# Pricing FX Options in the Heston/CIR Jump-Diffusion Model with Log-Normal and Log-Uniform Jump Amplitudes

Authors: Rehez Ahlip; Ante Prodan
Journal: International Journal of Stochastic Analysis
 (2015)

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## Abstract

We examine foreign exchange options in the jump-diffusion version of the Heston stochastic volatility
model for the exchange rate with log-normal jump amplitudes and the volatility model with log-uniformly distributed jump amplitudes. We assume that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semianalytical formula for the price of the foreign exchange European call option.


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## Body

## 1. Introduction

We extend the results from Ahlip and Rutkowski [1] by deriving a closed-form pricing formula for the foreign exchange (FX) options in a model where the spot exchange rate and its volatility are jump-diffusions, with log-normal and log-uniform jump amplitudes, respectively, whereas the domestic and foreign interest rates are governed by the Cox–Ingersoll–Ross (CIR) dynamics postulated in [2]. In particular, our model allows for correlation between the exchange rate process and its instantaneous volatility. The interest rate processes are independent of one another, and they are also independent of the foreign exchange rate and its volatility.In the seminal paper by Heston [3], the author noted that increasing the volatility of volatility only increases the kurtosis of spot returns and does not capture skewness. In order to capture the skewness, it is crucial to include also the properly specified correlation between the volatility and the spot exchange rate returns. In papers by Bakshi et al. [4], Bates [5], and Duffie et al. [6], the authors showed that stochastic volatility models do not offer reliable prices for close to expiration derivatives. This motivated Bates [5] and Bakshi et al. [4] to introduce jumps to the dynamics of the exchange rate. However, as observed by Andersen and Andreasen [7] and Alizadeh et al. [8], the addition of jumps to the dynamics of the exchange rate is not sufficient to capture the sudden increase in volatility due to market turbulence. Since the overall volatility in financial markets consists of a highly persistent slow moving and rapid moving components, Eraker et al. [9] proposed to introduce jump process to the dynamics of the volatility process in order to enhance the cross-sectional impact on option prices.More recently, D’Ippoliti et al. [10] obtained closed-form solutions, in the spirit of Heston, in a model with jumps in both spot returns of the underlying asset and its volatility. Yan and Hanson [11] consider a model in which the stock prices follow a jump-diffusion process with log-uniformly distributed jump amplitudes under the Heston volatility model. In the above-mentioned papers, the authors assume constant interest rates. Although the assumption of constant interest rates leads to highly tractable FX models, empirical results have confirmed that such models do not reflect the market reality, especially for long-dated hybrid FX products. For these products, the fluctuations of both the exchange rate and the interest rates are critical, so that the constant interest rates assumption is clearly inappropriate for reliable valuation and hedging.Let us comment briefly on the existing literature in the same vein. van Haastrecht et al. [12] have extended the stochastic volatility model of Schobel and Zhu [13] to equity/currency derivatives by including stochastic interest rates and assuming all driving model factors to be instantaneously correlated. Since their model is based on the Gaussian processes, it enjoys analytical tractability even in the most general case of a full correlation structure. By contrast, when the squared volatility is driven by the CIR process and the interest rate is driven either by the Vasicek [14] or the Cox et al. [2] process, a full correlation structure leads to intractability of equity options even under a partial correlation of the driving factors, as have been documented by, among others, van Haastrecht and Pelsser [15] and Grzelak and Oosterlee [16, 17], who examined, in particular, the Heston/Vasicek and Heston/CIR hybrid models (see also Grzelak et al. [18], where the Schobel–Zhu/Hull–White and Heston/Hull–White models for equity derivatives are studied).Our goal is to derive semianalytical solutions for prices of plain-vanilla FX options in a model in which the instantaneous volatility component is specified by the extended Heston model with log-normally and log-uniformly distributed jump amplitudes for the exchange rate and the volatility process, respectively, whereas the short-term interest rates for the domestic and foreign economies are governed by the independent CIR processes. The model thus incorporates important empirical characteristics of exchange rate return variability: (a) the correlation between the exchange rate and its stochastic volatility, (b) the presence of jumps in the exchange rate and volatility processes, and (c) the random character of interest rates. The practical importance of this feature of newly developed FX models is rather clear in view of the existence of complex FX products that have a long lifetime and are sensitive to smiles or skews in the market. The results obtained in this paper extend results obtained by Guoqing et al. In their model only the stock price process is subject to jumps, but the volatility of volatility is modeled by the Heston dynamics.The paper is organized as follows. In Section2, we set the foreign exchange model examined in this work. The options pricing problem is introduced in Section 3. The main result, Theorem 3 of Section 4, furnishes the pricing formula for FX options. It is worth stressing that the independence of volatility and interest rates appears to be a crucial assumption from the point of view of analytical tractability and thus it cannot be relaxed. Numerical illustrations of our method are provided in Section 5 where the diffusion and jump-diffusion models are compared.


## 2. The Heston/CIR Jump-Diffusion Foreign Exchange Model

Let(Ω,F,P) be an underlying probability space. Let the exchange rate Q=(Qt)t∈[0,T], its instantaneous squared volatility v=(vt)t∈[0,T], the domestic short-term interest rates r=(rt)t∈[0,T], and the foreign short-term interest rate r^=(r^t)t∈[0,T] be governed by the following system of SDEs:(1)dQt=rt-r^t-λμQQtdt+QtvtdWtQ+Qt-dZtQ,dvt=θ-κvtdt+σvvtdWtv+dZtv,drt=ad-bdrtdt+σdrtdWtd,dr^t=af-bfr^tdt+σfr^tdWtf.We work under the following standing assumptions:(A.1)
ProcessesWQ=(WtQ)t∈[0,T] and Wv=(Wtv)t∈[0,T] are correlated Brownian motions with a constant correlation coefficient, so that the quadratic covariation between the processes WQ and Wv satisfies d[WQ,Wv]t=ρdt for some constant ρ∈[-1,1].(A.2)
ProcessesWd=(Wtd)t∈[0,T] and Wf=(Wtf)t∈[0,T] are independent Brownian motions and they are also independent of the Brownian motions WQ and Wv; hence the processes Q, r, and r^ are independent.(A.3)
The processZtQ=∑k=1NtQJkQ is the compound Poisson process; specifically, the Poisson process NQ has the intensity λQ>0 and the random variables ln(1+JkQ), k=1,2,…, have the probability distribution N(ln⁡[1+μQ]-1/2σQ2,σQ2); hence the jump sizes (JkQ)k=1∞ are log-normally distributed on (-1,∞) with mean μQ>-1.(A.4)
The processZtv=∑k=1NtvJkv is the compound Poisson process; specifically, the Poisson process Nv has the intensity λv>0 and the jump sizes Jkv are uniformly distributed.(A.5)
The Poisson processesNQ, Nv and sequences of random variables (JkQ)k=1∞ and (Jkv)k=1∞ are mutually independent, as well as independent of the Brownian motions WQ, Wv, Wd, Wf.(A.6)
The model’s parameters satisfy the stability conditions:2θ>σv2>0, 2ad>σd2>0, and 2af>σf2>0 (see, e.g., Wong and Heyde [19]).Note that we postulate that the instantaneous squared volatility processv, the domestic short-term interest rate r, and the foreign interest rate r^ are independent stochastic processes. We will argue in what follows that this assumption is indeed crucial for analytical tractability. For brevity, we refer to the foreign exchange model given by SDEs (1) under Assumptions (A.1)–(A.6) as the Heston/CIR jump-diffusion FX model.


## 3. Foreign Exchange Call Option

We will first establish the general representation for the value of the foreign exchange (i.e., currency) European call option with maturityT>0 and a constant strike level K>0. The probability measure P is interpreted as the domestic spot martingale measure (i.e., the domestic risk-neutral probability). We denote by F=(Ft)t∈[0,T] the filtration generated by the Brownian motions WQ, Wv, Wd, Wf and the compound Poisson processes ZQ and Zv. We write EtP(·) and Pt(·) to denote the conditional expectation and the conditional probability under P with respect to the σ-field Ft, respectively. In our computations, we will adopt the “domestic” point of view, which will frequently be represented by the subscript d. Similarly, we will use the subscript f when referring to a foreign denominated variable. Hence the arbitrage price Ct(T,K) of the foreign exchange call option at time t∈[0,T] is given as the conditional expectation with respect to the σ-field Ft of the option’s payoff at expiration discounted by the domestic money market account; that is, (2)CtT,K=EtPexp⁡-∫tTruduCTT,K=EtPexp⁡-∫tTruduQT-K+,or, equivalently,(3)CtT,K=EtPexp⁡-∫tTruduQT1QT>K-KEtPexp⁡-∫tTrudu1QT>K.Similarly, the arbitrage price of the domestic discount bond maturing at time T equals, for every t∈[0,T], (4)Bdt,T=EtPexp⁡-∫tTrudu,and an analogous formula holds for the price process Bf(t,T) of the foreign discount bond under the foreign spot martingale measure (see, e.g., Chapter 14 in Musiela and Rutkowski [20]).As a preliminary step towards the general valuation result presented in Section4, we state the following well-known proposition (see, e.g., Cox et al. [2] or Chapter 10 in Musiela and Rutkowski [20]). It is worth stressing that we use here, in particular, the postulated independence of the foreign interest rate r^ and the exchange rate process Q. Under this standing assumption, the dynamics of the foreign bond price Bf(t,T) under the domestic spot martingale measure P can be seen as an immediate consequence of formula (14.3) in Musiela and Rutkowski [20]. The simple form of the dynamics of Bf(t,T) under P is a consequence of the postulated independence of Wf and WQ (see Assumption (A.2)). This crucial feature underpins our further calculations and thus it cannot be easily relaxed.Proposition 1.
The prices at datet of the domestic and foreign discount bonds maturing at time T>t in the CIR model are given by the following expressions: (5)Bdt,T=exp⁡mdt,T-ndt,Trt,Bft,T=exp⁡mft,T-nft,Tr^t,where for i∈{d,f}(6)mit,T=2aiσi2·log⁡γie1/2biT-tγicosh⁡γiT-t+1/2bisinh⁡γiT-t,nit,T=sinh⁡γiT-tγicosh⁡γiT-t+1/2bisinh⁡γiT-t,γi=12bi2+2σi2.The dynamics of the domestic and foreign bond prices under the domestic spot martingale measure P are given by (7)dBdt,T=Bdt,Trtdt-σdndt,TrtdWtd,dBft,T=Bft,Tr^tdt-σfnft,Tr^tdWtf.The following result is also well known (see, e.g., Section 14.1.1 in Musiela and Rutkowski [20]).Lemma 2.
The forward exchange rateF(t,T) at time t for settlement date T equals (8)Ft,T=Bft,TBdt,TQt.Since manifestlyQT=F(T,T), the option’s payoff at expiration can also be expressed as follows: (9)CTT,K=FT,T1FT,T>K-K1FT,T>K.Consequently, the option’s value at time t∈[0,T] admits the following representation: (10)CtT,K=EtPexp⁡-∫tTruduFT,T1FT,T>K-KEtPexp-∫tTrudu1FT,T>K.In what follows, we will frequently use the notation xt=ln⁡F(t,T), where t∈[0,T].


## 4. Pricing Formula for the FX Call Option

We are in a position to state the main result of the paper, which furnishes a semianalytical formula for the arbitrage price of the FX call option of European style under the Heston stochastic volatility for the exchange rate combined with the independent CIR models for the domestic and foreign short-term rates. Since the proof of Theorem3 relies on the derivation of the conditional characteristic function of the logarithm of the exchange rate, any suitable version of the Fourier inversion technique or simulation technique can be applied to obtain the option price. The interested reader is referred to, for instance, Carr and Madan [21, 22] or Lord and Kahl [23, 24] and the references therein, as well as the recent papers by Bernard et al. [25] and Levendorskii [26] who developed and examined in detail methods with essential improvements in accuracy and/or efficiency.Theorem 3.
Let the foreign exchange model be given by SDEs (1) under Assumptions (A.1)–(A.6). Then the price of the European FX call option equals, for every t∈[0,T], (11)CtT,K=QtBft,TP1t,Qt,vt,rt,r^t,K-KBdt,TP2t,Qt,vt,rt,r^t,K,where the bond prices Bd(t,T) and Bf(t,T) are given in Proposition 1 and the functions P1 and P2 are given by, for j=1,2, (12)Pjt,Qt,vt,rt,r^t,K=12+1π∫0∞Refjϕexp⁡-iϕln⁡Kiϕdϕ,where the Ft-conditional characteristic functions fj(ϕ)=fj(ϕ,t,Qt,vt,rt,r^t),j=1,2, of the random variable xT=ln(QT) under the probability measure P^T (see Definition 8) and PT (see Definition 6), respectively, are given by (13)f1ϕ=ctexp⁡λQτ1+μQiϕe-1/2ϕ2+iϕσQ2-1·exp⁡-iϕλQμQτ+λvτσve-ρ1+iϕ/σva-e-ρ1+iϕ/σvbρ1+iϕb-a-1+1+iϕρσvvt+θτexp⁡-iϕndt,Trt+∫tTadndu,Tduexp⁡1+iϕnft,Tr^t+∫tTafnfu,Tduexp⁡-G1τ,s1,s2vt-G2τ,s3,s4rt-G3τ,s5,s6r^t·exp⁡-θH1τ,s1,s2-adH2τ,s3,s4-afH3τ,s5,s6,(14)f2ϕ=ctexp⁡λQτ1+μQiϕe-1/2ϕ2+iϕσQ2-1·exp⁡-iϕλQμQτ+λvτσve-ρiϕ/σva-e-ρiϕ/σvbρiϕb-a-1+iϕρσvvt+θτexp⁡1-iϕndt,Trt+∫tTadndu,Tduexp⁡iϕnft,Tr^t+∫tTafnfu,Tduexp⁡-G1τ,q1,q2vt-G2τ,q3,q4rt-G3τ,q5,q6·exp⁡-θH1τ,q1,q2-adH2τ,q3,q4-afH3τ,q5,q6,where the functions G1, G2, G3, H1, H2, H3 are given in Lemma 5 and ct equals (15)ct=exp⁡iϕxt=exp⁡iϕln⁡Ft,T.Moreover, the constants s1, s2, s3, s4, s5, s6 are given by (16)s1=-1+iϕρσv,s2=-1+iϕ21-ρ22-1+iϕρκσv+1+iϕ2,s3=0,s4=-iϕ,s5=0,s6=1+iϕ,and the constants q1, q2, q3, q4, q5, q6 equal (17)q1=-iϕρσv,q2=-iϕ21-ρ22-iϕρκσv+iϕ2,q3=0,q4=1-iϕ,q5=0,q6=iϕ.


### 4.1. Auxiliary Results

The proof of Theorem3 hinges on a number of lemmas. We start by stating the well-known result, which can be easily obtained from Proposition 8.6.3.4 in Jeanblanc et al. [27]. Let us denote τ=T-t and let us set, for all 0≤t<T, (18)JQt,T≔∑k=NtQ+1NTQln⁡1+JkQ.Note that we use here Assumptions (A.3)–(A.5). The property (A.3) (resp., (A.4)) implies that the random variable JQ(t,T) (resp., ZTv-Ztv) is independent of the σ-field Ft. Let ν1 stand for the Gaussian distribution N(ln⁡(1+μQ)-1/2σQ2,σQ2) and let ν2 stand for the uniform distribution with density: (19)ν2z=1b-a1,a<z<b,0,else,where 0<a<b.Lemma 4.
(i) Under Assumptions (A.3) and (A.5), the following equalities are valid:(20)EtPexp⁡iϕJQt,T=EtPexp⁡iϕ∑k=NtQ+1NTQln⁡1+JkQ=exp⁡λQτ∫-∞+∞eiϕz-1ν1dz=exp⁡λQτ1+μQiϕe-1/2σQ2ϕ2+iϕ-1.(ii) Under Assumptions (A.4) and (A.5), the following equalities are valid for c=a+bi with a≤0: (21)EtPexp⁡cZTv-Ztv=EtPexp⁡c∑k=Ntv+1NTvJkv=exp⁡λvτ∫-∞+∞ecz-1ν2dz=exp⁡λvτecb-ecacb-a-1.The next result extends Lemma 6.1 in Ahlip and Rutkowski [28] (see also Duffie et al. [6]) where the model without the jump component in the dynamics of v was examined.Lemma 5.
Let the dynamics of processesv, r, and r^ be given by SDEs (1) with independent Brownian motions Wv, Wd, and Wf. For any complex numbers μ, λ, μ~, λ~, μ^, λ^, we set (22)Fτ,vt,rt,r^t=EtPexp⁡-λvT-μ∫tTvudu-λ~rT-μ~∫tTrudu-λ^r^T-μ^∫tTr^udu.Then(23)Fτ,vt,rt,r^t=exp⁡-G1τ,λ,μvt-G2τ,λ~,μ~rt-G3τ,λ^,μ^r^t-θH1τ,λ,μ-adH2τ,λ~,μ~-afH3τ,λ^,μ^,where(24)G1τ,λ,μ=λγ+κ+eγτγ-κ+2μeγτ-1σv2λeγτ-1+γ-κ+eγτγ+κ,G2τ,λ~,μ~=λ~γ~+bd+eγ~τγ~-bd+2μ~eγ~τ-1σd2λ~eγ~τ-1+γ~-bd+eγ~τγ~+bd,G3τ,λ^,μ^=λ^γ^+bf+eγ^τγ^-bf+2μ^eγ^τ-1σf2λ^eγ^τ-1+γ^-bf+eγ^τγ^+bf,H1τ,λ,μ=∫0τG1t,λ,μ+λvθ1+e-bG1t,λ,μ-e-aG1t,λ,μG1t,λ,μb-adt,H2τ,λ~,μ~=-2σd2·ln⁡2γ~eγ~+bdτ/2σd2λ~eγ~τ-1+γ~-bd+eγ~τγ~+bd,H3τ,λ^,μ^=-2σf2·ln⁡2γ^eγ^+bfτ/2σf2λ^eγ^τ-1+γ^-bf+eγ^τγ^+bf,where one denotes γ=κ2+2σv2μ, γ~=bd2+2σd2μ~, and γ^=bf2+2σf2μ^.Proof.
For the reader’s convenience, we sketch the proof of the lemma. Let us set, fort∈[0,T], (25)Mt=Fτ,vt,rt,r^t·exp⁡-μ∫0tvudu-μ~∫0trudu-μ^∫0tr^udu.Then the process M=(Mt)t∈[0,T] satisfies (26)Mt=EtPexp⁡-λvT-μ∫0Tvudu-λ~rT-μ~∫0Trudu-λ^r^T-μ^∫0Trudu,and thus it is an F-martingale under P. By applying the Itô formula to the right-hand side in (25) and by setting the drift term in the dynamics of M to be zero, we deduce that the function F(τ,v,r,r^) satisfies the following partial integrodifferential equation (PIDE): (27)-∂F∂τ+12σv2v∂2F∂v2+λv∫0∞Fτ,v+z,r,r^-Fτ,v,r,r^ν2dz+12σd2r∂2F∂r2+12σf2r^∂2F∂r^2+θ-κv∂F∂v+ad-bdr∂F∂r+af-bfr^∂F∂r^-μv+μ~r+μ^r^F=0,with the initial condition F(0,v,r,r^)=exp⁡(-λv-λ~r-λ^r^). We search for a solution to this PIDE in the form (28)Fτ,v,r,r^=exp⁡-G1τ,λ,μv-G2τ,λ~,μ~r-G3τ,λ^,μ^r^-θH1τ,λ,μ-adH2τ,λ~,μ~-afH3τ,λ^,μ^,with(29)G10,λ,μ=λ,G20,λ~,μ~=λ~,G30,λ^,μ^=λ^,H10,λ,μ=H20,λ~,μ~=H30,λ^,μ^=0.By substituting this expression in the PIDE and using part (ii) in Lemma 4, we obtain the following system of ODEs for the functions G1, G2, G3, H1, H2, H3 (for brevity, we suppress the last three arguments): (30)∂G1τ∂τ=-12σv2G12τ-κG1τ+μ,∂H1τ∂τ=G1τ+λvθ1+e-bG1τ-e-aG1τG1τb-a,∂G2τ∂τ=-12σd2G22τ-bdG2τ+μ~,∂H2τ∂τ=G2τ,∂G3τ∂τ=-12σf2G32τ-bfG3τ+μ^,∂H3τ∂τ=G3τ.By solving these equations, we obtain the stated formulae.Under the assumptions of Lemma5, it is possible to factorize F as a product of two conditional expectations. This means that the functions G1 (H1), G2 (H2), and G3 (H3) are of the same form, except that they correspond to different sets of parameters, θ, κ, σv for G1, H1, ad, bd, σr for G2, H2, and af, bf, σf for G3, H3. Note, however, that the roles played by the processes v, r, and r^ in our model are clearly different.It should also be stressed that no closed-form analytical expression forF(τ,vt,rt,r^t) is available in the case of correlated Brownian motions Wv, Wr, Wf. Brigo and Alfonsi [29], who deal with this issue in a different context, propose to use a simple Gaussian approximation, instead of the exact solution. More recently, Grzelak and Oosterlee [16] proposed more sophisticated approximations in the framework of the Heston/CIR hybrid model. We do not follow this path here, however, and we focus instead on finding a semianalytical solution, since this goal can be achieved under Assumptions (A.1)–(A.6).Let us now introduce a convenient change of the underlying probability measure, from the domestic spot martingale measureP to the domestic forward martingale measure PT.Definition 6.
Thedomestic forward martingale measurePT, equivalent to P on (Ω,FT), is defined by the Radon-Nikodým derivative process η=(ηt)t∈[0,T], where(31)ηt=dPTdPFt=exp⁡-∫0tσdndu,TrudWud-12∫0tσd2nd2u,Trudu.An application of the Girsanov theorem shows that the processWT=(WtT)t∈[0,T], which is given by the equality (32)WtT=Wtd+∫0tσdndu,Trudu,is the Brownian motion under the domestic forward martingale measure PT. Using the standard change of a numéraire technique, one can check that the price of the European foreign exchange call option admits the following representation under the probability measure PT: (33)CtT,K=Bdt,TEtPTFT,T1FT,T>K-KBdt,TEtPT1FT,T>K.The following auxiliary result is easy to establish and thus its proof is omitted. Recall thatJQ(t,T) is given by equality (18).Lemma 7.
Under Assumptions (A.1)–(A.6), the dynamics of the forward exchange rateF(t,T) under the domestic forward martingale measure PT are given by the SDE: (34)dFt,T=Ft,TdZtQ-λQμQdt+vtdWtQ+σdndt,TrtdWtT-σfnft,Tr^tdWtf,or, equivalently, (35)FT,T=Ft,Texp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du,where the dot · denotes the inner product in R3, (σ~F(t,T))t∈[0,T] is the R3-valued process (row vector) given by (36)σ~Ft,T=vt,σdndt,Trt,-σfnft,Tr^t,and W~T=(W~tT)t∈[0,T] stands for the R3-valued process (column vector) given by (37)W~tT=WtQ,WtT,Wtf∗.It is easy to check that, under Assumptions (A.1)–(A.6), the processW~T is the three-dimensional standard Brownian motion under PT. In view of Lemma 7, we have that (38)Bdt,TEtPTFT,T1FT,T>K=Bdt,T·EtPTFt,Texp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du·1FT,T>K=QtBft,TEtPTexp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du1FT,T>K.To deal with the first term in the right-hand side of (33), we introduce another auxiliary probability measure.Definition 8.
Themodified domestic forward martingale measureP^T, equivalent to PT on (Ω,FT), is defined by the Radon-Nikodým derivative process η^=(η^t)t∈[0,T], where(39)η^t=dP^TdPTFt=exp⁡∫0tσ~Fu,T·dW~uT-12∫0tσ~Fu,T2du.
Using Lemma7 and (8), we obtain (40)Bdt,TEtPTFT,T1FT,T>K=QtBft,TEtPT1FT,T>Kη^TEtPTη^Tand thus the Bayes formula and Definition 8 yield (41)Bdt,TEtPTFT,T1FT,T>K=QtBft,TEtP^T1FT,T>K.This shows that P^T is a martingale measure associated with the choice of the price process QtBf(t,T) as a numéraire asset. We are now in a position to state the following lemma.Lemma 9.
The price of the FX call option satisfies(42)CtT,K=QtBft,TP^TQT>K∣Ft-KBdt,TPTQT>K∣Ft,or, equivalently,(43)CtT,K=QtBft,TP^TxT>ln⁡K∣Ft-KBdt,TPTxT>ln⁡K∣Ft.To complete the proof Theorem3, it remains to evaluate the conditional probabilities arising in formula (43). By another application of the Girsanov theorem, one can check that the process (Q,v,r,r^) has the Markov property under the probability measures PT and P^T. In view of Proposition 1 and Lemma 2, the random variable xT is a function of QT, rT, and r^T. We thus conclude that (44)CtT,K=QtBft,TP1t,Qt,vt,rt,r^t,K-KBdt,TP2t,Qt,vt,rt,r^t,K,where we denote(45)P1t,Qt,vt,rt,r^t,K=P^TxT>ln⁡K∣Qt,vt,rt,r^t,P2t,Qt,vt,rt,r^t,K=PTxT>ln⁡K∣Qt,vt,rt,r^t.To obtain explicit formulae for the conditional probabilities above, it suffices to derive the corresponding conditional characteristic functions:(46)f1ϕ,t,Qt,vt,rt,r^t=EtP^Texp⁡iϕxT,f2ϕ,t,Qt,vt,rt,r^t=EtPTexp⁡iϕxT.The idea is to use the Radon-Nikodým derivatives in order to obtain convenient expressions for the characteristic functions in terms of conditional expectations under the domestic spot martingale measure P. The following lemma will allow us to achieve this goal.Lemma 10.
The following equality holds:(47)dP^TdPFt=exp⁡∫0tvudWuQ-∫0tσfnfu,Tr^udWuf-12∫0tvu+σf2nf2u,Tr^udu.Proof.
Straightforward computations show that(48)dP^TdPFt=dP^TdPTFtdPTdPFt=exp⁡∫0tσ~Fu,T·dW~uT-12∫0tσ~Fu,T2du·exp⁡-∫0tσdndu,TrudWud-12·∫0tσd2nd2u,Trudu=exp⁡∫0tvudWuQ+∫0tσdndu,TrudWuT-∫0tσfnfu,Tr^udWufexp⁡-12·∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu·exp⁡-∫0tσdndu,TrudWud-12·∫0tσd2nd2u,Trudu.Using (32), we now obtain (49)dP^TdPFt=exp⁡∫0tvudWuQ-∫0tσfnfu,Tr^udWuf-12∫0tvu+σf2nf2u,Tr^udu,which is the desired expression.In view of the formula established in Lemma10 and the abstract Bayes formula, to compute f1(ϕ)=f1(ϕ,t,Qt,vt,rt,r^t), it suffices to focus on the following conditional expectation under P: (50)f1ϕ=EtPexp⁡iϕxTexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf-12∫tTvu+σf2nf2u,Tr^udu.Similarly, in view of formula (31), we obtain for f2(ϕ)=f2(ϕ,t,Qt,vt,rt,r^t)(51)f2ϕ=EtPexp⁡iϕxT·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Trudu.To proceed, we will need the following result, which is an immediate consequence of Lemma 7.Corollary 11.
Under Assumptions (A.1)–(A.4), the processxt=ln⁡F(t,T) admits the following representation under the domestic forward martingale measure PT: (52)xT=xt+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du+JQt,T-λQμQT-t,or, more explicitly,(53)xT=xt+∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf-12∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu+∑k=NtQ+1NTQln⁡1+JkQ-λQμQT-t.Using equality (50) and Corollary 11, we obtain (54)f1ϕ=EtPexp⁡iϕxTexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf-12∫tTvu+σf2nf2u,Tr^udu=EtPexp⁡iϕxt+∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf·exp⁡-iϕ2∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^uduexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf·exp⁡-12∫tTvu+σf2nf2u,Tr^uduexp⁡iϕJQt,T-iϕλQμQT-t.For the sake of conciseness, we denoteα=1+iϕ, β=iϕ, and ct=exp(iϕxt). After simplifications and rearrangement, the formula above becomes (55)f1ϕ=ctEtPexp⁡α∫tTvudWuQ-12∫tTvudu·exp⁡β∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu·exp⁡-α∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu·exp⁡βJQt,T-βλQμQT-t.In view of Assumptions (A.1)–(A.6), we may use the following representation for the Brownian motionWQ: (56)WtQ=ρWtv+1-ρ2Wt,where W=(Wt)t∈[0,T] is a Brownian motion under P independent of the Brownian motions Wv, Wd, and Wf. Consequently, the conditional characteristic function f1(ϕ) can be represented in the following way: (57)f1ϕ=ctEtPexp⁡αρ∫tTvudWuv+α1-ρ2∫tTvudWu-α2∫tTvudu·exp⁡β∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu·exp⁡-α∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu·exp⁡βJQt,T-βλQT-tμQ.By combining Proposition1 with Definition 6, we obtain the following auxiliary result, which will be helpful in the proof of Theorem 3.Lemma 12.
Given the dynamics (1) of processes v, r, and r^ and formula (32), we obtain the following equalities: (58)∫tTvudWuv=1σvvT-vt-θτ+κ∫tTvudu-ZTv-Ztv,∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu=-ndt,Trt-∫tTadndu,Tdu+∫tTrudu,∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu=-nft,Tr^t-∫tTafnfu,Tdu+∫tTr^udu.Proof.
The first asserted formula is an immediate consequence of (1). For the second, we recall that the function nd(t,T) is known to satisfy the following differential equation, for any fixed T>0: (59)∂ndt,T∂t-12σd2nd2t,T-bdndt,T+1=0,with the terminal condition nd(T,T)=0. Therefore, using the Itô formula and equality (32), we obtain (60)dndt,Trt=rtdndt,T+ndt,Tdrt=rt12σd2nd2t,T+bdndt,T-1dt+ndt,Tad-bdrtdt+ndt,TσdrtdWtd=12σd2nd2t,Trtdt-rtdt+ndt,Taddt+ndt,TσdrtdWtd=-12σd2nd2t,Trtdt-rtdt+ndt,Taddt+ndt,TσdrtdWtT.This yields the second asserted formula, upon integration between t and T. The derivation of the last one is based on the same arguments and thus it is omitted.


### 4.2. Proof of Theorem3

We split the proof of Theorem3 into two steps in which we deal with f1(ϕ) and f2(ϕ).Step 1.
We will first computef1(ϕ). By combining (57) with the equalities derived in Lemma 12, we obtain the following representation for f1(ϕ): (61)f1ϕ=ctEtPexp⁡-αρσvvt+θτ+αρκσv-α2∫tTvudu+α1-ρ2∫tTvudWu+αρσvvT·exp⁡-βndt,Trt+∫tTadndu,Tdu+β∫tTrudu·exp⁡αnft,Tr^t+∫tTafnfu,Tdu-α∫tTr^uduexp⁡βJQt,T-βλQμQT-t-αρσvZTv-Ztv.Recall the well-known property that if ζ has the standard normal distribution then E(ezζ)=ez2/2 for any complex number z∈C.
Consequently, by conditioning first on the sample path of the process(v,r,r^) and using the independence of the processes (v,r,r^) and W under P and Lemma 4, we obtain (62)f1ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσve-ρ1+iϕ/σva-e-ρ1+iϕ/σvbρ1+iϕb-a+1+αρσvvt+θτexp⁡-βndt,Trt+∫tTadndu,Tduexp⁡αnft,Tr^t+∫tTafnfu,TduEtPexp⁡αρσvvT+α21-ρ22+αρκσv-α2∫tTvudu·exp⁡β∫tTrudu-α∫tTr^udu,where we denote γ=1-iϕ. This in turn implies that the following equality holds:(63)f1ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσve-ρ1+iϕ/σva-e-ρ1+iϕ/σvbρ1+iϕb-a+1+αρσvvt+θτexp⁡-βndt,Trt+∫tTadndu,Tduexp⁡αnft,Tr^t+∫tTafnfu,TduEtPexp⁡-s1vT-s2∫tTvudu-s3rT-s4∫tTrudu-s5r^T-s6∫tTr^udu,where the constants s1, s2, s3, s4, s5, s6 are given by (16). A direct application of Lemma 5 furnishes an explicit formula for f1(ϕ), as reported in the statement of Theorem 3.Step 2.
In order to compute the conditional characteristic function(64)f2ϕ=f2ϕ,t,Qt,vt,rt,r^t=EtPTexp⁡iϕxT,we proceed in an analogous manner as for f1(ϕ). We first recall that (see (51)) (65)f2ϕ=EtPexp⁡iϕxT·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Trudu.Therefore, using Corollary 11, we obtain (66)f2ϕ=ctEtPexp⁡iϕ∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf·exp⁡-iϕ12∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Truduexp⁡iϕJQt,T.Consequently, using formulae (32) and (56) and Lemma 4, we obtain the following expression for f2(ϕ): (67)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1-βλQμQτ·EtPexp⁡βρ∫tTvudWuv+1-ρ2∫tTvudWu-∫tTσfnfu,Tr^udWuf·exp⁡-β12∫tTvu+σf2nf2u,Tr^uduexp⁡-γ∫tTσdndu,TrudWud+12∫tTσd2nd2u,Trudu.Similarly as in the case of f1(ϕ), we condition on the sample path of the process (v,r,r^) and we use the postulated independence of the processes (v,r,r^) and W under P. By invoking also Lemma 4, we obtain (68)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1-βλQμQτ·EtPexp⁡βρ∫tTvudWuv+β21-ρ2-β2∫tTvudu·exp⁡-γ∫tTσdndu,TrudWud+12∫tTσd2nd2u,Trudu·exp-β∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu.Using Lemma 12, we conclude that(69)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσveρiϕ/σva-e-ρiϕ/σvbρiϕb-a+1+βρσvvt+θτexp⁡γndt,Trt+∫tTadndu,Tduexp⁡βnft,Tr^t+∫tTafnfu,TduEtPexp⁡-q1vT-q2∫tTvudu-q3rT-q4∫tTrudu-q5r^T-q6∫tTr^udu,with the coefficients q1, q2, q3, q4, q5, q6 reported in formula (17). Another straightforward application of Lemma 5 yields the closed-form expression (14) for the conditional characteristic function f2(ϕ).
To complete the proof of Theorem3, it suffices to combine formula (44) with the standard inversion formula (12) providing integral representations for the conditional probabilities: (70)P1t,Qt,vt,rt,r^t,K=P^TxT>ln⁡K∣Qt,vt,rt,r^t,P2t,Qt,vt,rt,r^t,K=PTxT>ln⁡K∣Qt,vt,rt,r^t.
This ends the derivation of the pricing formula for the foreign exchange call option. The price of the corresponding put option is readily available as well, due to the put-call parity relationship for FX options (see formula (72) in Section 5).


## 4.1. Auxiliary Results

The proof of Theorem3 hinges on a number of lemmas. We start by stating the well-known result, which can be easily obtained from Proposition 8.6.3.4 in Jeanblanc et al. [27]. Let us denote τ=T-t and let us set, for all 0≤t<T, (18)JQt,T≔∑k=NtQ+1NTQln⁡1+JkQ.Note that we use here Assumptions (A.3)–(A.5). The property (A.3) (resp., (A.4)) implies that the random variable JQ(t,T) (resp., ZTv-Ztv) is independent of the σ-field Ft. Let ν1 stand for the Gaussian distribution N(ln⁡(1+μQ)-1/2σQ2,σQ2) and let ν2 stand for the uniform distribution with density: (19)ν2z=1b-a1,a<z<b,0,else,where 0<a<b.Lemma 4.
(i) Under Assumptions (A.3) and (A.5), the following equalities are valid:(20)EtPexp⁡iϕJQt,T=EtPexp⁡iϕ∑k=NtQ+1NTQln⁡1+JkQ=exp⁡λQτ∫-∞+∞eiϕz-1ν1dz=exp⁡λQτ1+μQiϕe-1/2σQ2ϕ2+iϕ-1.(ii) Under Assumptions (A.4) and (A.5), the following equalities are valid for c=a+bi with a≤0: (21)EtPexp⁡cZTv-Ztv=EtPexp⁡c∑k=Ntv+1NTvJkv=exp⁡λvτ∫-∞+∞ecz-1ν2dz=exp⁡λvτecb-ecacb-a-1.The next result extends Lemma 6.1 in Ahlip and Rutkowski [28] (see also Duffie et al. [6]) where the model without the jump component in the dynamics of v was examined.Lemma 5.
Let the dynamics of processesv, r, and r^ be given by SDEs (1) with independent Brownian motions Wv, Wd, and Wf. For any complex numbers μ, λ, μ~, λ~, μ^, λ^, we set (22)Fτ,vt,rt,r^t=EtPexp⁡-λvT-μ∫tTvudu-λ~rT-μ~∫tTrudu-λ^r^T-μ^∫tTr^udu.Then(23)Fτ,vt,rt,r^t=exp⁡-G1τ,λ,μvt-G2τ,λ~,μ~rt-G3τ,λ^,μ^r^t-θH1τ,λ,μ-adH2τ,λ~,μ~-afH3τ,λ^,μ^,where(24)G1τ,λ,μ=λγ+κ+eγτγ-κ+2μeγτ-1σv2λeγτ-1+γ-κ+eγτγ+κ,G2τ,λ~,μ~=λ~γ~+bd+eγ~τγ~-bd+2μ~eγ~τ-1σd2λ~eγ~τ-1+γ~-bd+eγ~τγ~+bd,G3τ,λ^,μ^=λ^γ^+bf+eγ^τγ^-bf+2μ^eγ^τ-1σf2λ^eγ^τ-1+γ^-bf+eγ^τγ^+bf,H1τ,λ,μ=∫0τG1t,λ,μ+λvθ1+e-bG1t,λ,μ-e-aG1t,λ,μG1t,λ,μb-adt,H2τ,λ~,μ~=-2σd2·ln⁡2γ~eγ~+bdτ/2σd2λ~eγ~τ-1+γ~-bd+eγ~τγ~+bd,H3τ,λ^,μ^=-2σf2·ln⁡2γ^eγ^+bfτ/2σf2λ^eγ^τ-1+γ^-bf+eγ^τγ^+bf,where one denotes γ=κ2+2σv2μ, γ~=bd2+2σd2μ~, and γ^=bf2+2σf2μ^.Proof.
For the reader’s convenience, we sketch the proof of the lemma. Let us set, fort∈[0,T], (25)Mt=Fτ,vt,rt,r^t·exp⁡-μ∫0tvudu-μ~∫0trudu-μ^∫0tr^udu.Then the process M=(Mt)t∈[0,T] satisfies (26)Mt=EtPexp⁡-λvT-μ∫0Tvudu-λ~rT-μ~∫0Trudu-λ^r^T-μ^∫0Trudu,and thus it is an F-martingale under P. By applying the Itô formula to the right-hand side in (25) and by setting the drift term in the dynamics of M to be zero, we deduce that the function F(τ,v,r,r^) satisfies the following partial integrodifferential equation (PIDE): (27)-∂F∂τ+12σv2v∂2F∂v2+λv∫0∞Fτ,v+z,r,r^-Fτ,v,r,r^ν2dz+12σd2r∂2F∂r2+12σf2r^∂2F∂r^2+θ-κv∂F∂v+ad-bdr∂F∂r+af-bfr^∂F∂r^-μv+μ~r+μ^r^F=0,with the initial condition F(0,v,r,r^)=exp⁡(-λv-λ~r-λ^r^). We search for a solution to this PIDE in the form (28)Fτ,v,r,r^=exp⁡-G1τ,λ,μv-G2τ,λ~,μ~r-G3τ,λ^,μ^r^-θH1τ,λ,μ-adH2τ,λ~,μ~-afH3τ,λ^,μ^,with(29)G10,λ,μ=λ,G20,λ~,μ~=λ~,G30,λ^,μ^=λ^,H10,λ,μ=H20,λ~,μ~=H30,λ^,μ^=0.By substituting this expression in the PIDE and using part (ii) in Lemma 4, we obtain the following system of ODEs for the functions G1, G2, G3, H1, H2, H3 (for brevity, we suppress the last three arguments): (30)∂G1τ∂τ=-12σv2G12τ-κG1τ+μ,∂H1τ∂τ=G1τ+λvθ1+e-bG1τ-e-aG1τG1τb-a,∂G2τ∂τ=-12σd2G22τ-bdG2τ+μ~,∂H2τ∂τ=G2τ,∂G3τ∂τ=-12σf2G32τ-bfG3τ+μ^,∂H3τ∂τ=G3τ.By solving these equations, we obtain the stated formulae.Under the assumptions of Lemma5, it is possible to factorize F as a product of two conditional expectations. This means that the functions G1 (H1), G2 (H2), and G3 (H3) are of the same form, except that they correspond to different sets of parameters, θ, κ, σv for G1, H1, ad, bd, σr for G2, H2, and af, bf, σf for G3, H3. Note, however, that the roles played by the processes v, r, and r^ in our model are clearly different.It should also be stressed that no closed-form analytical expression forF(τ,vt,rt,r^t) is available in the case of correlated Brownian motions Wv, Wr, Wf. Brigo and Alfonsi [29], who deal with this issue in a different context, propose to use a simple Gaussian approximation, instead of the exact solution. More recently, Grzelak and Oosterlee [16] proposed more sophisticated approximations in the framework of the Heston/CIR hybrid model. We do not follow this path here, however, and we focus instead on finding a semianalytical solution, since this goal can be achieved under Assumptions (A.1)–(A.6).Let us now introduce a convenient change of the underlying probability measure, from the domestic spot martingale measureP to the domestic forward martingale measure PT.Definition 6.
Thedomestic forward martingale measurePT, equivalent to P on (Ω,FT), is defined by the Radon-Nikodým derivative process η=(ηt)t∈[0,T], where(31)ηt=dPTdPFt=exp⁡-∫0tσdndu,TrudWud-12∫0tσd2nd2u,Trudu.An application of the Girsanov theorem shows that the processWT=(WtT)t∈[0,T], which is given by the equality (32)WtT=Wtd+∫0tσdndu,Trudu,is the Brownian motion under the domestic forward martingale measure PT. Using the standard change of a numéraire technique, one can check that the price of the European foreign exchange call option admits the following representation under the probability measure PT: (33)CtT,K=Bdt,TEtPTFT,T1FT,T>K-KBdt,TEtPT1FT,T>K.The following auxiliary result is easy to establish and thus its proof is omitted. Recall thatJQ(t,T) is given by equality (18).Lemma 7.
Under Assumptions (A.1)–(A.6), the dynamics of the forward exchange rateF(t,T) under the domestic forward martingale measure PT are given by the SDE: (34)dFt,T=Ft,TdZtQ-λQμQdt+vtdWtQ+σdndt,TrtdWtT-σfnft,Tr^tdWtf,or, equivalently, (35)FT,T=Ft,Texp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du,where the dot · denotes the inner product in R3, (σ~F(t,T))t∈[0,T] is the R3-valued process (row vector) given by (36)σ~Ft,T=vt,σdndt,Trt,-σfnft,Tr^t,and W~T=(W~tT)t∈[0,T] stands for the R3-valued process (column vector) given by (37)W~tT=WtQ,WtT,Wtf∗.It is easy to check that, under Assumptions (A.1)–(A.6), the processW~T is the three-dimensional standard Brownian motion under PT. In view of Lemma 7, we have that (38)Bdt,TEtPTFT,T1FT,T>K=Bdt,T·EtPTFt,Texp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du·1FT,T>K=QtBft,TEtPTexp⁡JQt,T-λQμQT-t+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du1FT,T>K.To deal with the first term in the right-hand side of (33), we introduce another auxiliary probability measure.Definition 8.
Themodified domestic forward martingale measureP^T, equivalent to PT on (Ω,FT), is defined by the Radon-Nikodým derivative process η^=(η^t)t∈[0,T], where(39)η^t=dP^TdPTFt=exp⁡∫0tσ~Fu,T·dW~uT-12∫0tσ~Fu,T2du.
Using Lemma7 and (8), we obtain (40)Bdt,TEtPTFT,T1FT,T>K=QtBft,TEtPT1FT,T>Kη^TEtPTη^Tand thus the Bayes formula and Definition 8 yield (41)Bdt,TEtPTFT,T1FT,T>K=QtBft,TEtP^T1FT,T>K.This shows that P^T is a martingale measure associated with the choice of the price process QtBf(t,T) as a numéraire asset. We are now in a position to state the following lemma.Lemma 9.
The price of the FX call option satisfies(42)CtT,K=QtBft,TP^TQT>K∣Ft-KBdt,TPTQT>K∣Ft,or, equivalently,(43)CtT,K=QtBft,TP^TxT>ln⁡K∣Ft-KBdt,TPTxT>ln⁡K∣Ft.To complete the proof Theorem3, it remains to evaluate the conditional probabilities arising in formula (43). By another application of the Girsanov theorem, one can check that the process (Q,v,r,r^) has the Markov property under the probability measures PT and P^T. In view of Proposition 1 and Lemma 2, the random variable xT is a function of QT, rT, and r^T. We thus conclude that (44)CtT,K=QtBft,TP1t,Qt,vt,rt,r^t,K-KBdt,TP2t,Qt,vt,rt,r^t,K,where we denote(45)P1t,Qt,vt,rt,r^t,K=P^TxT>ln⁡K∣Qt,vt,rt,r^t,P2t,Qt,vt,rt,r^t,K=PTxT>ln⁡K∣Qt,vt,rt,r^t.To obtain explicit formulae for the conditional probabilities above, it suffices to derive the corresponding conditional characteristic functions:(46)f1ϕ,t,Qt,vt,rt,r^t=EtP^Texp⁡iϕxT,f2ϕ,t,Qt,vt,rt,r^t=EtPTexp⁡iϕxT.The idea is to use the Radon-Nikodým derivatives in order to obtain convenient expressions for the characteristic functions in terms of conditional expectations under the domestic spot martingale measure P. The following lemma will allow us to achieve this goal.Lemma 10.
The following equality holds:(47)dP^TdPFt=exp⁡∫0tvudWuQ-∫0tσfnfu,Tr^udWuf-12∫0tvu+σf2nf2u,Tr^udu.Proof.
Straightforward computations show that(48)dP^TdPFt=dP^TdPTFtdPTdPFt=exp⁡∫0tσ~Fu,T·dW~uT-12∫0tσ~Fu,T2du·exp⁡-∫0tσdndu,TrudWud-12·∫0tσd2nd2u,Trudu=exp⁡∫0tvudWuQ+∫0tσdndu,TrudWuT-∫0tσfnfu,Tr^udWufexp⁡-12·∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu·exp⁡-∫0tσdndu,TrudWud-12·∫0tσd2nd2u,Trudu.Using (32), we now obtain (49)dP^TdPFt=exp⁡∫0tvudWuQ-∫0tσfnfu,Tr^udWuf-12∫0tvu+σf2nf2u,Tr^udu,which is the desired expression.In view of the formula established in Lemma10 and the abstract Bayes formula, to compute f1(ϕ)=f1(ϕ,t,Qt,vt,rt,r^t), it suffices to focus on the following conditional expectation under P: (50)f1ϕ=EtPexp⁡iϕxTexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf-12∫tTvu+σf2nf2u,Tr^udu.Similarly, in view of formula (31), we obtain for f2(ϕ)=f2(ϕ,t,Qt,vt,rt,r^t)(51)f2ϕ=EtPexp⁡iϕxT·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Trudu.To proceed, we will need the following result, which is an immediate consequence of Lemma 7.Corollary 11.
Under Assumptions (A.1)–(A.4), the processxt=ln⁡F(t,T) admits the following representation under the domestic forward martingale measure PT: (52)xT=xt+∫tTσ~Fu,T·dW~uT-12∫tTσ~Fu,T2du+JQt,T-λQμQT-t,or, more explicitly,(53)xT=xt+∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf-12∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu+∑k=NtQ+1NTQln⁡1+JkQ-λQμQT-t.Using equality (50) and Corollary 11, we obtain (54)f1ϕ=EtPexp⁡iϕxTexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf-12∫tTvu+σf2nf2u,Tr^udu=EtPexp⁡iϕxt+∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf·exp⁡-iϕ2∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^uduexp⁡∫tTvudWuQ-∫tTσfnfu,Tr^udWuf·exp⁡-12∫tTvu+σf2nf2u,Tr^uduexp⁡iϕJQt,T-iϕλQμQT-t.For the sake of conciseness, we denoteα=1+iϕ, β=iϕ, and ct=exp(iϕxt). After simplifications and rearrangement, the formula above becomes (55)f1ϕ=ctEtPexp⁡α∫tTvudWuQ-12∫tTvudu·exp⁡β∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu·exp⁡-α∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu·exp⁡βJQt,T-βλQμQT-t.In view of Assumptions (A.1)–(A.6), we may use the following representation for the Brownian motionWQ: (56)WtQ=ρWtv+1-ρ2Wt,where W=(Wt)t∈[0,T] is a Brownian motion under P independent of the Brownian motions Wv, Wd, and Wf. Consequently, the conditional characteristic function f1(ϕ) can be represented in the following way: (57)f1ϕ=ctEtPexp⁡αρ∫tTvudWuv+α1-ρ2∫tTvudWu-α2∫tTvudu·exp⁡β∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu·exp⁡-α∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu·exp⁡βJQt,T-βλQT-tμQ.By combining Proposition1 with Definition 6, we obtain the following auxiliary result, which will be helpful in the proof of Theorem 3.Lemma 12.
Given the dynamics (1) of processes v, r, and r^ and formula (32), we obtain the following equalities: (58)∫tTvudWuv=1σvvT-vt-θτ+κ∫tTvudu-ZTv-Ztv,∫tTσdndu,TrudWuT-12∫tTσd2nd2u,Trudu=-ndt,Trt-∫tTadndu,Tdu+∫tTrudu,∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu=-nft,Tr^t-∫tTafnfu,Tdu+∫tTr^udu.Proof.
The first asserted formula is an immediate consequence of (1). For the second, we recall that the function nd(t,T) is known to satisfy the following differential equation, for any fixed T>0: (59)∂ndt,T∂t-12σd2nd2t,T-bdndt,T+1=0,with the terminal condition nd(T,T)=0. Therefore, using the Itô formula and equality (32), we obtain (60)dndt,Trt=rtdndt,T+ndt,Tdrt=rt12σd2nd2t,T+bdndt,T-1dt+ndt,Tad-bdrtdt+ndt,TσdrtdWtd=12σd2nd2t,Trtdt-rtdt+ndt,Taddt+ndt,TσdrtdWtd=-12σd2nd2t,Trtdt-rtdt+ndt,Taddt+ndt,TσdrtdWtT.This yields the second asserted formula, upon integration between t and T. The derivation of the last one is based on the same arguments and thus it is omitted.


## 4.2. Proof of Theorem3

We split the proof of Theorem3 into two steps in which we deal with f1(ϕ) and f2(ϕ).Step 1.
We will first computef1(ϕ). By combining (57) with the equalities derived in Lemma 12, we obtain the following representation for f1(ϕ): (61)f1ϕ=ctEtPexp⁡-αρσvvt+θτ+αρκσv-α2∫tTvudu+α1-ρ2∫tTvudWu+αρσvvT·exp⁡-βndt,Trt+∫tTadndu,Tdu+β∫tTrudu·exp⁡αnft,Tr^t+∫tTafnfu,Tdu-α∫tTr^uduexp⁡βJQt,T-βλQμQT-t-αρσvZTv-Ztv.Recall the well-known property that if ζ has the standard normal distribution then E(ezζ)=ez2/2 for any complex number z∈C.
Consequently, by conditioning first on the sample path of the process(v,r,r^) and using the independence of the processes (v,r,r^) and W under P and Lemma 4, we obtain (62)f1ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσve-ρ1+iϕ/σva-e-ρ1+iϕ/σvbρ1+iϕb-a+1+αρσvvt+θτexp⁡-βndt,Trt+∫tTadndu,Tduexp⁡αnft,Tr^t+∫tTafnfu,TduEtPexp⁡αρσvvT+α21-ρ22+αρκσv-α2∫tTvudu·exp⁡β∫tTrudu-α∫tTr^udu,where we denote γ=1-iϕ. This in turn implies that the following equality holds:(63)f1ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσve-ρ1+iϕ/σva-e-ρ1+iϕ/σvbρ1+iϕb-a+1+αρσvvt+θτexp⁡-βndt,Trt+∫tTadndu,Tduexp⁡αnft,Tr^t+∫tTafnfu,TduEtPexp⁡-s1vT-s2∫tTvudu-s3rT-s4∫tTrudu-s5r^T-s6∫tTr^udu,where the constants s1, s2, s3, s4, s5, s6 are given by (16). A direct application of Lemma 5 furnishes an explicit formula for f1(ϕ), as reported in the statement of Theorem 3.Step 2.
In order to compute the conditional characteristic function(64)f2ϕ=f2ϕ,t,Qt,vt,rt,r^t=EtPTexp⁡iϕxT,we proceed in an analogous manner as for f1(ϕ). We first recall that (see (51)) (65)f2ϕ=EtPexp⁡iϕxT·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Trudu.Therefore, using Corollary 11, we obtain (66)f2ϕ=ctEtPexp⁡iϕ∫tTvudWuQ+∫tTσdndu,TrudWuT-∫tTσfnfu,Tr^udWuf·exp⁡-iϕ12∫tTvu+σd2nd2u,Tru+σf2nf2u,Tr^udu·exp⁡-∫tTσdndu,TrudWud-12∫tTσd2nd2u,Truduexp⁡iϕJQt,T.Consequently, using formulae (32) and (56) and Lemma 4, we obtain the following expression for f2(ϕ): (67)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1-βλQμQτ·EtPexp⁡βρ∫tTvudWuv+1-ρ2∫tTvudWu-∫tTσfnfu,Tr^udWuf·exp⁡-β12∫tTvu+σf2nf2u,Tr^uduexp⁡-γ∫tTσdndu,TrudWud+12∫tTσd2nd2u,Trudu.Similarly as in the case of f1(ϕ), we condition on the sample path of the process (v,r,r^) and we use the postulated independence of the processes (v,r,r^) and W under P. By invoking also Lemma 4, we obtain (68)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1-βλQμQτ·EtPexp⁡βρ∫tTvudWuv+β21-ρ2-β2∫tTvudu·exp⁡-γ∫tTσdndu,TrudWud+12∫tTσd2nd2u,Trudu·exp-β∫tTσfnfu,Tr^udWuf+12∫tTσf2nf2u,Tr^udu.Using Lemma 12, we conclude that(69)f2ϕ=ctexp⁡λQτ1+μQβe-1/2βγσQ2-1·exp⁡-βλQμQτ+λvτσveρiϕ/σva-e-ρiϕ/σvbρiϕb-a+1+βρσvvt+θτexp⁡γndt,Trt+∫tTadndu,Tduexp⁡βnft,Tr^t+∫tTafnfu,TduEtPexp⁡-q1vT-q2∫tTvudu-q3rT-q4∫tTrudu-q5r^T-q6∫tTr^udu,with the coefficients q1, q2, q3, q4, q5, q6 reported in formula (17). Another straightforward application of Lemma 5 yields the closed-form expression (14) for the conditional characteristic function f2(ϕ).
To complete the proof of Theorem3, it suffices to combine formula (44) with the standard inversion formula (12) providing integral representations for the conditional probabilities: (70)P1t,Qt,vt,rt,r^t,K=P^TxT>ln⁡K∣Qt,vt,rt,r^t,P2t,Qt,vt,rt,r^t,K=PTxT>ln⁡K∣Qt,vt,rt,r^t.
This ends the derivation of the pricing formula for the foreign exchange call option. The price of the corresponding put option is readily available as well, due to the put-call parity relationship for FX options (see formula (72) in Section 5).


## 5. Numerical Results

The goal of the final section is to illustrate our approach by means of numerical examples in which we apply our FX market model, that is, the Heston/CIR jump-diffusion model, and we compare this approach with other related models that were proposed in Moretto et al. [30] and Ahlip and Rutkowski [1] to deal with the exchange rate derivatives.Let us start by noting that the foreign exchange market differs from equity markets in that quotes for options are not made in terms of strikes. Indeed, the FX option prices are quoted in terms of the associated implied volatilities for a fixedforward deltaΔF and a fixed time to expiry τ=T-t. For more information about the market conventions, the interested reader is referred to, for instance, Moretto et al. [30] or Reiswich and Uwe [31].For a quoted volatilityσ, the corresponding strike price K is obtained using the following conversion formula, which is based on the classic Garman-Kohlhagen lognormal model for the exchange rate: (71)K=Ft,Texp⁡-δστN-1δΔF+12σ2τ,where N-1 is the inverse of the standard normal cumulative distribution function and the auxiliary parameter δ satisfies δ=1 (δ=-1, resp.) for the call (put, resp.) option. Formula (71) makes it clear that market quotations prices based on the implied volatility for fixed deltas are in fact equivalent to quoting prices for fixed strikes. For more details, the interested reader is referred to Hakala and Wystup [32] or Reiswich and Uwe [31].Another relevant feature is that currency derivatives are based on the notion ofat-the-money forward (ATMF) rate, that is, the forward exchange rate F(t,T) obtained by exploiting the interest rate parity implicit in (8). Recall that the universal put-call parity formula for plain-vanilla foreign exchange options reads (72)CtT,K-PtT,K=QtBft,T-KBdt,T,where Ct(T,K) and Pt(T,K) are prices of currency call and put options, respectively. In particular, the prices of ATMF call and put options are equal in any arbitrage-free market model.


### 5.1. Market Data

In the numerical results presented in Tables1, 2, and 3, we make use (with the kind permission of the authors) of the data for the USD/EUR exchange rate derivatives and interest rates from the paper by Moretto et al. [30] (see page 469 therein).Table 1
Market volatilityσMKT for USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Delta
−10%
−15%
−25%
ATMF (50%)
25%
15%
10%
1 M
10.36%
10.09%
9.73%
9.30%
9.15%
9.18%
9.25%
2 M
10.28%
10.01%
9.65%
9.25%
9.15%
9.22%
9.31%
3 M
10.22%
9.95%
9.62%
9.25%
9.19%
9.28%
9.39%
6 M
10.23%
9.95%
9.64%
9.35%
9.39%
9.55%
9.74%
9 M
10.22%
9.96%
9.96%
9.40%
9.49%
9.68%
9.88%
1 Y
10.24%
9.98%
9.69%
9.45%
9.56%
9.77%
9.99%
2 Y
10.28%
10.02%
9.74%
9.55%
9.72%
9.98%
10.24%Table 2
Market strike prices for USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Strike
−10%
−15%
−25%
ATMF (50%)
25%
15%
10%
1 M
1.1651
1.1745
1.1877
1.2101
1.2317
1.2435
1.2519
2 M
1.1496
1.1626
1.1807
1.2116
1.2421
1.2591
1.2712
3 M
1.1370
1.1529
1.1752
1.2134
1.2518
1.2735
1.2891
6 M
1.1129
1.1350
1.1660
1.2189
1.2753
1.3081
1.3324
9 M
1.0968
1.1233
1.1609
1.2246
1.2951
1.3369
1.3680
1 Y
1.0843
1.1147
1.1579
1.2307
1.3140
1.3638
1.4013
2 Y
1.0561
1.0984
1.1596
1.2562
1.3826
1.4606
1.5205Table 3
Market domestic (USD) and foreign (EUR) interest rates on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Rates
r
d
r
f
1 M
3.14%
2.09%
2 M
3.22%
2.09%
3 M
3.32%
2.10%
6 M
3.50%
2.09%
9 M
3.60%
2.09%
1 Y
3.68%
2.09%
2 Y
4.02%
2.19%


### 5.2. Comparison of Model Prices

The dynamics of the exchange rate and volatility, as given by (1), involve the parameters: λQ, μQ, σQ, λv, κ, θ, and σv. In addition, there are three parameters for each of the interest rates. In our numerical examples, the values of parameters κ, θ, and σv are borrowed from Moretto et al. [30], who proposed an extension of the Heston model for the exchange rate under the assumption of constant interest rates, as represented by the market yield curve. It should be acknowledged that the choice of interest rate parameters in our model is rather artificial and it was made for illustrative purposes only. We used the following values of parameters for the Heston/CIR (HCIR) model and the Heston/CIR/Log Normal/Log Uniform Jump-Diffusion (HCIR-LN-LU) model: ad=0.0332; bd=0.03; af=0.021; bf=0.024; σd=0.25; σf=0.24; Q0=1.2087; and ρ=0.9786. For each maturity date, the initial value v0=0.0078. The parameters given in Table 4 were taken from D’Ippoliti et al. [10] and were used for illustrative purposes only. The Heston model, the HCIR model examined in Ahlip and Rutkowski [1], and the HCIR-LN-LU model put forward in this paper were compared. Although the numerical results presented here are only preliminary, they nevertheless make it clear that jumps in exchange rate and volatility dynamics and the uncertain character of interest rates affect the valuation of foreign exchange derivatives.Table 4
Values of parameters of the HCIR/LN/LU model (1).
a
b
θ
κ
λ
Q
μ
Q
λ
v
ρ
σ
Q
0.1
0.2
0.02606
0.091
0.1000
0.00258
0.1000
0.9786
0.0644In Table5, we report prices of ATM calls for expiries ranging from one month to one year. We use here the ATM volatilities for different maturities given in Table 1, the corresponding ATM strike prices from Table 2, and the interest rates from Table 3. As one can see, the prices obtained using our model (HCIR-LN-LU) are higher than the prices for HCIR model and substantially higher than the prices obtained for the Heston model. In Table 6, we report 25% USD/EUR currency call option prices computed in the Heston model (refer to Figures 1 and 4), the HCIR model, and the present model using data of June 13, 2005. In the next three examples (see Tables 7, 8, and 9) we consider prices for ATM and 25% and 15% volatilities (refer to Figures 3, 2, and 5), respectively. For each maturity, the corresponding strike level was obtained using (71). Prices for ATM, 25%, and 15% USD/EUR currency call options are computed in the Heston, HCIR, and HCIR-LN-LU models using data of June 13, 2005, and parameter values given in Table 4.Table 5
Prices of ATM USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0128496
0.0128912
0.0129512
2 M
0.0190734
0.0192465
0.0233037
3 M
0.0245511
0.0249469
0.0322193
6 M
0.0390888
0.0407209
0.0585503
9 M
0.0526127
0.0556335
0.0816888
12 M
0.0656178
0.0699566
0.102082Table 6
Prices of 25% USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0005469
0.0054274
0.0062094
2 M
0.0088177
0.0089194
0.0140153
3 M
0.0116882
0.0119536
0.0224628
6 M
0.0205735
0.0216388
0.0457036
9 M
0.0297421
0.0323061
0.0663260
12 M
0.0389762
0.0437765
0.0849682Table 7
Prices for ATM USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0128496
0.0128546
0.0128697
(1.21019)
(1.21028)
(1.21028)
2 M
0.0190734
0.0190999
0.0222887
(1.21184)
(1.21217)
(1.21217)
3 M
0.024422
0.02449323
0.0319434
(1.21369)
(1.21428)
(1.21428)
6 M
0.0386608
0.0390158
0.0573285
(1.21992)
(1.22289)
(1.22289)
9 M
0.0518264
0.0527189
0.0790228
(1.22652)
(1.23329)
(1.23329)
12 M
0.0644786
0.0681417
0.0988909
(1.23356)
(1.24071)
(1.24071)Table 8
Prices for 25% USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0054139
0.0054181
0.0058999
(1.23193)
(1.23201)
(1.23201)
2 M
0.0086633
0.0086839
0.0138257
(1.24274)
(1.24308)
(1.24308)
3 M
0.0116882
0.0117434
0.0232394
(1.25188)
(1.25267)
(1.25267)
6 M
0.0204432
0.0207368
0.0480355
(1.27581)
(1.27892)
(1.27892)
9 M
0.0293928
0.0301153
0.0706473
(1.29652)
(1.30367)
(1.30367)
12 M
0.0385139
0.0399506
0.0909937
(1.31587)
(1.32884)
(1.32884)Table 9
Prices for 15% USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0031477
0.0031503
0.0032142
(1.24388)
(1.24397)
(1.24397)
2 M
0.0005302
0.0053178
0.0120455
(1.26005)
(1.27429)
(1.27429)
3 M
0.0074126
0.0074553
0.0208664
(1.27349)
(1.27429)
(1.27429)
6 M
0.0138051
0.0140366
0.0433107
(1.30848)
(1.311672
(1.31167)
9 M
0.0207329
0.0213553
0.0616101
(1.33813)
(1.34551)
(1.34551)
1 Y
0.0280719
0.0293209
0.0437222
(1.36544)
(1.33788)
(1.33788)Figure 1
Graphs for ATM options prices given in Table5.Figure 2
Graphs for 25% options prices given in Table6.Figure 3
Graphs for options prices given in Table7.Figure 4
Graphs for options prices given in Table8.Figure 5
Graph for option 15% prices given in Table9.


## 5.1. Market Data

In the numerical results presented in Tables1, 2, and 3, we make use (with the kind permission of the authors) of the data for the USD/EUR exchange rate derivatives and interest rates from the paper by Moretto et al. [30] (see page 469 therein).Table 1
Market volatilityσMKT for USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Delta
−10%
−15%
−25%
ATMF (50%)
25%
15%
10%
1 M
10.36%
10.09%
9.73%
9.30%
9.15%
9.18%
9.25%
2 M
10.28%
10.01%
9.65%
9.25%
9.15%
9.22%
9.31%
3 M
10.22%
9.95%
9.62%
9.25%
9.19%
9.28%
9.39%
6 M
10.23%
9.95%
9.64%
9.35%
9.39%
9.55%
9.74%
9 M
10.22%
9.96%
9.96%
9.40%
9.49%
9.68%
9.88%
1 Y
10.24%
9.98%
9.69%
9.45%
9.56%
9.77%
9.99%
2 Y
10.28%
10.02%
9.74%
9.55%
9.72%
9.98%
10.24%Table 2
Market strike prices for USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Strike
−10%
−15%
−25%
ATMF (50%)
25%
15%
10%
1 M
1.1651
1.1745
1.1877
1.2101
1.2317
1.2435
1.2519
2 M
1.1496
1.1626
1.1807
1.2116
1.2421
1.2591
1.2712
3 M
1.1370
1.1529
1.1752
1.2134
1.2518
1.2735
1.2891
6 M
1.1129
1.1350
1.1660
1.2189
1.2753
1.3081
1.3324
9 M
1.0968
1.1233
1.1609
1.2246
1.2951
1.3369
1.3680
1 Y
1.0843
1.1147
1.1579
1.2307
1.3140
1.3638
1.4013
2 Y
1.0561
1.0984
1.1596
1.2562
1.3826
1.4606
1.5205Table 3
Market domestic (USD) and foreign (EUR) interest rates on June 13, 2005 (original source of data: Banca Caboto S.p.A., Gruppo Intesa, Milano).
Rates
r
d
r
f
1 M
3.14%
2.09%
2 M
3.22%
2.09%
3 M
3.32%
2.10%
6 M
3.50%
2.09%
9 M
3.60%
2.09%
1 Y
3.68%
2.09%
2 Y
4.02%
2.19%


## 5.2. Comparison of Model Prices

The dynamics of the exchange rate and volatility, as given by (1), involve the parameters: λQ, μQ, σQ, λv, κ, θ, and σv. In addition, there are three parameters for each of the interest rates. In our numerical examples, the values of parameters κ, θ, and σv are borrowed from Moretto et al. [30], who proposed an extension of the Heston model for the exchange rate under the assumption of constant interest rates, as represented by the market yield curve. It should be acknowledged that the choice of interest rate parameters in our model is rather artificial and it was made for illustrative purposes only. We used the following values of parameters for the Heston/CIR (HCIR) model and the Heston/CIR/Log Normal/Log Uniform Jump-Diffusion (HCIR-LN-LU) model: ad=0.0332; bd=0.03; af=0.021; bf=0.024; σd=0.25; σf=0.24; Q0=1.2087; and ρ=0.9786. For each maturity date, the initial value v0=0.0078. The parameters given in Table 4 were taken from D’Ippoliti et al. [10] and were used for illustrative purposes only. The Heston model, the HCIR model examined in Ahlip and Rutkowski [1], and the HCIR-LN-LU model put forward in this paper were compared. Although the numerical results presented here are only preliminary, they nevertheless make it clear that jumps in exchange rate and volatility dynamics and the uncertain character of interest rates affect the valuation of foreign exchange derivatives.Table 4
Values of parameters of the HCIR/LN/LU model (1).
a
b
θ
κ
λ
Q
μ
Q
λ
v
ρ
σ
Q
0.1
0.2
0.02606
0.091
0.1000
0.00258
0.1000
0.9786
0.0644In Table5, we report prices of ATM calls for expiries ranging from one month to one year. We use here the ATM volatilities for different maturities given in Table 1, the corresponding ATM strike prices from Table 2, and the interest rates from Table 3. As one can see, the prices obtained using our model (HCIR-LN-LU) are higher than the prices for HCIR model and substantially higher than the prices obtained for the Heston model. In Table 6, we report 25% USD/EUR currency call option prices computed in the Heston model (refer to Figures 1 and 4), the HCIR model, and the present model using data of June 13, 2005. In the next three examples (see Tables 7, 8, and 9) we consider prices for ATM and 25% and 15% volatilities (refer to Figures 3, 2, and 5), respectively. For each maturity, the corresponding strike level was obtained using (71). Prices for ATM, 25%, and 15% USD/EUR currency call options are computed in the Heston, HCIR, and HCIR-LN-LU models using data of June 13, 2005, and parameter values given in Table 4.Table 5
Prices of ATM USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0128496
0.0128912
0.0129512
2 M
0.0190734
0.0192465
0.0233037
3 M
0.0245511
0.0249469
0.0322193
6 M
0.0390888
0.0407209
0.0585503
9 M
0.0526127
0.0556335
0.0816888
12 M
0.0656178
0.0699566
0.102082Table 6
Prices of 25% USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0005469
0.0054274
0.0062094
2 M
0.0088177
0.0089194
0.0140153
3 M
0.0116882
0.0119536
0.0224628
6 M
0.0205735
0.0216388
0.0457036
9 M
0.0297421
0.0323061
0.0663260
12 M
0.0389762
0.0437765
0.0849682Table 7
Prices for ATM USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0128496
0.0128546
0.0128697
(1.21019)
(1.21028)
(1.21028)
2 M
0.0190734
0.0190999
0.0222887
(1.21184)
(1.21217)
(1.21217)
3 M
0.024422
0.02449323
0.0319434
(1.21369)
(1.21428)
(1.21428)
6 M
0.0386608
0.0390158
0.0573285
(1.21992)
(1.22289)
(1.22289)
9 M
0.0518264
0.0527189
0.0790228
(1.22652)
(1.23329)
(1.23329)
12 M
0.0644786
0.0681417
0.0988909
(1.23356)
(1.24071)
(1.24071)Table 8
Prices for 25% USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0054139
0.0054181
0.0058999
(1.23193)
(1.23201)
(1.23201)
2 M
0.0086633
0.0086839
0.0138257
(1.24274)
(1.24308)
(1.24308)
3 M
0.0116882
0.0117434
0.0232394
(1.25188)
(1.25267)
(1.25267)
6 M
0.0204432
0.0207368
0.0480355
(1.27581)
(1.27892)
(1.27892)
9 M
0.0293928
0.0301153
0.0706473
(1.29652)
(1.30367)
(1.30367)
12 M
0.0385139
0.0399506
0.0909937
(1.31587)
(1.32884)
(1.32884)Table 9
Prices for 15% USD/EUR European exchange rate call options using data of June 13, 2005. Values in brackets are strike levels obtained using formula (71).
Maturity
Heston price
HCIR price
HCIR-LN-LU price
1 M
0.0031477
0.0031503
0.0032142
(1.24388)
(1.24397)
(1.24397)
2 M
0.0005302
0.0053178
0.0120455
(1.26005)
(1.27429)
(1.27429)
3 M
0.0074126
0.0074553
0.0208664
(1.27349)
(1.27429)
(1.27429)
6 M
0.0138051
0.0140366
0.0433107
(1.30848)
(1.311672
(1.31167)
9 M
0.0207329
0.0213553
0.0616101
(1.33813)
(1.34551)
(1.34551)
1 Y
0.0280719
0.0293209
0.0437222
(1.36544)
(1.33788)
(1.33788)Figure 1
Graphs for ATM options prices given in Table5.Figure 2
Graphs for 25% options prices given in Table6.Figure 3
Graphs for options prices given in Table7.Figure 4
Graphs for options prices given in Table8.Figure 5
Graph for option 15% prices given in Table9.



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