium

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