Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates

Foreign exchange options are studied in the Heston stochastic volatility model for the exchange rate combined with the Cox et al. dynamics for the domestic and foreign stochastic interest rates. 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. The main result furnishes a semi-analytical formula for the price of the foreign exchange European call option. The FX options pricing formula is derived using the probabilistic approach, which leads to explicit expressions for conditional characteristic functions. Stylized numerical examples show that the dynamics of interest rates are important for the valuation of foreign exchange options. We argue that the model examined in this paper is the only analytically tractable version of the foreign exchange market model that combines the Heston stochastic volatility model for the exchange rate with the CIR dynamics for interest rates.

Keywords: Options pricing; Stochastic volatility; Currency derivatives; Continuous time models

Our goal is to derive a closed-form pricing formula for foreign exchange (FX) options in a model combining the Heston stochastic volatility for the exchange rate with the Cox–Ingersoll–Ross (CIR) dynamics for domestic and foreign interest rates, and thus it is termed the Heston/CIR model of the exchange rate. In particular, the model allows for correlation between the exchange rate process and its instantaneous volatility. In the seminal paper of Heston ([14]), the author notes 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 correlation between the volatility and the spot exchange rate returns and to properly specify this correlation. Hakala and Wystup ([13]) developed the Heston stochastic volatility model in the FX setting under the postulate of constant domestic and foreign interest rates. Although the assumption of constant interest rates is definitely very appealing, due to its simplicity, empirical results have confirmed that such models do not reflect the market reality, especially in the case of a new generation of 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 related research by other authors. Van Haastrecht et al. ([24]) extended the stochastic volatility model of Schöbel and Zhu ([22]) to equity/currency derivatives by including stochastic interest rates and assuming all driving model factors to be instantaneously correlated. It is notable that their model is based on Gaussian processes and thus 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 ([25]) or the Cox et al. ([8]) process, a full correlation structure leads to the intractability of equity options even under a partial correlation of the driving factors, as has been documented by, among others, Van Haastrecht and Pelsser ([23]) and Grzelak and Oosterlee ([10]), who examined, in particular, the Heston/Vasicek and Heston/CIR hybrid models (see also Grzelak et al. ([12]), who study the Schöbel–Zhu/Hull–White and Heston/Hull–White models for equity derivatives).

Our goal is to derive semi-analytical solutions for prices of plain-vanilla FX options in a model in which the instantaneous volatility component is specified by the Heston model, whereas the short-term rates for the domestic and foreign economies are governed by independent CIR processes. The model thus incorporates important empirical characteristics of exchange rate return variability: (a) correlations with stochastic volatility and (b) stochastic 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. In a recent study of Grzelak and Oosterlee ([11]) (see also the discussion and the references therein), the authors develope a model similar to ours, but with the short-term rates driven by Hull–White models. Their approach departs from the modeling approach adopted by the financial industry, which is based on a log-normally distributed foreign exchange rate combined with the Hull–White dynamics for short-term interest rates, but still preserves the Gaussian property of short-term rates. This implausible feature of interest rates is also relaxed in the present paper. Also, the model proposed by Grzelak and Oosterlee ([11]) postulates a full correlation matrix between the exchange rate, its Heston's volatility and interest rates, so that the model is not affine, in the usual sense. Hence it is not analytically tractable and thus the authors propose resorting to an ad hoc approximation of non-affine terms in the pricing PDE in order to facilitate the valuation of derivatives. By contrast, we focus here on a particular class of models in which the closed-form pricing formulae for plain-vanilla FX options are available.

The paper is organized as follows. In section 2, we set out the foreign exchange model examined in this work. The options pricing problem is introduced in section 3. The main result, theorem 4.1 of section 4, furnishes the pricing formula for FX options. In section 5, we present some preliminary numerical results illustrating the computational technique proposed in the preceding section. Section 6 concludes the paper by providing a brief discussion of the postulated model and the limitations of the method employed in this work. 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. For an extension of the valuation formula derived in this paper to forward-start foreign exchange options, we refer to the follow-up work of Ahlip and Rutkowski ([3]).

Let be an underlying probability space. We postulate that the joint dynamics of the exchange rate , its instantaneous squared volatility , the domestic short-term interest rates , and the foreign short-term interest rate are governed by the following SDEs:

(1)

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where

• A. and are correlated Brownian motions with a constant correlation coefficient, so that the quadratic covariation between and satisfies for some constant ,
• B. and are independent Brownian motions and they are also independent of the Brownian motions and (hence the processes , and are independent), and
• C. the model's parameters satisfy the stability conditions , , (see, for instance, Wong and Heyde ([26])).

Note that we postulate here that the volatility process , the domestic short-term interest rate, denoted as , and its foreign counterpart, denoted as , are independent stochastic processes. We will argue below (see, in particular, section 6) that this assumption is indeed crucial. For brevity, we refer to the FX model given by the SDEs (1) under assumptions (A.1)–(A.3) as the Heston/CIR foreign exchange model.

We will first establish the general representation for the value of the foreign exchange (i.e. currency) European call option with maturity and a constant strike level . The probability measure is interpreted as the domestic spot martingale measure (also known as the domestic risk-neutral probability). We denote by the filtration generated by the Brownian motions , , , and and we write and to denote the conditional expectation and the conditional probability under with respect to the -field , 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 of the foreign exchange call option at time is given as the conditional expectation with respect to the -field of the option's payoff at expiration discounted by the domestic money market account, that is

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or, equivalently,

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Similarly, the arbitrage price of the domestic discount bond maturing at time equals, for every ,

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and an analogous formula holds for the price process of the foreign discount bond under the foreign spot martingale measure (see, e.g. chapter 14 of Musiela and Rutkowski ([20])).

As a preliminary step towards the general valuation result presented in section 4, we state the following well-known proposition (see, e.g. Cox et al. ([8]) or chapter 10 of Musiela and Rutkowski ([20])). It is worth stressing that we use here, in particular, the postulated independence of the foreign interest rate and the exchange rate process . Under this standing assumption, the dynamics of the foreign bond price under the domestic spot martingale measure can be seen as an immediate consequence of formula (14.3) of Musiela and Rutkowski ([20]). The simple form of the dynamics of under is a consequence of the postulated independence of and (see assumption (A.2)). This crucial feature underpins further calculations and it cannot readily be relaxed (for more detailed comments on the analytical tractability of the model and its extensions, see also section 6).

The prices at date of the domestic and foreign discount bonds maturing at time in the CIR model are given by the following expressions:

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where, for

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and

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The dynamics of the domestic and foreign bond prices under the domestic spot martingale measure are given by

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The following result is also well known (see, for instance, section 14.1.1 of Musiela and Rutkowski ([20])).

The forward exchange rate at time for settlement date equals

(2)

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Since manifestly , the option's payoff at expiration can also be expressed as

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Consequently, the option's value at time admits the following representation:

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In the following we will frequently use the notation , where .

We are in a position to state the main result of the paper. It furnishes a semi-analytical 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 theorem 4.1 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 ([6], [7]) or Lord and Kahl ([17], [18]) and the references therein, as well as the recent papers by Bernard et al. ([4]) and Levendorskiĭ ([15]), who developed and examined in detail methods with essential improvements in accuracy and/or efficiency.

Assume that the foreign exchange model is given by the SDEs (1) under assumptions –. Then the price of the European FX call option equals, for every ,

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where the bond prices and are given in proposition 3.1, and the functions and are given by, for ,

(3)

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where the -conditional characteristic functions , , of the random variable under the probability measure see definition  and see definition 4.3), respectively, are given by

(4)

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and

(5)

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where the functions , , , , , and are given in lemma 4.2 and equals

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Moreover, the constants , , , , , and are given by

(6)

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and the constants , , , , , and equal

(7)

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The proof of theorem 4.1 hinges on a number of lemmas. We start with the following well-known technical result (see, e.g. Ahlip and Rutkowski ([2]) or Duffie et al. ([9])). For brevity, we write hereafter .

Let the dynamics of processes , , and be given by the SDEs (1) with independent Brownian motions , , and . For any complex numbers , , , , , and , we set

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Then

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where

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and

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where we denote , , and .

For the reader's convenience, we sketch the proof of the lemma. Let us set, for ,

(8)

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Then the process satisfies

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and thus it is an -martingale under . By applying the Itô formula to the right-hand side of (8) and by setting the drift term in the dynamics of to be zero, we deduce that the function satisfies the following PDE:

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subject to the initial condition . We search for a solution to this PDE in the form

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with

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and

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By substituting this expression in the PDE, we obtain the following system of ODEs for the functions , , , , , and (for brevity, we suppress the last three arguments):

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By solving these equations, we obtain the stated formulae.

Under the assumptions of lemma 4.2, it is possible to factorize as a product of two conditional expectations. This means that the functions (), (), and () are of the same form, except that they correspond to different sets of parameters, , , and for and , , , and for and , and , , and for and . Note, however, that the roles played by the processes , and in our model are clearly different. It should also be stressed that no closed-form analytical expression for is available in the case of correlated Brownian motions , , and . Brigo and Alfonsi ([5]), who deal with this issue in a different context, propose using a simple Gaussian approximation instead of the exact solution. More recently, Grzelak and Oosterlee ([10]) 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 the exact solution, which appears to be available under assumptions (A.1)–(A.3). For further comments regarding these standing assumptions, we refer to section 6.

Let us now introduce a convenient change of the underlying probability measure, from the domestic spot martingale measure to the domestic forward martingale measure .

The domestic forward martingale measure, equivalent to on , is defined by the Radon–Nikodým derivative process , where

(9)

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The Girsanov theorem shows that the process , which is given by the equality

(10)

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is the Brownian motion under the domestic forward martingale measure . 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 :

(11)

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The following auxiliary result is easy to establish and thus its proof is omitted.

Under assumptions –, the dynamics of the forward exchange rate under the domestic forward martingale measure are given by the SDE

(12)

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or, equivalently,

(13)

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where denotes the inner product in , is the -valued process row vector given by

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and is the -valued process column vector given by

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It is easy to check that, under assumptions (A.1)–(A.3), the process is the three-dimensional standard Brownian motion under . In view of lemma 4.4, we have

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To deal with the first term on the right-hand side of formula (11), we introduce another auxiliary probability measure, denoted by .

The modified domestic forward martingale measure, equivalent to on , is defined by the Radon–Nikodým derivative process , where

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Using equations (2) and (13), we obtain

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and thus the Bayes formula and definition yield

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This shows that is a martingale measure associated with the choice of the price process as a numéraire asset. We are now in a position to state the following lemma.

The price of the FX call option satisfies

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or, equivalently,

(14)

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To complete the proof of theorem 4.1 it thus remains to evaluate the conditional probabilities arising in the pricing formula (14). To this end, we first note that, due to Girsanov's theorem, it is possible to check that the process enjoys the Markov property under the probability measures and . In view of proposition 3.1 and lemma 3.2, the random variable is a function of and . We thus conclude that

(15)

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where we denote

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To obtain explicit formulae for the above conditional probabilities, it suffices to derive the corresponding conditional characteristic functions

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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 . The following lemma will serve to achieve this goal.

The following equality holds:

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Straightforward computations show that

Graph

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Using (10), we now obtain

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which is the desired expression.

In view of the formula established in lemma 4.7 and the abstract Bayes formula, to compute it suffices to focus on the following conditional expectation under :

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We will need the following observation, which is an immediate consequence of (12).

Under assumptions –, the process admits the following representation under the domestic forward martingale measure :

(16)

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or, more explicitly,

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From corollary 4.8, we obtain

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For conciseness, we denote , , and . After simplification and rearrangement, the above formula becomes

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In view of assumptions (A.1)–(A.3), we may use the following representation for the Brownian motion :

(17)

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where is a Brownian motion under independent of , , and .

Consequently, the conditional characteristic function can be represented in the following way:

(18)

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By combining proposition 3.1 with definition 4.3, we obtain the following auxiliary result, which will be used below.

Given the dynamics (1) of processes , , and and formula (10), we obtain the following equalities:

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The first formula is an immediate consequence of (1). For the second, let us recall that, for a fixed , the function is known to satisfy the following differential equation:

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with the terminal condition . Hence, using the Itô formula and equality (10), we obtain

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This yields the second formula upon integration between and . The derivation of the last formula is based on the same arguments and is thus omitted.

We now proceed to the proof of the main result.

(proof of theorem 4.1) By combining (18) with the equalities derived in lemma 4.9, we obtain the following representation for :

Graph

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Recall the well-known property that if has the standard normal distribution, then for any complex number . Therefore, by conditioning first on the sample path of the process and using the independence of the processes and under , we obtain

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This in turn implies that the following equality holds:

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where the constants , , , , , and are given by (6). A direct application of lemma 4.2 furnishes an explicit formula for , as reported in the statement of theorem 4.1. In order to compute the conditional characteristic function

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we proceed in an analogous manner as for . We first observe that, in view of (9),

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Therefore, by virtue of corollary 4.8,

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Consequently, using formulae (10) and (17), we obtain the following expression for :

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where we denote and . Similarly as in the case of , by conditioning on the sample path of the process and using the postulated independence of the processes and under , we obtain

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Using lemma 4.9, we conclude that

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Table 1 Market volatility (%) for the USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A. – Gruppo Intesa, Milano).

Table 2 Prices of ATM USD/EUR European exchange rate call options using data of June 13, 2005.

Table 3 Market strike prices for the USD/EUR derivative exchange rate on June 13, 2005 (original source of data: Banca Caboto S.p.A. – Gruppo Intesa, Milano).

Table 4 Market domestic (USD) and foreign (EUR) interest rates on June 13, 2005 (original source of data: Banca Caboto S.p.A. – Gruppo Intesa, Milano).

Table 5 Prices for 25% USD/EUR European exchange rate call options using data of June 13, 2005.

where the constants , , , , , and are given by (7). Another straightforward application of lemma 4.2 yields the closed-form expression (5) for the conditional characteristic function . To complete the proof of theorem 4.1, it suffices to combine formula (15) with the standard inversion formula (3) providing integral representations for the conditional probabilities

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This completes the derivation of the pricing formula for the foreign exchange call option. The price of the corresponding put option is also readily available due to the put–call parity relationship (20) for FX options.

The goal of this section is to illustrate our approach by numerical examples in which we apply our FX market model, that is the Heston/CIR model, and related models proposed in the past by other authors 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. Rather, option prices are quoted in terms of volatilities for a fixed forward delta and a fixed time to expiry (see, for instance, Moretto et al. ([19]) or Reiswich and Wystup ([21])). For a quoted volatility , the corresponding strike price is obtained using the following conversion formula, which is based on the classic Garman–Kohlhagen log-normal model for the exchange rate (for details, the interested reader is referred to Hakala and Wystup ([13]) or Reiswich and Wystup ([21]))

(19)

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where is the inverse of the standard normal cumulative distribution function and the auxiliary parameter satisfies (resp. ) for the call (resp. put) option. Formula (19) makes it clear that quoting prices in terms of volatility for fixed deltas is in fact equivalent to quoting prices for fixed strikes.

Another relevant feature is that currency derivatives are based on the notion of the at-the-money forward (ATMF) rate, that is the forward exchange rate obtained by exploiting the so-called interest rate parity implicit in equation (2). It is worth recalling that the put–call parity formula for plain-vanilla foreign exchange options reads

(20)

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where and are prices of currency call and put options, respectively. Hence the prices of ATMF call and put options are equal in any arbitrage-free market model.

In the numerical results presented below, 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 of (Moretto et al. ([19]), p. 469).

The dynamics of the exchange rate volatility, as given by equation (1), involve three parameters: , , and . In addition, there are three parameters for each of the interest rates, domestic and foreign. In our numerical examples, the values of parameters , , and are borrowed from Moretto et al. ([19]), 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. Although the numerical examples presented here are only preliminary, they nevertheless make it clear that the uncertain character of interest rates affects the valuation of foreign exchange derivatives.

The following values of the parameters were used for the Heston/CIR model in the computations reported below: , , , , , , , , , , and . For the MPT model, we use the calibration results from Moretto et al. ([19]) and we set and . For each maturity date, the initial value is obtained from the equality , where the market volatility is reported in table 1. The Heston model, the model put forward in this paper (the Heston/CIR model), and the model examined by Moretto et al. ([19]) (that is, the MPT model) were compared.

Table 2 reports prices of at-the-money calls for expiries ranging from one month to two years. We use here the ATM volatilities for the different maturities given in table 1, the corresponding ATM strike prices from table 3, and the interest rates from table 4. As one can see, the prices obtained using the Heston/CIR model are typically marginally higher than the prices obtained for the Heston model, as well as those obtained by Moretto et al. ([19]).

For the next example, we consider prices for 25% USD/EUR currency call options computed using the Heston model, the present model, and those obtained by Moretto et al. ([19]). From table 5 we see that, for short-dated options, the prices derived using the Heston/CIR model are marginally lower than the Heston and MPT prices, whereas for longer maturities the option prices computed using the present model seem to be higher that those obtained by the Heston and MPT models.

Let us comment briefly on plausible variants of the foreign exchange model examined in this paper. One could argue that it would be more natural to postulate that the exchange rate and interest rates are governed by (1), where , , , and are possibly correlated Brownian motions under a martingale measure with constant instantaneous correlations.

In general, it is possible to consider the model given by the SDEs (1) in which the quadratic covariations between the driving Brownian motions are

(21)

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Under the postulated full correlation structure (21), one can check that the following processes follow Brownian motions under the domestic forward martingale measure introduced in definition 4.3:

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This in turn implies, in particular, that the dynamics of , , and under are given by the following SDEs:

(22)

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Due to the presence of the terms and in the SDEs (22), it is rather clear that the model given by (1) would no longer be tractable using the approach presented in this paper if at least one of the correlation coefficients , or would be non-zero. In addition, a thorough inspection of the proof of theorem 4.1 shows that even under the postulate that , model (1) still remains intractable by the method elaborated here, unless one assumes, in addition, that the equalities hold.

It is worth noting that the postulated independence property of processes , , and was also crucial for all computations performed in section 4 within the model given by (1). Indeed, the closed-form expression for , obtained in lemma 4.2, is no longer available if the assumption of the independence of , , and is relaxed. Put anot her way, the assumption that is a necessary condition for the validity of lemma 4.2. To conclude, the foreign exchange model specified in section 2 by means of assumptions (A.1)–(A.3) is apparently the only analytically tractable version of the market model described by the set of SDEs (1).

It is fair to say that our numerical experiments are only preliminary. In future research, we intend to focus on numerical examples that specifically require stochastic interest rates and FX volatility, such as, for instance, dual currency bonds and other complex foreign exchange products with long maturities. We also hope to calibrate a whole matrix of option prices with different strikes and maturities and compare the improvement in fitting the market data with respect to the Heston model with deterministic interest rates and other recently developed foreign exchange models, such as, for instance, the models proposed in the recent papers of Van Haastrecht et al. ([24]) and Grzelak and Oosterlee ([11]).

The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460. The authors express their gratitude to Enrico Moretto and Uwe Wystup who kindly advised them with regard to the numerical examples presented in section 5. All remaining errors are our own.