Economic Modelling 37 (2014) 296–305

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Economic Modelling

j ourna l homepage: www.e lsev ie r .com/ locate /ecmod

Pricing foreign equity options with regime-switching

Kun Fan a,d, Yang Shen b,d, Tak Kuen Siu c,d,⁎, Rongming Wang a

a School of Finance and Statistics, East China Normal University, Shanghai 200241, Chinab School of Risk and Actuarial Studies and CEPAR, Australian School of Business, University of New South Wales, Sydney, NSW 2052, Australiac Cass Business School, City University London, 106 Bunhill Row, London EC1Y 8TZ, United Kingdomd Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

⁎ Corresponding author at: Cass Business School, CityRow, London EC1Y 8TZ, United Kingdom; Department oStudies, Faculty of Business and Economics, MacquarieAustralia. Tel.: +61 2 98508573; fax: +61 2 98509481.

E-mail address: ktksiu2005@gmail.com (T.K. Siu).

0264-9993/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.econmod.2013.11.009

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 1 November 2013

JEL classification:F31G13

Keywords:Foreign equity optionRegime-switchingMean-reversionFast Fourier transform

In this paper, we investigate the valuation of two types of foreign equity options under a Markovian regime-switching mean-reversion lognormal model, where some key model parameters in the dynamics of the foreignequity price and the foreign exchange rate are modulated by a continuous-time, finite-stateMarkov chain. A fastFourier transform (FFT) approach is applied to provide an efficient way to evaluate the option prices. Numericalanalysis and empirical studies are provided to illustrate the practical implementation of the proposed pricingmodel.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Due to recent technological advance and trade liberalization, thegrowth of globalization has been accelerated and the economic growthhas been boosted unprecedentedly. In the global financial markets, for-eign exchange risk arising from fluctuations in foreign exchange ratehas received much attention, especially since the currency crises inemerging markets. To hedge and manage foreign exchange risk, bothacademic researchers and industry practitioners have proposed a varie-ty of currency options. Partly attributed to globalization,many firms andhouseholds are massively involved in investment activities of foreignassets. There are two key sources of risk arising in investment on foreignassets, namely foreign exchange (FX) risk and asset's price risk. Effectivemanagement of these two sources of risk is the key to successes in for-eign assets investments. Foreign equity options provide a possible waytomanage or hedge both the FX risk and the equity price risk. Accordingto the definition in Kwok and Wong (2000), “the currency-translatedforeign equity options are contingent claims whose payoffs are deter-mined by financial prices or indices denominated in one currency butthe actual payouts are settled in another currency”. As its name implies,the underlying asset of a foreign equity option is a foreign equity. There

University London, 106 Bunhillf Applied Finance and ActuarialUniversity, Sydney, NSW 2109,

ghts reserved.

are twomain tempting features of foreign equity options. Firstly, foreignequity options provide investors with a variety of flexible ways to dealwith the multidimensional risks, mainly the foreign equity price fluctu-ation risk and the foreign exchange risk. There exist a variety of types offoreign equity optionswith different payoff functions. Seen from this as-pect, foreign equity options could provide investors with more invest-ment and risk management choices. Exchange-trade is the secondadvantage of foreign equity options, which means this kind of financialproduct enjoys a higher degree of liquidity. Furthermore, the regula-tions of clearinghouse help investors reduce or avoid some risks, suchas counterparty risk.

Since the pricingmodel of foreign equity options needs to depict thejoint dynamics of the exchange rate and foreign equity prices, there aresome literature about the valuation of foreign equity options underdifferent models. Early works usually consider the valuation of foreignequity options in the Black–Scholes framework. Kwok and Wong(2000) investigated the valuation of foreign equity options with pathdependent features. Examples of pricing foreign equity options beyondthe traditional BS framework include a multi-dimensional Lévy processto depict the dynamics of both the exchange rate and the foreign equityprices in Huang and Hung (2005). Xu et al. (2011a) considered thevaluation of foreign equity option under a stochastic volatility modelwith double jumps. To incorporate the impacts of skewness and kurtosison foreign equity option prices, the Gram–Charlier series expansionapproach was adopted by Xu et al. (2011b).

It is known that certain vital features of financial time series cannotbe depicted by the classical Black-Scholes models. Among the modelsextending the classical Black–Scholes model, the ability to incorporate

297K. Fan et al. / Economic Modelling 37 (2014) 296–305

structural changes in economic conditions makes regime-switchingmodels one of themost practically useful models in financial economet-rics. These changes, which may be attributed to changes in economicfundamentals or business cycles, represent an additional source of riskto which an additional amount of risk premium may be required tocompensate. Furthermore, the risk brought by these changes can behardly diversified since it is more likely to be regarded as a systematicrisk. Since regime-switching models provide a natural and convenientchoice to model the structural changes in economic conditions, espe-cially due to financial crises, this class of models will enjoy more andmore popularity. The seminal work of Hamilton (1989) popularized ap-plications of regime-switching models in financial econometrics. Typi-cally, the so-called “modulated by a Markov chain” means the modeldynamics or parameters will change when the underlying Markovchain changes from one state to another. The states of the Markovchain represent the states of an economy. Since the last decade or so,there has been an interest on studying option valuation problems inregime-switching models (see Buffington and Elliott (2002), Elliottet al. (2005), Siu (2008), Yuen and Yang (2010), Shen et al. (2013),Shen and Siu (2013), etc.). Considering the increasingly changing for-eign exchange market, there is a considerable interest to investigatethe valuation of currency options under regime-switching models, in-cluding Bollen et al. (2000), Siu et al. (2008), Bo et al. (2010). Empiricalstudies in Bollen et al. (2000) verified that trading strategies underregime-switching models can gain higher profit and be more attractiveto investors. That also indicates the potential practical value of regime-switching models.

However, relatively little attention has been given to pricing foreignequity options in the context of regime-switchingmodels. In this paper,we investigate the valuation of foreign equity options under aMarkovianregime-switching mean-reversion lognormal model, which extendsthe mean-reversion lognormal model for foreign exchange rate.More specifically, the model parameters, including the risk-free do-mestic interest rate, the volatility of the foreign equity, the mean-reversion level and the volatility of the foreign exchange rate, aswell as the instantaneous correlation coefficient between the foreignequity and the exchange rate, are modulated by a continuous-time,finite-state, observableMarkov chain. To apply the fast Fourier trans-form (FFT) approach to discretize the integral pricing formula, weneed to first calculate the characteristic function of the logarithmicunderlying equity price. For the valuation of the foreign equity op-tion with strike price in the foreign currency (FEOF

1), we firstapply a measure change technique and use a version of the Bayes'rule to derive the conditional characteristic function of the logarith-mic equity price under the newmeasure. For the valuation of the for-eign equity option with strike price in the domestic currency (FEOD),we calculate the characteristic function of the summation of the log-arithmic foreign equity price and the logarithmic foreign exchangerate under the risk-neutral probability measure. Then, we derivethe Fourier transform of the foreign equity option price in thesetwo cases. To illustrate the pricing of foreign equity options, we pro-vide a numerical analysis using the FFT method. Finally, an empiricalapplication is provided, revealing that the regime-switching modeloutperforms the model with a single regime in terms of lower fittingerrors and prediction errors. Themain contributions of this paper areas follows. (1) We investigate the valuation of foreign equity optionsunder a regime-switching mean-reversion lognormal model. Themain feature of our model is that it combines the advantages ofboth regime-switching models and mean-reversion lognormalmodels. The mean reversion feature of foreign exchange rates hasbeen well-documented. (Jorion and Sweeney (1996); Sweeney(2006)) Wong and Lau (2008); Wong and Lo (2009); Wong and

1 Following the notation inXuet al. (2011b), let FEOF and FEOD represent the foreign eq-uity option with strike price in the foreign currency and the foreign equity option withstrike price in the domestic currency, respectively.

Zhao (2010); Leung et al. (2013)). (2) By applying a measure changetechnique, the Fourier transform of the FEOF option price can be cal-culated more easily. Then, we adopt the FFT approach in Carr andMadan (1999) and Liu et al. (2006) to derive a pricing formulae forthe foreign equity options.

The rest of the paper is organized as follows. The next section pre-sents the model dynamics. In Section 3, we derive the pricing formulaeof FEOF and FEOD under the Markovian, regime-switching, mean-reversion lognormal model, respectively. Section 4 presents numericalexamples. An empirical application of our model is provided inSection 5. The final section concludes the paper.

2. The model dynamics

In this section,we consider a continuous-time economywith a finitetime horizon T :¼ 0; T½ �, where T b ∞. Let (Ω, F P) be a complete prob-ability space. In the literature about foreign exchange ratemodeling, it iscustomary to assume that P is a risk-neutral probability measure (SeeWong and Lau (2008)). To describe the evolution of the state of an econ-omy over time, we consider a continuous-time, N-state, observableMarkov Chain X :¼ X tð Þjt∈Tf g . The N different states of the chainmay represent N observable different states of an economy or differentstages of a business cycle. Without loss of generality, using the conven-tion in Elliott et al. (1994), we assume the chain X has a canonical statespace E : = {e1,e2,…,eN} ⊂ ℜN, where the j-th component of ei is theKronecker delta δij, for each i,j = 1,2,…,N. Let Q: = [qij]i,j = 1,2,…,N de-note the generator or rate matrix of the chain X under P , where qij isthe transition intensity of the chain X from state ei to state ej. Then thefollowing semimartingale dynamics for the chain X were obtained inElliott et al. (1994):

X tð Þ ¼ X 0ð Þ þZ t

0QX sð ÞdsþM tð Þ; t∈T :

Here M tð Þjt∈Tf g is an ℜN-valued, (FX, P)-martingale, where FX isthe right-continuous, P -complete, natural filtration generated by thechain X.

We now specify the Markovian regime-switching models forthe dynamics of the foreign equity and the foreign exchangerate. Let S :¼ S tð Þjt∈Tf g and Z :¼ Z tð Þjt∈Tf g denote the price processof the foreign equity and the logarithmic foreign exchange rate processrespectively. Let y′ be the transpose of a vector or amatrix y, ⟨⋅,⋅ ⟩ be thescalar product in ℜN, and diag(y) be the diagonal matrix with diagonalelements being given by the components of the vector y. For each t∈T ,let r(t) andσ(t) be thedomestic, instantaneous continuously compounded,interest rate and the volatility of the equity at time t, respectively. Weassume that r(t) and σ(t) are determined by the value X(t) of thechain at time t as:

r tð Þ :¼ r;X tð Þh i;σ tð Þ :¼ σ;X tð Þh i;

where r: = (r1,r2,…,rN)′ ∈ ℜNwith ri N 0 andσ : = (σ1,σ2,…,σN)′ ∈ ℜN

with σi N 0 for each i = 1,2,…,N.Let α tð Þjt∈Tf g and γ tð Þjt∈Tf gbe themean-reversion level and vol-

atility of the process Z. Again we suppose that

α tð Þ :¼ α;X tð Þh i;γ tð Þ :¼ γ;X tð Þh i;

where α : = (α1,α2,…,αN)′ ∈ ℜN and γ : = (γ1,γ2,…,γN)′ ∈ ℜN

with γi N 0, for each i = 1,2,…,N. The parameter β, controlling thespeed of mean reversion for the logarithmic foreign exchange rate pro-cess, is assumed to be a positive constant.

298 K. Fan et al. / Economic Modelling 37 (2014) 296–305

Then, under the risk-neutral probability P , the dynamics of S and Zare given by

dS tð Þ ¼ r tð ÞS tð Þdt þ σ tð ÞS tð ÞdW1 tð Þ ; ð1Þ

and

dZ tð Þ ¼ β α tð Þ−Z tð Þð Þdt þ γ tð ÞdW2 tð Þ ; ð2Þ

where W1 tð Þjt∈Tf g and W2 tð Þjt∈Tf g are two standard Brownian mo-tionswith respect to their respective right-continuous,P-complete, nat-ural filtrations under P . Furthermore, we suppose that the twoBrownian motions W1, W2 are correlated and that the instantaneouscorrelation coefficient at time t is given by:

W1;W2h i tð Þ ¼Z t

0ρ sð Þds ;

where ρ(t) = ⟨ρ,X(t)⟩ andρ : = (ρ1,ρ2,…,ρN)′ ∈ ℜNwith−1 b ρj b 1for j = 1,…,N;with a slight abuse of the notation, W1;W2h i tð Þjt∈T gf de-notes the predictable quadratic covariation process betweenW1 andW2.

For each t∈T , write Y(t): = ln(S(t)) and F(t): = eZ(t) for the loga-rithmic foreign equity price and the foreign exchange rate at time t, re-spectively. Note that in Shen and Siu (2013), a Markovian regime-switching Hull-White model was used for modeling stochastic interestrate, so there is a positive probability that the interest rate goes negative.Here the logarithmic foreign exchange rate is modelled by a Markovianregime-switching, mean-reverting process, so that foreign exchangerate stays positive. Applying Itô's differentiation rule, the risk-neutraldynamics of Y and F are given by

dY tð Þ ¼ r tð Þ−12σ2 tð Þ

� �dt þ σ tð ÞdW1 tð Þ ;

and

dF tð ÞF tð Þ ¼ βα tð Þ þ 1

2γ2 tð Þ−βlnF tð Þ

� �dt þ γ tð ÞdW2 tð Þ :

Let FS :¼ F S tð Þjt∈T� �

, FZ :¼ F Z tð Þjt∈T� �

and FX :¼ FX tð Þjt∈T� �

be the right-continuous, P -complete, natural filtrations generated byprocesses S, Z and X, respectively. Furthermore, we define the enlargedfiltration G :¼ G tð Þjt∈Tf g by the minimal σ-field containing F S(t),FZ(t) and FX(t). That is,

G tð Þ :¼ F S tð Þ∨F Z tð Þ∨FX tð Þ ; t∈T :

For each t∈T , G tð Þ represents publicly available market informationup to time t.

Remark 2.1. The economic motivation of using the regime-switchingmodel may be illustrated by a utility maximization problem of a rep-resentative agent's intertemporal consumption. If the agent receivesa stream of Markov-modulated dividends from holding a single secu-rity, it can be shown that the maximization of an expected utility ofthe agent leads to a regime-switching model dynamics for the secu-rity. Interested readers may refer to Di Graziano and Rogers (2009)for details. Due to various economic and financial factors, the streamof dividends may change with the current economic conditions ormarket modes. The regime-switching model provides a natural wayto describe these changes, so it may provide a better matching ofthe agent's expectation towards market trends than its non-regimeswitching counterpart.

3. Pricing foreign equity options

Typically, there are two types of foreign equity options, which areclassified by the strike prices of the underlying foreign equities. The

payoff functions of the two kinds of foreign equity options aregiven by

FEOF ¼ F Tð Þ S Tð Þ−K Fð Þþ ;

and

FEOD ¼ F Tð ÞS Tð Þ−KDð Þþ ;

where KF and KD are the strike prices in the foreign currency and indomestic currency, respectively. Note that the payoff is representedin the domestic currency for both the FEOF option and the FEOD

option.

3.1. Valuation of an FEOF option

Considering the particular payoff function of an FEOF option, the fol-lowing pricing formula is standard:

C F 0; T;K Fð Þ ¼ E e−

Z T

0r tð Þdt

F Tð Þ S Tð Þ−K Fð Þþ

264

375 ; ð3Þ

where E is an expectation under the risk-neutral measure P . LetkF = ln(KF) be the logarithmic strike price. The modified FEOF optionprice is defined by

cF 0; T ; kFð Þ ¼ eaFk F C F 0; T ;K Fð Þ ;

where aF is a predetermined positive constant such that cF(0, T, kF) issquare integrable in kF over the entire real line. As in Carr and Madan(1999), the Fourier transform of cF(0, T, kF) is as follows:

ψF 0; T;uð Þ ¼Z ∞

−∞eiuk F cF 0; T; kFð ÞdkF : ð4Þ

The following proposition gives an integral representation for theprice of the FEOF option.

Proposition 3.1. For each j = 1, 2,∙∙∙,N, let

g j t;uð Þ :¼ −r j þ βe−β T−tð Þα j þ12e−2β T−tð Þγ2

j−12

u−i aF þ 1ð Þð Þ2σ2j

þ i u−i aF þ 1ð Þð Þ r j−12σ2

j þ e−β T−tð Þρ jσ jγ j

� �:

Write

g t;uð Þ :¼ g1 t;uð Þ; g2 t;uð Þ;…; gN t;uð Þð Þ′∈ℭN;

where ℭ is the complex space and ℭN is the N-fold product of ℭ.Then under the Markovian regime-switching mean-reversion log-

normal model, the price of the FEOF option is given by the following in-tegral formula:

C F 0; T;K Fð Þ ¼ e−a Fk F

π

Z ∞

0e−iuk FψF 0; T;uð Þdu ;

where

ψF 0; T;uð Þ ¼exp e−βTZ 0ð Þ þ iuþ aF þ 1ð ÞY 0ð Þn oa2F þ aF−u2 þ i 2aF þ 1ð Þu

X 0ð ÞexpZ T

0diag g t;uð Þð Þdt þ QT

� �;1

� :

299K. Fan et al. / Economic Modelling 37 (2014) 296–305

Before proving Proposition 3.1, several useful results are given. First-ly, we introduce a probability measureQ equivalent toP on G Tð Þ by thefollowing Radon–Nikodým derivative:

dQdP

G Tð Þ :¼eZ Tð Þ

E eZ Tð ÞjFX Tð Þ� � :

The idea of introducing a probability measure Q here may not be unlikethat of introducing a forward measure in a Markovian regime-switching, mean-reverting process, (see, for example, Shen and Siu(2013)).

Then, by a version of the Bayes' rule, Eq. (3) becomes

C F 0; T;K Fð Þ ¼ E E e−

Z T

0r tð Þdt

F Tð Þ S Tð Þ−K Fð ÞþFX Tð Þ

264

375

264

375

¼ E e−

Z T

0r tð Þdt

E F Tð ÞjFX Tð Þh i

EQ S Tð Þ−K Fð ÞþjFX Tð Þ

h i264

375;

where EQ represents an expectation under the measure Q.

Lemma 3.1. The Radon–Nikodým derivative is given by:

dQdP

G Tð Þ ¼ exp −12

Z T

0e−2β T−tð Þγ2 tð Þdt þ

Z T

0e−β T−tð Þγ tð ÞdW2 tð Þ

� �:

Then,

WQ1 tð Þ :¼ W1 tð Þ−

Z t

0ρ sð Þe−β t−sð Þγ sð Þds

and

WQ2 tð Þ :¼ W2 tð Þ−

Z t

0e−β t−sð Þγ sð Þds

are two standard Brownian motions underQ. The instantaneous corre-lation coefficient between WQ

1 and WQ2 at time t is still ρ(t).

Proof. A direct calculation to Eq. (2) gives

Z Tð Þ ¼ e−βTZ 0ð Þ þZ T

0βe−β T−tð Þα tð Þdt þ

Z T

0e−β T−tð Þγ tð ÞdW2 tð Þ :

It is easy to see that given FX(T), the conditional distribution of Z(T)is a normal distribution with the following mean and variance:

E Z Tð ÞjFX Tð Þh i

¼ e−βTZ 0ð Þ þZ T

0βe−β T−tð Þα tð Þdt ;

Var Z Tð ÞjFX Tð Þh i

¼Z T

0e−2β T−tð Þγ2 tð Þdt ;

so that

E eZ Tð ÞjFX Tð Þh i¼ exp e−βTZ 0ð Þ þ

Z T

0βe−β T−tð Þα tð Þdt þ 1

2

Z T

0e−2β T−tð Þγ2 tð Þdt

� �:

Consequently,

dQdP ¼ eZ Tð Þ

E eZ Tð ÞjFX Tð Þ� �

¼ exp −12

Z T

0e−2β T−tð Þγ2 tð Þdt þ

Z T

0e−β T−tð Þγ tð ÞdW2 tð Þ

� �:

By Girsanov's theorem, W1Q(t): = W1(t) − ∫ 0

t ρ(s)e−β(t − s)γ(s)dsandW2

Q(t): = W2(t) − ∫ 0t e−β(t − s)γ(s)ds are two standard Brownian

motions underQ. It is also obvious that WQ1 ;W

Q2

D Etð Þ ¼ W1;W2h i tð Þ ¼

∫t

0ρ uð Þdu. □

To apply the fast Fourier transform approach (Carr and Madan(1999)), we need to calculate the characteristic function of the logarith-mic terminal spot price of the foreign equity. Here, due to the measurechange, the conditional characteristic function of the logarithmic termi-nal foreign equity price under the probability measureQ should be de-rived first.

Lemma 3.2. The conditional characteristic function of Y(T) given F X(T)under Q is calculated as:

ϕQY Tð ÞjFX Tð Þ uð Þ :¼ EQ eiuY Tð ÞjFX Tð Þ

h i¼ exp

�iuY 0ð Þ þ iu

Z T

0r tð Þ−1

2σ2 tð Þ

� �dt−1

2u2Z T

0σ2 tð Þdt

þ iuZ T

0e−β T−tð Þρ tð Þσ tð Þγ tð Þdt

�

where EQ denotes an expectation under the probability measure Q.

Proof. It is easy to find that under the probability measure Q,

S Tð Þ ¼ S 0ð Þexp�Z T

0r tð Þ−1

2σ2 tð Þ

� �dt þ

Z T

0e−β T−tð Þρ tð Þσ tð Þγ tð Þdt

þZ T

0σ tð ÞdWQ

1 tð Þ�

:

Consequently, conditional on F X(T), Y(T) is normally distributedwith the following mean and variance:

EQ Y Tð ÞjFX Tð Þh i

¼ Y 0ð Þ þZ T

0r tð Þ−1

2σ2 tð Þ

� �dt

þZ T

0e−β T−tð Þρ tð Þσ tð Þγ tð Þdt ;

and

VarQ Y Tð ÞjFX Tð Þh i

¼Z T

0σ2 tð Þdt :

Then the conditional characteristic function of Y(T) is easy to com-pute. □

For notational simplicity, write:

RT :¼Z T

0r tð Þdt ;

LT :¼Z T

0βe−β T−tð Þα tð Þdt þ 1

2

Z T

0e−2β T−tð Þγ2 tð Þdt :

The result presented in the following lemma resembles to those inLemma 4.1 and Lemma 4.2 in Shen and Siu (2013).

Lemma 3.3. Let FQY Tð ÞjFX Tð Þ yð Þ be the conditional distribution function of

Y(T) given FX(T) under Q. Then, the Fourier transform of the price of theFEOF option is given by:

ψF 0; T;uð Þ

¼E e−RTþLTþe−βT z 0ð ÞϕQ

Y Tð ÞjFX Tð Þ u−i aF þ 1ð Þð Þ�

a2F þ aF−u2 þ i 2aF þ 1ð Þu

¼exp e−βTZ 0ð Þ þ iuþ aF þ 1ð ÞY 0ð Þ

n oX 0ð Þ exp

Z T

0diag g t;uð Þð Þdt þ QT

� �;1

� a2F þ aF−u2 þ i 2aF þ 1ð Þu

:

300 K. Fan et al. / Economic Modelling 37 (2014) 296–305

whereϕQY Tð ÞjFX Tð Þ uð Þ denotes the conditional characteristic function of Y(T)

given FX(T) under the probability measureQ.

Proof. The proof resembles to the proofs of Lemma 4.1 and Lemma 4.2in Shen and Siu (2013), which are standard. We present the proof herefor the sake of completeness. Let kF = ln(KF),

ψF 0; T ;uð Þ ¼Z ∞

−∞eiukF c 0; T ; kFð ÞdkF

¼Z ∞

−∞eaFk F eiuk F C 0; T ;K Fð ÞdkF

¼Z ∞

−∞eaFk F eiuk FE e−RT F Tð Þ eY Tð Þ−ekF

� �þ

�dkF

¼ E Z ∞

−∞eaFk F eiukFE e−RT F Tð Þ eY Tð Þ−ekF

� �þjFX Tð Þ

�dkF

�

¼ EZ ∞

−∞eaFk F eiuk F e−RTE F Tð ÞjFX Tð Þ

h iEQ eY Tð Þ−ekF

� �þjFX Tð Þ

�dkF

�

¼ EZ ∞

−∞eaFk F eiukF e−RTþLTþe−βT Z 0ð ÞEQ eY Tð Þ−ekF

� �þjFX Tð Þ

�dkF

�

¼ EZ ∞

−∞eaFk F eiukF e−RTþLTþe−βT Z 0ð Þ

Z ∞

k F

ey−ekF� �

FQY Tð ÞjFX Tð Þ dyð ÞdkF

" #

¼ EZ ∞

−∞e−RTþLTþe−βT Z 0ð Þ

Z y

−∞eyþ a Fþiuð Þk F−e 1þa Fþiuð Þk F� �

dkF FQY Tð ÞjFX Tð Þ dyð Þ

�

¼ E e−RTþLTþe−βT Z 0ð ÞZ ∞

−∞

e 1þa Fþiuð Þy

aF þ iu− e 1þa Fþiuð Þy

1þ aF þ iu

!FQY Tð ÞjFX Tð Þ dyð Þ

" #

¼E e−RTþLTþe−βT Z 0ð ÞϕQ

Y Tð ÞjFX Tð Þ u−i aF þ 1ð Þð Þh i

a2F þ aF−u2 þ i 2aF þ 1ð Þu

¼exp e−βTZ 0ð Þ þ iuþ aF þ 1ð ÞY 0ð Þn o

E expZ T

0g t;uð Þ;X tð Þh idt

� � �a2F þ aF−u2 þ i 2aF þ 1ð Þu

:

Define

Γ tð Þ :¼ X tð ÞexpZ t

0g s;uð Þ;X sð Þh ids

� �; t∈T :

Applying Itô's differentiation rule to Γ(t),

dΓ tð Þ ¼ g t;uð Þ;X tð Þh iΓ tð Þdt þ expZ t

0g s;uð Þ;X sð Þh ids

� �dX tð Þ

¼ diag g t;uð Þð Þ þ Qð ÞΓ tð Þdt þ expZ t

0g s;uð Þ;X sð Þh ids

� �dM tð Þ :

ð5Þ

Taking expectation on both sides of Eq. (5) under P gives:

dE Γ tð Þ½ � ¼ diag g t;uð Þð Þ þ Qð ÞE Γ tð Þ½ �dt :

Solving gives

E X Tð ÞexpZ T

0g t;uð Þ;X tð Þh idt

� � �

¼ X 0ð ÞexpZ T

0diag g t;uð Þð Þdt þQT

� �:

Consequently,

E expZ T

0g t;uð Þ;X tð Þh idt

� � �

¼ E Γ Tð Þ½ �;1h i

¼ X 0ð ÞexpZ T

0diag g t;uð Þð Þdt þ QT

� �;1

� :

□

Proof of Proposition 3.1. The proof is standard. Applying the inverseFourier transform to Eq. (4), the following equation can be derived:

C F 0; T;K Fð Þ ¼ e−a Fk F cF 0; T; kFð Þ ¼ e−a Fk F

π

Z ∞

0e−iuk FψF 0; T ;uð Þdu :

The result can be obtained from Lemma 3.3 immediately. □

3.2. Valuation of an FEOD option

The pricing formula for an FEOD option is given by

CD 0; T ;KDð Þ ¼ E e−

Z T

0r tð Þdt

F Tð ÞS Tð Þ−KDð Þþ

264

375 ; ð6Þ

where E denotes an expectation under the risk-neutral mea-sure P. Let kD = ln(KD). The modified FEOD option price is de-fined by

cD 0; T; kDð Þ ¼ eaDkDCD 0; T; kDð Þ ;

where aD is a predetermined positive constant such that cD(0, T,kD) is square integrable in kD over the entire real line. As in Carrand Madan (1999), the Fourier transform of cD(0, T, kD) is asfollows:

ψD 0; T; vð Þ ¼Z ∞

−∞eivkDcD 0; T; kDð ÞdkD : ð7Þ

Define a process G tð Þjt∈Tf gwith G(t): = ln(F(t)S(t)) for each t∈T .By direct calculation,

G Tð Þ ¼ e−βTZ 0ð Þ þ lnS 0ð Þ þZ T

0r tð Þ−1

2σ2 tð Þ þ βe−β T−tð Þα tð Þ

� �dt

þZ T

0σ tð ÞdW1 tð Þ þ

Z T

0e−β T−tð Þγ tð ÞdW2 tð Þ :

Then, the conditional characteristic function of G(T) given F X(T)under the risk-neutral probability measure P is given by

ϕG Tð ÞjFX Tð Þ vð Þ ¼ E exp ivG Tð Þf gjFX Tð Þh i

:

The following proposition gives an integral representation for theprice of the FEOD option. This result resembles to that of Proposition 3.1.

Proposition 3.2. For each j = 1, 2,∙∙∙,N, let

h j t; vð Þ :¼ −r j þ i v−i aD þ 1ð Þð Þ r j−12σ2

j þ βe−β T−tð Þα j

� �

−12

v−i aD þ 1ð Þð Þ2 σ2j þ e−2β T−tð Þγ2

j þ 2e−β T−tð Þρ jσ jγ j

� �:

Write

h t; vð Þ :¼ h1 t; vð Þ; h2 t; vð Þ;…; hN t; vð Þð Þ′∈ℭN:

Then under the Markovian regime-switching mean-reversion log-normal model, the price of the FEOD option is given by

CD 0; T ;KDð Þ ¼ e−aDkD

π

Z ∞

0e−ivkDψD 0; T; vð Þdv ;

Table 2Option prices calculated via the FFT.

FEOF FEOD

k State 1 State 2 State 1 State 2

−0.3 0.5614 0.4554 0.9849 0.8450−0.2 0.4672 0.3611 0.9022 0.7713−0.1 0.3772 0.2698 0.8164 0.69320 0.2947 0.1882 0.7282 0.61120.1 0.2225 0.1225 0.6388 0.52690.2 0.1623 0.0753 0.5500 0.44260.3 0.1146 0.0450 0.4637 0.3610

0 0.2 0.4 0.6 0.8 1

0.35

0.4

0.45

0.5

q

pric

es

Option prices with modified strike k=−0.2

State1State2

0.4

0.5

s

Option prices with modified strike k=−0.1

State1State2

301K. Fan et al. / Economic Modelling 37 (2014) 296–305

where

ψD 0; T ; vð Þ

¼exp ivþ aDð Þ Y 0ð Þ þ e−βTZ 0ð Þ

� �n oa2D þ aD−v2 þ i 2aD þ 1ð Þv

X 0ð ÞexpZ T

0diag h t; vð Þð Þdt þ QT

� �;1

� :

Proof. The proof here resembles to that in Lemma 3.3. Let FG Tð ÞjFX Tð Þ gð Þdenote the conditional distribution function of G(T) given FX(T) underP. Then, the Fourier transform of the FEOD option price is given by:

ψD 0; T ; vð Þ ¼Z ∞

−∞eaDkDeivkDCD 0; T ;KDð ÞdkD

¼Z ∞

−∞eaDkDeivkDE e−RT eG Tð Þ−ekD

� �þ

�dkD

¼ EZ ∞

−∞eaDkDeivkDE e−RT eG Tð Þ−ekD

� �þjFX Tð Þ

�dkD

�

¼ EZ ∞

−∞eaDkD eivkD e−RT

Z ∞

kD

eg−ekD� �

FG Tð ÞjFX Tð Þ dgð ÞdkD

" #

¼ EZ ∞

−∞e−RT

Z g

−∞eaDkDeivkD eg−ekD

� �dkD FG Tð ÞjFX Tð Þ dgð Þ

�

¼ EZ ∞

−∞e−RT

Z g

−∞e aDþivð ÞkDþg−e aDþ1þivð ÞkD� �

dkD FG Tð ÞjFX Tð Þ dgð Þ �

¼E e−RTϕG Tð ÞjFX Tð Þ v−i aD þ 1ð Þð Þh ia2D þ aD−v2 þ i 2aD þ 1ð Þv

:

Since

G Tð Þ ¼ e−βTZ 0ð Þ þ lnS 0ð Þ þZ T

0r tð Þ−1

2σ2 tð Þ þ βe−β T−tð Þα tð Þ

� �dt

þZ T

0σ tð ÞdW1 tð Þ þ

Z T

0e−β T−tð Þγ tð ÞdW2 tð Þ ;

then

e−RTϕG Tð ÞjFX Tð Þ v−i aD þ 1ð Þð Þ

¼ exp�

ivþ aDð ÞZ T

0r tð Þdt þ ivþ aD þ 1ð Þ Y 0ð Þ þ e−βTZ 0ð Þ þ

Z T

0βe−β T−tð Þα tð Þdt

�

þ12

v−i aD þ 1ð Þð Þ2Z T

0e−2β T−tð Þγ2 tð Þdt þ 2

Z T

0ρ tð Þσ tð Þγ tð Þe−β T−tð Þdt

�

þ iv aD þ 12

� �−1

2v2 þ 1

2aD aD þ 1ð Þ

� �Z T

0σ2 tð Þdt

�:

Define

h t;X tð Þ; vð Þ ¼ −r tð Þ þ i v−i aD þ 1ð Þð Þ r tð Þ−12σ2 tð Þ þ βe−β T−tð Þα tð Þ

� �

−12

v−i aD þ 1ð Þð Þ2 σ2 tð Þ þ e−2β T−tð Þγ2 tð Þ þ 2e−β T−tð Þρ tð Þσ tð Þγ tð Þ� �

:

Then,

ψD 0; T ; vð Þ

¼exp ivþ aD þ 1ð Þ Y 0ð Þ þ e−βTZ 0ð Þ

� �n oa2D þ aD−v2 þ i 2aD þ 1ð Þv

E expZ T

0h t; vð Þ;X tð Þh idt

� � �

¼exp ivþ aD þ 1ð Þ Y 0ð Þ þ e−βTZ 0ð Þ

� �n oa2D þ aD−v2 þ i 2aD þ 1ð Þv

X 0ð ÞexpZ T

0diag h t; vð Þð Þdt þ QT

� �;1

� :

Table 1Assumptions of parameter values.

r β α γ σ ρ

State 1 0.02 1 0.8 0.2 0.4 0.4State 2 0.04 1 0.4 0.4 0.2 0.2

The derivation of the second equation is similar with the proof inLemma 3.3. □

4. Numerical examples

In this section, we perform a numerical analysis for pricing theFEOF option and the FEOD option under our regime-switchingmean-reversion lognormal model. To simplify our computation,we consider a two-state Markov chain X. For each t∈T ,X(t) = (1,0) and X(t) = (0,1) are State 1 and State 2,respectively.

The rate matrix of the Markov chain X under P is assumed to be

Q ¼ −q qq −q

� �;

where q takes values in [0,1]. Intuitively, with a larger q, the economywill display a more volatile feature. We consider the following configu-rations of other parameters values given in Table 1.

Here, the domestic interest rate, the mean-reversion level and vola-tility of the exchange rate, the volatility of the foreign equity as well asthe instantaneous correlation coefficient between the foreign equityprice and the exchange rate take different values when the states ofthe economy change. Table 2 presents the prices of the FEOF optionand the FEOD option with different modified strike levels under theregime-switching mean-reversion lognormal model, where we assumeS(0) = 1, F(0) = 1, T = 1 and q = 0.5. The FFT method is applied tocalculate the option prices (see also Carr and Madan (1999), Lee(2004), Liu et al. (2006), Wong and Guan (2011) and Kwok et al.(2012)).

0 0.2 0.4 0.6 0.8 10.2

0.3

q

pric

e

Fig. 1. Option prices corresponding to different q with k = −0.2,−0.1.

0 0.2 0.4 0.6 0.8 1

0.2

0.25

0.3

0.35

q

pric

es

Option prices with modified strike k=0State1State2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

q

pric

es

Option prices with modified strike k=0.1

State1State2

Fig. 2. Option prices corresponding to different q with k = 0,0.1.

−0.2 −0.1 0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

k

pric

es

Option prices under the RS model and the NRS model in State 2

RS model

NRS model

Fig. 4. Option prices calculated under the RSmodel and the NRSmodel in State 2.

302 K. Fan et al. / Economic Modelling 37 (2014) 296–305

As shown in Table 2, for both the FEOF option and the FEOD op-tion, the option prices in State 1 are systematically higher thanthose in State 2 when the strike level is fixed. If the option valuationis viewed from the perspective of a domestic investor, State 1 is a“Bad” state while State 2 is a “Good” one. Seen from the foreign ex-change rate aspect, the higher volatility of the foreign exchange ratemeans higher potential profits when the income, denominated inthe foreign currency, translated into the domestic currency. On theother hand, seen from the foreign equity aspect, the underlying for-eign equity has a higher interest rate and a lower volatility in State2. A lower volatility means less chance of the equity price beingvery high or very low. In this case, the option will be less valuable.Consequently, it is reasonable that the option prices in State 1 arehigher than the corresponding prices in State 2 due to the additionalamount of risk premium required to compensate for a disadvantageeconomic condition. Note that the option prices converge quickly. Inour illustration, we always adopt the number of discretizationM = 4096.

−0.2 −0.1 0 0.1 0.2 0.30.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

k

pric

es

Option prices under the RS model and the NRS model in State 1

RS modelNRS model

Fig. 3. Option prices calculated under the RSmodel and the NRSmodel in State 1.

In the sequel, we use the valuation of an FEOF option as an examplefor sensitivity analysis. The sensitivity analysis of the valuation of anFEOD option can be conducted similarly.

4.1. The impact of q on option prices

Under our model, we assume S(0) = 1, F(0) = 1 and T = 1. To il-lustrate the impact of q on option prices, we perform a sensitivity anal-ysis for the option prices with respect to the rate of transition q.

From Figs. 1 and 2, when q increases, the option prices in State 1 andState 2 display different trends. In State 1, the option prices decreasewhile increase with q in State 2. Note that the value of q calibrates theprobability of the chain X transiting between State 1 and State 2, As ex-plained earlier, the options aremore expensive in State 1 and cheaper inState 2. Consequently, the option pricewill decreasewhen q increases inState 1, while the opposite trend is displayed in State 2. When q = 0,the regime-switching effect will not exist. Under this degenerate case,option prices are the highest in State 1 and lowest in State 2.

From Figs. 1 and 2, a particular attention is given to the case q = 0,where the model dynamics for the foreign equity and the foreign ex-change rate have no switching regimes. For simplicity, we denote our

−0.4−0.2

00.2

0.4

0

0.5

10

0.2

0.4

0.6

0.8

k

Option prices against T and k in State 1

T

Opt

ion

pric

es

Fig. 5. Option prices corresponding to different T and k in State 1.

−0.4−0.2

00.2

0.4

0

0.5

10

0.1

0.2

0.3

0.4

0.5

k

Option prices against T and k in State 2

T

Opt

ion

pric

es

Fig. 6. Option prices corresponding to different T and k in State 2.

02

46

810

0

5

100

10

20

30

40

50

S(0)

Option prices against S(0) and F(0) in State 2

F(0)

Opt

ion

pric

es

Fig. 8. Option prices corresponding to different S(0) and k in State 2.

0.40.2

0.40

0.2

0.4

0.6

0.8

Option prices against ρ1 and k in State 1

Opt

ion

pric

es

303K. Fan et al. / Economic Modelling 37 (2014) 296–305

regime-switching mean-reversion lognormal model and the modelwithout regime-switching as the “RS” model and the “NRS” model re-spectively. Figs. 3 and 4 provide us with a visual comparison betweenthe option prices under the RS model and the NRS model, with the as-sumption that q = 0.5 in the RS model.

As indicated in Figs. 3 and 4, the foreign equity option prices arelower (higher) under the RS model than those under the NRSmodel in State 1 (State 2). This is intuitively clear if State 1 andState 2 are interpreted as a “Bad” state and a “Good” one, respective-ly. Compared with the NRS model, the possibility of regime shiftsfrom the current state to the opposite one in the RS model will inev-itably lower option prices in a “Bad” state while higher those in a“Good” one. In other words, ignoring the regime-switching effectwould result in the FEOF option being overpriced in State 1 andbeing underpriced in State 2.

−0.4−0.2

00.2

−0.4−0.2

0

kρ1

Fig. 9. Option prices corresponding to different ρ1 and k in State 1.

4.2. The impact of T and k on option prices

Figs. 5 and 6 depict the price of the FEOF option versus the modifiedstrike value k and thematurity time T. It is easy to see that the longer thematurity time is, the higher the option price when k is fixed in both thetwo states. On the other hand, when the maturity time remains thesame, the price of the FEOF option decreases when k increases.

02

46

810

0

5

100

10

20

30

40

50

60

S(0)

Option prices against S(0) and F(0) in State 1

F(0)

Opt

ion

pric

es

Fig. 7. Option prices corresponding to different S(0) and k in State 1.

−0.4−0.2

00.2

0.4

−0.4

−0.2

00.2

0.40

0.1

0.2

0.3

0.4

0.5

k

Option prices against ρ1 and k in State 2

ρ1

Opt

ion

pric

es

Fig. 10. Option prices corresponding to different ρ1 and k in State 2.

Fig. 11. Nikkei 225 index from September 2003 to September 2013.

304 K. Fan et al. / Economic Modelling 37 (2014) 296–305

4.3. The impact of initial equity price S(0) and F(0) on option prices

Figs. 7 and 8 illustrate the FEOF option prices versus different initialequity price S(0) and different initial foreign exchange rate F(0) withk = 1, q = 0.5 and T = 1.When the initial equity price S(0) or the ini-tial exchange rate F(0) increases, the price of the FEOF option is morelikely to be higher due to the possible higher payoff. Note that the in-creasing speed of the FEOF option price is faster against S(0) than thatagainst F(0).

Fig. 12. USD/JPY exchange rates from Se

4.4. The impact of the correlation coefficient ρ1 on option prices

Furthermore, we provide sensitivity analysis for the correlation co-efficient ρ1 with different k in both State 1 and State 2 under the as-sumption that S(0) = 1, T = 1, F(0) = 1 and q = 0.5. As illustratedin Figs. 9 and 10, the foreign equity option prices will increase as ρ1does given that other parameters are fixed. This indicates that an addi-tional amount of premium is required to compensate the correlationrisk between the foreign equity price and the exchange rate.

ptember 2003 to September 2013.

Table 3In-sample fitting errors and out-of-sample prediction errors.

Errors RS model NRS model

In-sample 0.4356% 1.0286%Out-of-sample 0.9529% 1.9859%

305K. Fan et al. / Economic Modelling 37 (2014) 296–305

5. Empirical studies

In this section, an empirical study of the regime-switching mean-reversion lognormal model is provided to illustrate the practical imple-mentation of the model. Here, we take the Nikkei 225 index as the for-eign equity and the US dollar as the domestic currency from theperspective of an US investor. Firstly we calibrate themodel parametersto the market prices of the European call options on Nikkei 225 indexand the exchange rates between the US dollar (USD) and the JapaneseYen (JPY). By comparing the in-sample fitting errors and out-of-sample prediction errors, we illustrate how well the RS model mightfit the market data and how the RS model might improve on the NRSmodel. As in the last section, we assume that the Markov chain hasonly two states and use the valuation of an FEOF option as an example.

Figs. 11 and 12describe the value of theNikkei 225 index and the ex-change rate USD/JPY from September 2003 to September 2013. Fromthese figures, one may see that the stock index and the exchange rateexhibit both the regime-switching and mean-reversion features. Ourdataset consists of European call option prices written on the Nikkei225 index and the exchange rate of USD/JPY for seventeen consecutivetrading days from 1 October 2013 to 17 October 2013, obtained fromthe Datastream Database of Reuters. For each trading day, the optionshave 9 strikes ranging from 12,000 to 16,000. The in-sample data in-clude the option prices and exchange rates from 1 October 2013 to 14October 2013 and the out-of-sample data include the rest option pricesand exchange rates from 15 October 2013 to 17 October 2013.

Without loss of generality,we assume the domestic risk-free interestrate to be r = (0.02, 0.04)́. In addition,we assume the ratematrix is notnecessarily symmetric. To test how well the RS model and the NRSmodel fit the market data, we employ the method of nonlinear leastsquares by minimizing the sum of squared errors between the marketprices and model prices. Denote the model parameters as Θ: = (α1,α2, γ1, γ2, β, σ1, σ2, ρ1, ρ2, q12, q21, p). Here the model prices are theweighted averages of the option prices in State 1 and State 2, withweights being given by p and 1-p, respectively. The parameter estimatesof the RS model are given by

Θ ¼ ð0:650;0:625;0:346;0:201;0:902;0:209;0:102;

−0:697;0:101;0:666;0:334;0:315Þ :

Furthermore, Table 3 reports the root mean square error (RMSE) forthe fitting and prediction errors of both the RS model and the NRSmodel based on the same in-sample and out-of-sample data. As indicat-ed in Table 3, the RSmodel has lower RMSEs for both fitting and predic-tion errors than the NRS model. The results verify that the RS modelimprove quite significantly the performance of the NRS model in the de-scriptionof thedynamics of the foreign equityprice and the exchange rate.

From the perspective of an investor, using the RS model for the val-uation of foreign equity options can providemore accurate option pricesand hence reduce model risks.

6. Conclusions

We considered the valuation of foreign equity options with strikeprices in both the foreign currency and the domestic foreign currencyunder a regime-switching mean-reversion lognormal model. The pa-rameters were assumed to be modulated by an observable, finite-stateMarkov chain. The FFT approach was applied to price these two kinds offoreign equity options. Numerical examples and empirical studies wereprovided to illustrate the practical implementation of our methods.

Acknowledgments

The authors thank an anonymous referee for his/her helpful com-ments. R. Wang and K. Fan would like to acknowledge the NationalNatural Science Foundation of China (11231005), Doctoral ProgramFoundation of the Ministry of Education of China (20110076110004),Program for New Century Excellent Talents in University (NCET-09-0356) and “the Fundamental Research Funds for the Central Universities”.

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