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Insurance: Mathematics and Economics 46 (2010) 461–469

Contents lists available at ScienceDirect

Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Markov-modulated jump–diffusions for currency option pricingI

Lijun Bo a, Yongjin Wang c,d, Xuewei Yang b,d,∗a Department of Mathematics, Xidian University, Xi’an 710071, PR Chinab School of Mathematical Sciences, Nankai University, Tianjin 300071, PR Chinac School of Business, Nankai University, Tianjin 300071, PR Chinad TEDA Institute of Computational Finance, Nankai University, Tianjin 300071, PR China

a r t i c l e i n f o

Article history:Received July 2009Received in revised formDecember 2009Accepted 15 January 2010

JEL classification:C13C15F31

MSC:60H15

Keywords:Spot foreign exchange rateRare eventTime-inhomogeneityEsscher transformCurrency option

a b s t r a c t

This paper introduces dynamic models for the spot foreign exchange rate with capturing both therare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, weuse a compound Poisson process with log-normal jump amplitude to describe the jumps. As for thetime-inhomogeneity in the market dynamics, we particularly stress the strong dependence of thedomestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on thetime-varying sovereign ratings in the currency market. The time-varying ratings are formulated by acontinuous-time finite-stateMarkov chain. Based on such a spot foreign exchange rate dynamics, we thenstudy the pricing of some currency options. Here we will adopt a so-called regime-switching Esschertransform to identify a risk-neutral martingale measure. By determining the regime-switching Esscherparameters we then get an integral expression on the prices of European-style currency options. Finally,numerical illustrations are given.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the currency market, for hedging the fluctuating risk, avariety of currency options have been invented. Among which, arequisite step is to set up a dynamicmodel for the currency fluctua-tions, and then the pricing of the foreign currency optionsmight begiven explicitly or practically. Recall the pioneering works by Bigerand Hull (1983) and Garman and Kohlhangen (1983), in which thedynamics of the spot foreign exchange (FX) rate is modelled by ageometric Brownian motion (GBM) with the constant drift and

I Supported by the LPMC at Nankai University and the Keygrant Project ofChinese Ministry of Education (No. 309009). The research of Bo and Yang is alsosupported by the Fundamental Research Funds for the Central Universities ofChina (Granted by Xidian University, Project Name: The Applications of Markov-Modulated Affine Jump–Diffusion Processes in Derivative Pricing). The authorswould like to thank the Editors and the anonymous referees for their valuablecomments and suggestions on the earlier version of this paper.∗ Corresponding author at: School of Mathematical Sciences, Nankai University,Tianjin 300071, PR China.E-mail addresses: bolijunnk@gmail.com (L. Bo), yjwang@nankai.edu.cn

(Y. Wang), xwyangnk@yahoo.com.cn (X. Yang).

volatility. That initiated the pricing of the currency options. How-ever, currently more and more empirical evidences have revealedthat the GBM benchmark is somehow not completely consistentwith the reality. Fortunately, some significant development hasbeen made recently, for instance,(1) Incorporating rare random shocks into the reference modelis necessary and significant (see, e.g., Aït-Sahalia (2002),Andersen et al. (2002), Carr et al. (2002), Johannes (2004),and Liu et al. (2005));

(2) The time-inhomogeneity should be allowed in the dynamicsmodels (see, e.g., Di Graziano and Rogers (2009), Jobert andRogers (2006), and Elliott and Osakwe (2006)).

In addition, we also note that, in the real spot FX rate market (seeFig. 1: the monthly data1 for USD/JPY spot FX rate),(1) The jumps do exist in the FX rate realizations;(2) The FX rate exhibits different behaviors in different timeperiods (compare the realizations before 1985 with those after1989).

1 We quoted the data from Sauder School of Business, University of BritishColumbia (http://fx.sauder.ubc.ca/data.html).

0167-6687/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2010.01.003

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462 L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469

Fig. 1. Monthly data for USD/JPY spot FX rate from January 1971 to May 2009.

In this paper, in order to incorporate both rare events andtime-inhomogeneity in the spot FX rate market simultaneouslyand parsimoniously, we model the spot FX rate by the so-calledMarkov-modulated jump–diffusions. In which, the rare events aredescribed as a compound Poisson process with log-normal jumpamplitude, and the time-inhomogeneity is captured by lettingthe market parameters2 depend on the sovereign rating, which ismodelled by a continuous-time finite-state Markov chain. In thiscontext, we say ‘‘these market parameters are modulated by thehidden Markov chain’’.The Markov-modulated jump–diffusion models combine two

classes of models for financial dynamics. The first class follows(Merton, 1976) in using compound Poisson processes tomodel rareevents, and (Heston, 1993) inmodelling the stochastic volatility bya Cox–Ingersoll–Ross (CIR) process (see Cox et al. (1985)). Someother known works include Shastri and Wetheyavivorn (1987),Jorion (1988), Melino and Turnbull (1990) and Bates (1996), andso on. The second class follows Elliott et al. (2005) and Elliott andOsakwe (2006) in using Markov-modulated (regime-switching)market parameters to capture the time-inhomogeneity generatedby the financial market. Examples of Markov-modulated modelsinclude the two-factor Markov-modulated stochastic volatilitymodel of Siu et al. (2008), and the Markov-modulated dividendsmodel of Di Graziano and Rogers (2009). The use of Markov-modulated jump–diffusions can extract the advantages of theabove two classes of models, since they generalize both classessimultaneously.Here we are aimed to the pricing of European-style currency

options under the spot FX dynamics modelled by some Markov-modulated jump–diffusions. We propose two models: one isthe Markov-modulated Merton jump–diffusion, and the otherincorporates Heston’s stochastic volatility into the foregoing one.3Specifically, we emphasize the jump risk components (suddenrandom shocks) as the Poisson jumps with Markov-modulatedarrival intensity and suppose that the jump amplitude follows alog-normal distribution (see, e.g., Merton (1976) and Bates (1996)).The choice of the jump structure is based on the following facts:

2 Such as domestic/foreign interest rates, the appreciation rate and the volatilityof the foreign currency, the annual jump intensity of the compound Poisson process,and so on.3 The latter can be viewed as the Markov-modulated version of the Bates (1996)model.

• The empirical studies showed that the jump structure (theconstant jump intensity and the log-normal jump amplitude)is appropriate to model the spot FX rate (see, e.g., Bates (1996),Johnson and Schneewels (1994) and Jorion (1988)).• Under the current assumptions, the jump structure allowsus to get the closed forms for the regime-switching Esscherparameters (see Corollaries 2.1 and 3.1); However we have topoint out that, under a general jump framework (e.g., as in thegeneralizedMarkov-modulated jump–diffusionmodel in Elliottet al. (2007)), the Esscher transform parameters do not admitthe closed forms (see, e.g., Hubaleka and Sgarra (2009)).

In the following, to explore the risk-neutral valuation, weemploy the randomEsscher transform (see, e.g., Elliott et al. (2005),Elliott et al. (2007) and Elliott and Osakwe (2006)) for the Markov-modulated jump–diffusions by decomposing the log spot FX rateinto a continuous diffusive part and a jump part, and then identifytheir respective risk premiums. Under the risk-neutral measure,the jump risk can be formulated by the random Esscher transformintensity and the mean percentage jump size. We then could givethe integral form of the pricing of the European-style currencyoptions. Finally we also give some numerical illustrations, whichwill be implemented by employing the Monte Carlo method.The rest of this paper is organized as follows. In Section 2,

a Markov-modulated Merton (1976) jump–diffusion for the FXrate is proposed and the valuation of the European-style currencyoptions is investigated by using the Esscher transform. Section 3is devoted to the analogue model as in Heston (1993) with thestochastic volatility. Section 4 presents the Monte Carlo methodfor the pricing of the European-style call options. The concludingremarks are contained in Section 5.

2. Markov-modulated Merton jump–diffusion for the FX rate

In this section, we model the spot FX rate dynamics by aMarkov-modulated Merton jump–diffusion, and then we aim toprice the European-style currency options via the so-called randomEsscher transform for the jump–diffusion.

2.1. The model description

Let (Ω,F , P) be a complete probability space, where P is areal-world probability measure. Recall the Merton jump–diffusionmodel for discontinuous stock returns in Merton (1976). Wehere propose a Markov-modulated Merton jump–diffusion formodelling the dynamics of the spot FX rate, which is formally givenby

dStSt−= (αt − kλt)dt + σtdWt + (eZt− − 1)dNt , (2.1)

where the appreciation rate (αt)0≤t≤T , the stochastic volatility(σt)0≤t≤T of the spot FX rate and the stochastic jump intensity(λt)0≤t≤T of the Poisson process N = (Nt)0≤t≤T are all modulatedby a common continuous-time, finite-state Markov chain ξ =(ξt)0≤t≤T as follows (see, e.g., Elliott et al. (2005) and Elliott andOsakwe (2006)):

αt = 〈α, ξt〉 , α = (α1, α2, . . . , αn) ∈ Rn,σt = 〈σ , ξt〉 , σ = (σ1, σ2, . . . , σn) ∈ (0,∞)n,λt = 〈λ, ξt〉 , λ = (λ1, λ2, . . . , λn) ∈ (0,∞)n,

where 〈·, ·〉 denotes the inner product on Rn. In addition, jumparrivals are indicated by the Poisson process N = (Nt)0≤t≤T withthe Markov-modulated intensity (λt)0≤t≤T . If the jump arrives attime t , the jump amplitude is controlled by Zt , which is normallydistributed with mean µJ and standard deviation σJ > 0. As aconsequence, the mean percentage jump of the FX rate is k =

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L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469 463

eµJ+12 σ2J − 1, given jump arrival. Specifically, we assume that for

t 6= s, Zt and Zs are independent, and all of the random shocksprocessesW ,N, ξ , and Z = (Zt)0≤t≤T are assumed to be mutuallyindependent.Throughout this paper, we take the state space of ξ to be a finite

set of unit vectors (e1, e2, . . . , en) with ei = (0, . . . , 1, . . . , 0) ∈Rn. Thus ξ has the following semi-martingale representation (see,e.g., Elliott et al. (2005))dξt = Q ξtdt + dMt , (2.2)where M = (Mt)0≤t≤T is an Rn-valued martingale with respect tothe natural filtration generated by (ξt)0≤t≤T , and Q = (qij)i,j=1,...,nis the Q -matrix of ξ .Next let (rDt )0≤t≤T and (r

Ft )0≤t≤T be respectively the domestic

and foreign instantaneous interest rates, and defined byrDt =

⟨rD, ξt

⟩, rD = (rD1 , r

D2 , . . . , r

Dn ) ∈ (0,∞)

n,

rFt =⟨rF , ξt

⟩, rF = (rF1 , r

F2 , . . . , r

Fn ) ∈ (0,∞)

n.

Furthermore, let Sdt be the spot FX rate at time t discounted by thecurrent value of the domestic money market account. Recall thespot FX rate St in (2.1). Then

Sdt = exp(∫ t

0(rFs − r

Ds )ds

)St , 0 ≤ t ≤ T .

Moreover, by Itô rule with jumps (see, e.g., Protter (1990)), we canget

dSdtSdt−= (rFt − r

Dt + αt − kλt)dt + σtdWt + (e

Zt− − 1)dNt . (2.3)

Here Sd = (Sdt )0≤t≤T will be used to derive the equivalentmartingale condition in the coming subsection.

2.2. Esscher transform for Markov-modulated jump–diffusions

We note that, a market described by the Markov-modulatedjump–diffusion is incomplete, hence the risk-neutral measuredetermined by the so-called martingale condition is not unique.In this subsection, we develop the random Esscher transform (see,e.g., Gerber and Shiu (1994, 1996), Elliott et al. (2005) and Elliottand Osakwe (2006)) for the jump–diffusion (2.1) to identify anequivalent domestic martingale measure.To identify an equivalent domestic martingale measure for the

Markov-modulated jump–diffusion, we decompose the log spot FXrate Yt = log StS0 (0 ≤ t ≤ T ) into a continuous diffusive part and ajump part (see also Elliott et al. (2007)):Yt = Ct + Jt , (2.4)where Ct and Jt are the continuous diffusive part and the jump partof Yt , and they admit the following forms:

Ct =∫ t

0

(αs − kλs −

12σ 2s

)ds+

∫ t

0σsdWs, Jt =

∫ t

0Zs−dNs.

Now let (F ξt )0≤t≤T and (F Y

t )0≤t≤T denote the P-augmentationsof the natural filtrations generated by ξ and Y , respectively. Foreach t ∈ [0, T ], setHt = F Y

t ∨ Fξ

T . Then we can define a randomEsscher transform Q θ

c ,θ J∼ P on Ht with respect to two families

of regime-switching parameters (θ cs )0≤s≤t and (θJs )0≤s≤t by

Lθc ,θ J

t =dQ θ

c ,θ J

dP

∣∣∣∣∣Ht

=:

exp(∫ t0 θ

cs dCs +

∫ t0 θ

Js−dJs

)E[exp

(∫ t0 θ

cs dCs +

∫ t0 θ

Js−dJs

)|F

ξt

] , (2.5)where θmt = 〈θ

m, ξt〉, and θm = (θm1 , θm2 , . . . , θ

mn ) ∈ R

n form ∈ c, J.The following proposition shows a concrete form of the random

Esscher transform density (Lθc ,θ J

t )0≤t≤T .

Proposition 2.1. For 0 ≤ t ≤ T , the random Esscher transformdensity Lθ

c ,θ J

t defined by (2.5) is given by

Lθc ,θ J

t = exp(∫ t

0θ cs σsdWs −

12

∫ t

0(θ cs σs)

2ds)

× exp(∫ t

0θJs−Zs−dNs −

∫ t

0λs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds).

Furthermore, the density process Lθc ,θ J= (Lθ

c ,θ J

t )0≤t≤T is an expo-nential (Ht)0≤t≤T -martingale and admits the following stochasticdifferential equation (SDE):

dLθc ,θ J

t

Lθc ,θ Jt−

= θ ct σtdWt + (eθJt−Zt− − 1)dNt

− λt

(eθJtµJ+

12 (θ

Jt σJ )

2− 1

)dt. (2.6)

Proof. From the mutual independence of random shocks W ,N ,and Z = (Zt)0≤t≤T , it follows that

E[exp

(∫ t

0θ cs dCs +

∫ t

0θJs−dJs

) ∣∣∣∣F ξt

]= E

[exp

(∫ t

0θ cs

(αs − kλs −

12σ 2s

)ds+

∫ t

0θ cs σsdWs

) ∣∣∣∣F ξt

]× E

[exp

(∫ t

0θJs−Zs−dNs

) ∣∣∣∣F ξt

]= exp

(∫ t

0θ cs

(αs − kλs −

12σ 2s

)ds+

12

∫ t

0(θ cs σs)

2ds)

× exp(∫ t

0λs

(eθJsµJ+

12 (θ

Js σJ )

2− 1

)ds). (2.7)

Thenwe apply (2.5) to conclude that Lθc ,θ J

t as in Box I. On the otherhand, the SDE (2.6) follows from the Itô rule with jumps. Then,by (2.6), the density Lθ

c ,θ J is a martingale. Thus the proof of theproposition is completed.

Based on the no-arbitrage pricing principal in Harrison andPliska (1981, 1983), in order to evaluate the currency options, weneed to deduce the so-called martingale condition: there is anequivalent domesticmartingalemeasure, underwhich the regime-switching jump–diffusion process Sd given in (2.3) is an (Ht)0≤t≤T -martingale.

Theorem 2.1. Let the random Esscher transform be defined by (2.5).Then the martingale condition holds if and only if the Markov-modulated parameters (θ ct , θ

Jt )0≤t≤T satisfy, for all 0 ≤ t ≤ T ,

rFt − rDt + αt − kλt + θ

ct σ2t + λ

θ,Jt k

θ,Jt = 0, (2.8)

where the random Esscher transform intensity λθ,Jt of the Poissonprocess, and themean percentage jump size kθ,Jt are respectively givenby

λθ,Jt = λte

θJtµJ+

12 (θ

Jt σJ )

2, kθ,Jt = (k+ 1)e

θJt σ2J − 1. (2.9)

Proof. Let Eθc ,θ J denote the mathematical expectation operatorwith respect to the Esscher transform Q θ

c ,θ J∼ P. Then the Bayes

formula and Proposition 2.1 jointly imply that, for u < t ,

Eθc ,θ J

[Sdt |Hu

]=

E[Lθc ,θ J

t Sdt |Hu]

E[Lθc ,θ Jt |Hu

] = E[ Lθc ,θ Jt

Lθc ,θ Ju

Sdt

∣∣∣∣∣Hu]. (2.10)

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464 L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469

Lθc ,θ J

t =

exp(∫ t0 θ

cs

(αs − kλs − 1

2σ2s

)ds+

∫ t0 θ

cs σsdWs +

∫ t0 θ

Js−Zs−dNs

)exp

(∫ t0 θ

cs

(αs − kλs − 1

2σ2s

)ds+ 1

2

∫ t0 (θ

cs σs)

2ds+∫ t0 λs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds)

= exp(∫ t

0θ cs σsdWs −

12

∫ t

0(θ cs σs)

2ds)exp

(∫ t

0θJs−Zs−dNs −

∫ t

0λs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds)

Box I.

On the other hand, by the Itô rule with jumps, we have

Sdt = Sdu exp

(∫ t

u

(rFs − r

Ds + αs − kλs −

12σ 2s

)ds

+

∫ t

uσsdWs +

∫ t

uZs−dNs

).

Thus

Lθc ,θ J

t

Lθc ,θ Ju

SdtSdu= exp

(∫ t

u(rFs − r

Ds + αs − kλs + θ

cs σ2s )ds+M

cu,t

)× exp

(∫ t

uλs

(e(1+θ

Js )µJ+

12 (1+θ

Js )2σ 2J − 1

)ds)

× exp(−

∫ t

uλs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds+M Ju,t

), (2.11)

where, for u < t ≤ T ,

Mcu,t =∫ t

u(1+ θ cs )σsdWs −

12

∫ t

u(1+ θ cs )

2σ 2s ds,

M Ju,t =∫ t

u(1+ θ Js−)Zs−dNs

−

∫ t

uλs

(e(1+θ

Js )µJ+

12 (1+θ

Js )2σ 2J − 1

)ds.

Hence, from (2.10), it follows that

Eθc ,θ J

[Sdt |Hu

]= Sdu exp

(∫ t

u(rFs − r

Ds + αs − kλs + θ

cs σ2s )ds

)× exp

(∫ t

uλs

(e(1+θ

Js )µJ+

12 (1+θ

Js )2σ 2J − 1

)ds)

× exp(−

∫ t

uλs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds).

As a consequence, the martingale condition Eθc ,θ J[Sdt |Hu

]= Sdu

holds if and only if the Markov-modulated parameters (θ c, θ J)satisfy that

rFt − rDt + αt − kλt + θ

ct σ2t + λ

θ,Jt k

θ,Jt = 0,

for all 0 ≤ t ≤ T . This proves the theorem.

The following corollary gives a pair of solutions (θ c,∗t , θJ,∗t )0≤t≤T

for the martingale condition (2.8).

Corollary 2.1. For each 0 ≤ t ≤ T , define the regime-switchingparameters by

θc,∗t =

kλt − αt + rDt − rFt

σ 2t, (2.12)

θJ,∗t = −

µJ +12σ2J

σ 2J. (2.13)

Then (θ c,∗t , θJ,∗t )0≤t≤T is a pair of Esscher parameters, i.e., it solves the

martingale condition (2.8). Here, the random Esscher transformmeanpercentage jump size becomes

kθ∗,Jt = (k+ 1)eθ

J,∗t σ 2J − 1 = 0,

and the random Esscher transform intensity of the Poisson process isgiven by

λθ∗,Jt = λt exp

(−µ2J

2σ 2J+σ 2J

8

).

The explicit Esscher parameters in (2.12) and in (2.13) (togetherwith the Esscher parameters given in Corollary 3.1) will be used tothe Monte Carlo simulations in Section 4.

2.3. Valuation of European currency options

As in Elliott et al. (2005, 2007) and Elliott and Osakwe (2006),an equivalent domestic martingale measure can be treated as therandom Esscher transform Q θ

c ,θ J with respect to the probabilitymeasure P. We begin with identifying the spot FX rates dynamicsSt under the equivalent domestic martingale measure Q θ

c ,θ J . Let(θ c,∗, θ J,∗) be the Esscher parameters of the risk-neutral regime-switching Esscher transform. Then

Proposition 2.2. Under the equivalent domestic martingale measureQ θ

c,∗,θ J,∗ and conditional on Fξ

T ,

W θc,∗

t = Wt −∫ t

0θ c,∗s σsds, 0 ≤ t ≤ T , (2.14)

is a Brownian motion. In addition, under Q θc,∗,θ J,∗ , the intensity λθ

∗,Jt

of Poisson process N and themean percentage jump size kθ∗,Jt are given

by

λθ∗,Jt = λteθ

J,∗t µJ+

12 (θ

J,∗t σJ )

2, (2.15)

and

kθ∗,Jt = (k+ 1)eθ

J,∗t σ 2J − 1, (2.16)

respectively.

Proof. In light of (2.6), apply the Girsanov theorem and by theindependence ofW and N , we immediately get thatW θc,∗

t definedin (2.14) is a Brownian motion under Q θ

c,∗,θ J,∗ . Next we prove(2.15) and (2.16) conditional on F

ξt . Let dMt = dNt − λtdt . Then it

is a P-martingale. The Girsanov theorem for point processes (see,e.g., Protter (1990)) yields that

dMθc,∗,θ J,∗= dMt − λt

(eθJ,∗t µJ+

12 (θ

J,∗t σJ )

2− 1

)dt

= dNt − λteθJ,∗t µJ+

12 (θ

J,∗t σJ )

2dt

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L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469 465

is a Q θc,∗,θ J,∗-martingale. Thus we proved (2.15). Define a P-

martingale (MZt )0≤t≤T by

dMZt = (eZt− − 1)St−dNt − kλtStdt.

Apply the Girsanov theorem, then we have

dMZ,θc,∗,θ J,∗

= dMZt − λtE[(eθJ,∗t Zt − 1

) (eZt − 1

)|F

ξt

]Stdt

= (eZt− − 1)St−dNt − λθ∗,Jt kθ

∗,Jt Stdt

is a Q θc,∗,θ J,∗-martingale and which implies (2.16). Thus the proof

of the proposition is complete.

As a consequence, Proposition 2.2 implies that the spot FXrate dynamics (St)0≤t≤T , under the equivalent domesticmartingalemeasure Q θ

c ,θ J , obeys that

dStSt−=

(αθ∗

t − kθ∗,Jt λ

θ∗,Jt

)dt + σtdW θc,∗

t + (eZt− − 1)dNt , (2.17)

where αθ∗

t = αt − kλt + kθ∗,Jt λ

θ∗,Jt + θ

c,∗t σ 2t , for 0 ≤ t ≤ T .

We note that, the risk-neutral spot FX rate in (2.17) is consistentwith the discontinuous stock return model in Merton (1976). Forthe European call currency options with the strike price K and thetime to expiration T , its price at time zero is given by

C0(S, K , T , ξ) = Eθc,∗,θ J,∗

[e−

∫ T0 (r

Ds −r

Fs )ds(ST − K)+|F

ξ

T

]. (2.18)

Let Ji(t, T ) denote the occupation time of ξ in state i (i = 1, 2,. . . , n) over the time period [t, T ]with t < T . Then we have

Rt,T =1T − t

∫ T

t(rDs − r

Fs )ds =

1T − t

n∑i=1

(rDi − rFi )Ji(t, T );

Ut,T =1T − t

∫ T

tσ 2s ds =

1T − t

n∑i=1

σ 2i Ji(t, T );

λθ∗,Jt,T =

1T − t

∫ T

tλθ∗,Js ds =

1T − t

n∑i=1

λθ∗,Ji Ji(t, T );

λθ∗

t,T =1T − t

∫ T

t(1+ kθ

∗,Js )λθ

∗,Js ds

=1T − t

n∑i=1

(1+ kθ∗,Ji )λ

θ∗,Ji Ji(t, T );

V 2t,T ,m = Ut,T +mσ 2JT − t

;

Rt,T ,m = Rt,T −1T − t

∫ T

tλθ∗,Js kθ

∗,Js ds

+mT − t

∫ T

t

log(1+ kθ∗,Js )

T − tds

= Rt,T −1T − t

n∑i=1

λθ∗,Ji kθ

∗,Ji Ji(t, T )

+mT − t

n∑i=1

log(1+ kθ∗,Ji )

T − tJi(t, T ),

where m denotes the number of jumps in the time interval [t, T ].Applying the pricing formula in Merton (1976) and define

C0(S, K , T ; R0,T ,U0,T , λθ∗,J0,T )

=

∞∑m=0

e−Tλθ∗

0,T (Tλθ∗

0,T )m

m!BS0(S, K , T , V 20,T ,m, R0,T ,m),

where BS0(S, K , T , σ 2, r) denotes the standard Black–Scholes op-tion pricing formulawith initial spot FX rate S0, strike price K , risk-

free rate r , volatility square σ 2 and time T to maturity. Thereforewe have the European-style call option pricing formula as follows:

C0(S, K , T ) =∫[0,T ]n

C0(S, K , T ; R0,T ,U0,T , λθ∗,J0,T )

×ψ(J1, J2, . . . , Jn)dJ1dJ2 · · · dJn,

where ψ(J1, J2, . . . , Jn) denotes the joint probability distributiondensity for the occupation time.(J1(0, T ), J2(0, T ), . . . , Jn(0, T )), which can be completely de-

termined by the following characteristic function

E[ei〈t,J(0,T )〉|F ξ

t

]=⟨ξte(Q+iD)T , 1

⟩, (2.19)

where i =√−1, 1 = (1, 1, . . . , 1) ∈ Rn and the matrix D

denotes a diagonal matrix consisting of the elements in the vectort = (t1, t2, . . . , tn) as its diagonal.

3. Markov-modulated Merton jump–diffusion model withHeston’s stochastic volatility

In this section, we model the spot FX rate dynamics by theMarkov-modulatedMerton jump–diffusionwithHeston’s stochas-tic volatility. Then we will price the European-style currencyoptions by employing the random Esscher transform for thejump–diffusion. First of all, we suggest the spot FX rate dynamicsbydStSt−= (αt − kλt)dt +

√VtdWt + (eZt− − 1)dNt ,

dVt = (γ − βVt)dt + σv√VtdW 1t ,

W 1t = ρWt +√1− ρ2W 2t ,

(3.1)

where γ , β, σv > 0 and |ρ| < 1. Here W and W 2 are twoindependent Brownian motions. This model can be regarded asa regime-switching jump version of Heston’s stochastic volatilitymodel (see Heston (1993, 1999)).Similarly as in Section 2, to price the European-style call cur-

rency options, we first define Yt = log StS0 with 0 ≤ t ≤ T . Then bythe Itô rule with jumps, we have

Yt = Ct + Jt , (3.2)

where the continuous part Ct and the jump part Jt of Yt are respec-tively written by

Ct =∫ t

0

(αs − kλs −

12Vs

)ds+

∫ t

0

√VsdWs, Jt =

∫ t

0Zs−dNs.

For 0 ≤ t ≤ T , set Gt = Fξt ∨F V

t , andHt = F Yt ∨GT . Then we

can define the random Esscher transform Q θc ,θ J∼ P on Ht with

respect to two families of regime-switching parameters (θ cs )0≤s≤tand (θ Js )0≤s≤t by

Lθc ,θ J

t =dQ θ

c ,θ J

dP

∣∣∣∣∣Ht

=

exp(∫ t0 θ

cs dCs +

∫ t0 θ

Js−dJs

)E[exp

(∫ t0 θ

cs dCs +

∫ t0 θ

Js−dJs

)|Gt

] , (3.3)where θmt = 〈θ

m, ξt〉 with θm = (θm1 , θm2 , . . . , θ

mn ) ∈ R

n andm ∈ c, J. A similar argument as in Proposition 2.1 shows that, therandom Esscher transform density Lθ

c ,θ J= (Lθ

c ,θ J

t )0≤t≤T definedvia (3.3) can be given by

Lθc ,θ J

t = exp(∫ t

0θ cs

√VsdWs −

12

∫ t

0(θ cs )

2Vsds)

× exp(∫ t

0θJs−Zs−dNs −

∫ t

0λs

(eθJsµJ+

12 (θ

JsσJ )

2− 1

)ds).

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466 L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469

Furthermore, Lθc ,θ J is an exponential (Ht)0≤t≤T -martingale and it

admits the following SDE:

dLθc ,θ J

t

Lθc ,θ Jt−

= θ ct

√VtdWt + (eθ

Jt−Zt− − 1)dNt

− λt

(eθJtµJ+

12 (θ

Jt σJ )

2− 1

)dt. (3.4)

Accordingly, we have

Lemma 3.1. Let the random Esscher transform be defined by (3.3).Then the martingale condition holds if and only if the Markov-modulated parameters (θ ct , θ

Jt )0≤t≤T satisfy, for all 0 ≤ t ≤ T ,

rFt − rDt + αt − kλt + θ

ct Vt + λ

θ,Jt k

θ,Jt = 0, (3.5)

where the random Esscher transform intensity λθ,Jt of Poisson processand the mean percentage jump size kθ,Jt are respectively given by

λθ,Jt = λte

θJtµJ+

12 (θ

Jt σJ )

2, kθ,Jt = (k+ 1)e

θJt σ2J − 1.

Now, let (θ c,∗, θ J,∗) be the parameters of the risk-neutral regime-switching Esscher transform. That is Esscher parameters (θ c,∗, θ J,∗)admit (3.5) in Lemma 3.1. Then we have

Lemma 3.2. Under the equivalent domestic martingale measureQ θ

c,∗,θ J,∗ and conditional on GT ,

W θc,∗

t = Wt −∫ t

0θ c,∗s

√Vsds, 0 ≤ t ≤ T , (3.6)

is a Brownian motion and the Poisson process N admits the intensityλθ∗,Jt and mean percentage jump size kθ

∗,Jt ,

λθ∗,Jt = λteθ

J,∗t µJ+

12 (θ

J,∗t σJ )

2; kθ

∗,Jt = (k+ 1)eθ

J,∗t σ 2J − 1.

The proofs of Lemmas 3.1 and 3.2 are similar to those in Section 2.A difference between them is that we here use the filtration(Ht)0≤t≤T instead of (F Y

t ∨ Fξ

T )0≤t≤T there. On the other hand,Lemmas 3.1 and 3.2 tell us that the stochastic volatility V =(Vt)0≤t≤T did not make any change on the form of the randomEsscher transform intensity (λθ

∗,Jt )0≤t≤T and mean percentage

jump amplitude (kθ∗,Jt )0≤t≤T (see Theorem 2.1). In addition, as in

Corollary 2.1, we also have

Corollary 3.1. For each 0 ≤ t ≤ T , define the regime-switchingparameters by

θc,∗t =

kλt − αt + rDt − rFt

Vt, (3.7)

θJ,∗t = −

µJ +12σ2J

σ 2J. (3.8)

Then (θ c,∗t , θJ,∗t )0≤t≤T is a pair of Esscher parameters, i.e., it solves the

martingale condition (3.5). Here, the random Esscher transformmeanpercentage jump size and the intensity of the Poisson process are thesame as those in Corollary 2.1.

Consequently, we have the dynamics of the spot FX rate S underthe equivalent domestic martingale measure Q θ

c,∗,θ J,∗ :dStSt−= (αθ

∗

t − kθ∗,Jt λ

θ∗,Jt )dt +

√VtdW θc,∗

t + (eZt− − 1)dNt ,

dVt = (γ − βθ∗

t Vt)dt + σv√VtdW

θc,∗,1t ,

W θc,∗,1t = ρW θc,∗

t +

√1− ρ2W 2t , ρv = ρσv,

(3.9)

where αθ∗

t = αt − kλt + kθ∗,Jt λ

θ∗,Jt + θ

c,∗t Vt and β

θ,∗t = β − ρvθ

c,∗t

for 0 ≤ t ≤ T .For the European-style call currency optionwith the strike price

K and time to expiration T , we have the following price formula:

C0(S, K , T , ξ) = Eθc,∗,θ J,∗

[e−

∫ T0 rDs −r

Fs ds(ST − K)+|F

ξ

T

]= e−

∫ T0 rDs −r

Fs dsEθ

c,∗,θ J,∗[(ST − K)1ST≥K|F

ξ

T

]= e−

∫ T0 rDs −r

Fs ds [FP1 − KP2] , (3.10)

where

F = Eθc,∗,θ J,∗

[ST |Fξ

T ],

P1 =∫∞

K

xQ θc,∗,θ J,∗(ST ∈ dx|F

ξ

T )

Eθc,∗,θ J,∗ [ST |Fξ

T ],

P2 = Q θc,∗,θ J,∗(ST > K |F

ξ

T ).

In the following, we use the results in Bates (1996) to deduce thepricing formula. For this purpose, let φ ∈ R and

F2(φ, V , T , λθ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

= exp(C2(φ, T , λ

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )+ D2(φ, T , β

θ,∗0,T )V

+ Tλθ∗,J0,T [(1+ k

θ∗,J0,T )e

12 σ2J (φ

2−φ)− 1]

), (3.11)

where

C2(φ, T , λθ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

= dθ∗

0,TφT −γ Tσ 2v[ρvφ − β

θ,∗0,T − Γ2(φ, β

θ,∗0,T )]

−2γσ 2vlog

[1+

12(ρvφ − β

θ,∗0,T − Γ2(φ, β

θ,∗0,T ))

1− eΓ2(φ,βθ,∗0,T )T

Γ2(φ, βθ,∗0,T )

],

D2(φ, T , βθ,∗0,T ) =

φ − φ2

ρvφ − βθ,∗0,T + Γ2(φ, β

θ,∗0,T )

1+eΓ2(φ,β

θ,∗0,T )T

1−eΓ2(φ,β

θ,∗0,T )T

,

Γ2(φ, βθ,∗0,T ) =

√(ρvφ − β

θ,∗0,T )

2 + σ 2v (φ − φ2),

and

λθ∗,J0,T =

1T

∫ T

0λθ∗,Js ds =

1T

n∑i=1

λθ∗,Ji Ji(0, T ),

kθ∗,J0,T =

1T

∫ T

0kθ∗,Js ds =

1T

n∑i=1

kθ∗,Ji Ji(0, T ),

dθ∗

0,T =1T

∫ T

0(αθ

∗

s − kθ∗,Js λθ

∗,Js )ds

=1T

n∑i=1

[αθ∗

i − kθ∗,Ji λ

θ∗,Ji ]Ji(0, T ),

βθ,∗0,T =

1T

∫ T

0βθ,∗s ds =

1T

n∑i=1

(β − ρvθc,∗i )Ji(0, T ). (3.12)

Then we have

P2 = P2(T , λθ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

=12+1π

∫∞

0

Im[F2(iφ; V0, T , λ

θ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )e

−iφ log KS0

]φ

dφ.

(3.13)

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L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469 467

As for P1, it also follows from Bates (1996) that

F1(φ, V , T , λθ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

= exp(C1(φ, T , λ

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )+ D1(φ, T , β

θ,∗0,T )V

+ Tλθ∗,J0,T [(1+ k

θ∗,J0,T )e

12 σ2J (φ

2+φ)− 1]

), (3.14)

where

C1(φ, T , λθ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

= dθ∗

0,TφT −γ Tσ 2v[ρvφ − β

θ,∗0,T + ρv − Γ1(φ, β

θ,∗0,T )]

−2γσ 2vlog

[1+

12

(ρvφ − β

θ,∗0,T + ρv − Γ1(φ, β

θ,∗0,T )

)

×1− eΓ1(φ,β

θ,∗0,T )T

Γ1(φ, βθ,∗0,T )

],

D1(φ, T , βθ,∗0,T ) =

−φ − φ2

ρvφ − βθ,∗0,T + ρv + Γ1(φ, β

θ,∗0,T )

1+eΓ1(φ,β

θ,∗0,T )T

1−eΓ1(φ,β

θ,∗0,T )T

,

Γ1(φ, βθ,∗0,T ) =

√(ρvφ − β

θ,∗0,T + ρv)

2 − σ 2v (φ + φ2),

and λθ∗,J0,T , k

θ∗,J0,T , d

θ∗

0,T and βθ,∗0,T are defined in (3.12).

As a consequence, we obtain

P1 = P1(T , λθ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

=12+1π

∫∞

0

Im[F1(iφ; V0, T , λ

θ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )e

−iφ log KS0

]φ

dφ.

(3.15)

Finallywe can get the European-style call currency pricing formula

C0(S, K , T ) =∫[0,T ]n

e−R0,T [FP1 − KP2] (T , λθ∗,J0,T , k

θ∗,J0,T , β

θ,∗0,T , d

θ∗,J0,T )

×ψ(J1, J2, . . . , Jn)dJ1dJ2 · · · dJn, (3.16)

where ψ(J1, J2, . . . , Jn) is uniquely determined by the characteris-tic function in (2.19).

4. Numerical illustrations

In this section, we employ Monte Carlo simulation and performa numerical experiment for the currency options pricing under theMarkov-modulated jump–diffusion (2.1) and (3.1). Furthermore,we compare the numerical results of the European-style calloptions implied by the jump–diffusion models (2.1) and (3.1) withthat implied by the Markov-modulated Black–Scholes model (thatis the model (2.1) with the jump intensity λt ≡ 0).Here we use the Esscher parameters in Corollary 2.1 for model

(2.1) and the parameters in Corollary 3.1 for model (3.1). FromItô formula with jumps, it follows that the log spot FX rate Y =(Yt)0≤t≤T admits

dYt =(rDt − r

Ft −

12σ 2t

)dt + σtdW θc,∗

t + Zt−dNθJ,∗

t , (4.1)

for the model (2.1). Similarly, Y = (Yt)0≤t≤T admitsdYt =

(rDt − r

Ft −

12Vt

)dt +

√VtdW θc,∗

t + Zt−dNθJ,∗

t ,

dVt = (γ − βθ∗

t Vt)dt + σv√VtdW

θc,∗,1t ,

W θc,∗,1t = ρW θc,∗

t +

√1− ρ2W 2t , β

θ,∗t = β − ρσvθ

c,∗t ,

(4.2)

Table 1The parameter values.

Parameter name Value in state e1 Value in state e2

Annual domestic interest rate rD1 = 0.04 rD2 = 0.01

Annual foreign interest rate rF1 = 0.025 rF2 = 0.03Volatility in model (2.1) σ1 = 0.1 σ2 = 0.3Appreciation rate of S α1 = 0.05 α2 = 0.02Annual jump intensity λ1 = 10 λ2 = 20Mean jump size µJ = 0Standard deviation of the jump size σJ = 0.1Initial FX rate S0 = 1Initial value of V V0 = 0.01Volatility of V σv = 0.1Speed of reversion of V β = 2Long-term average of V γ /β = 0.01Correlation coefficient ρ ρ = 0.5

for the model (3.1). In addition, under the equivalent domesticmartingale measure Q θ

c,∗,θ J,∗ , the random shocks W θc,∗=

(W θc,∗t )0≤t≤T andW 2 = (W 2t )0≤t≤T are two independent Brownian

motions, and the Poisson process NθJ,∗= (Nθ

J,∗t )0≤t≤T has the

Esscher transform intensity

λθ∗,Jt = λt exp

(−µ2J

2σ 2J+σ 2J

8

).

In the numerical simulations, we parsimoniously assume thatthe hidden Markov chain ξ has two states, which means thatwe consider the domestic macroeconomic shifts between the twostates: e1 (‘‘boom’’ or ‘‘good’’) and e2 (‘‘recession’’ or ‘‘bad’’). Wesuppose that the transition probability matrix of the two-stateMarkov chain ξ is given by(p11 p12p21 p22

)=

(0.7 0.30.2 0.8

). (4.3)

Moreover, we shall adopt the specimen values for the modelparameters, which are given in Table 1. The parameter valuesrelated to the continuous diffusive component and the stochasticvolatility are quoted from Siu et al. (2008) (except the ‘‘Speed ofreversion of V ’’ and the ‘‘Long-term average of V ’’). Those valuesrelated to the jump component are comparable to the estimatedvalues in Bates (1996).For each of the fixed maturity years T = 0.5, 1.0, 1.5, we are

concerned with a range of spot-to-strike ratio S0/K from 0.8 to 1.2,with an increment of 0.05. Suppose one year has 252 trading days,then the sample interval is 1/252. We perform 10000 simulationsfor computing the option price.From Figs. 2 and 3, we find that the jump risk implied by the

Markov-modulated jump–diffusion model seems to have a moresignificant impact on the option prices than that of the stochasticvolatility, since the two plots from the jump–diffusion models(2.1) and (3.1) are close to each other while the two plots arerelatively far apart from the plot from the Markov-modulatedBlack–Scholes model. Moreover, the jump risk becomes more andmore pronounced as the maturity increases, since the two plotsfrom (2.1) and (3.1) are pushing farther and farther away from theplot of theMarkov-modulated Black–Scholesmodel. However, thiseffect seems to be independent of the strike prices since the plotsin Fig. 2 are almost parallel.Finally, we investigate how the option prices vary with the

changes of the volatility σ or the annual jump intensity λ. Fig. 4depicts the option prices varying with the volatility and Fig. 5depicts the counterpart as the annual jump intensity varies. Fromthese plots, we observe that the effect of the annual jump intensityis more remarkable than that of the volatility, since we change theparameters by the same percentagewhile the distances among the

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468 L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469

Fig. 2. Option price against spot-to-strike ratio. The solid, dashed and dottedlines correspond tomodels (2.1), (3.1) and to theMarkov-modulated Black–Scholesmodel, respectively.

Fig. 3. Option price against time to maturity. The solid, dashed and dotted linescorrespond tomodel (2.1),model (3.1) and to theMarkov-modulated Black–Scholesmodel, respectively.

lines in Fig. 5 is farther than that in Fig. 4. Fig. 6 also displays a largechange on the option price due to the varying of the annual jumpintensity.We also observe that the deviations in Figs. 4 and 5 are

asymmetric. We point out that the asymmetry is caused by theexistence of the two sources of risk: the continuous Brownian risk

Fig. 4. Option price against spot-to-strike ratio for the model (2.1) with the annualjump intensities λ1 = 10 and λ2 = 20. The solid, dashed and dotted linescorrespond to (σ1 = 0.1, σ2 = 0.3), (σ1 = 0.05, σ2 = 0.15) and (σ1 = 0.15, σ2 =0.45), respectively.

Fig. 5. Option price against spot-to-strike ratio for the model (2.1) with thevolatilities σ1 = 0.1 and σ2 = 0.3. The solid, dashed and dotted lines correspondto (λ1 = 10, λ2 = 20), (λ1 = 5, λ2 = 10) and (λ1 = 15, λ2 = 30), respectively.

Fig. 6. Option price against spot-to-strike ratio for the model (3.1). The solid,dashed and dotted lines correspond to (λ1 = 10, λ2 = 20), (λ1 = 5, λ2 = 10)and (λ1 = 15, λ2 = 30), respectively.

and the jump risk. Concretely, due to the existence of the jump risk,the option price will not approach zero but to a strictly positivevalue when the volatilities tend to zero. That in turn leads to theasymmetry in Fig. 4. Similarly, the asymmetry in Fig. 5 is caused bythe existence of the continuous Brownian risk. The asymmetry inFig. 6 is not apparent, which may be due to the small value of thestochastic volatility (V0 = 0.01 and γ = 0.02).

5. Conclusion

The currency option pricing problemwas studiedwhile the spotFX rate was driven by some Markov-modulated jump–diffusions.Compared with most of the existing Markov-modulated diffusionmodels, the main advantage is that we incorporated jump riskinto themodels. We employed the Esscher transform to determinean equivalent martingale measure in the incomplete marketdescribed by the regime-switching jump–diffusion model. Underthe assumptions of the constant jump intensity and log-normaljump amplitude, we found the closed-form solutions for theregime-switching Esscher parameters (risk premiums) and thusthe equivalent domesticmartingalemeasurewas identified. Underthe measure, the explicit forms of the pricing for European-stylecurrency options was derived and the simulation results were alsopresented and discussed.For the numerical illustrations, we compared our Markov-

modulated jump–diffusion models with the Markov-modulatedBlack–Scholes model. We found that jump risk implied by our

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L. Bo et al. / Insurance: Mathematics and Economics 46 (2010) 461–469 469

Markov-modulated jump–diffusion has a more significant impacton the option prices. To some extent, the comparison has indicatedthe importance of incorporating jump risk into the spot FX ratemodels. We also found that the existence of two sources of risk(the continuous Brownian risk and the jump risk) can result inasymmetry with respect to each single risk in the pricing of theEuropean-style currency options.As a potential future work, we might consider incorporating

jump risk into the two-factor model studied in Siu et al. (2008).Comparing the effect of jump risk on the option prices with thatof the long-term volatility (described by a hidden Markov chain inSiu et al. (2008)) may be a new interesting topic.

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