We examine foreign exchange options in the jumpdiffusion version of the Heston stochastic volatility model for the exchange rate with lognormal jump amplitudes and the volatility model with loguniformly 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 shortterm 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.
We extend the results from Ahlip and Rutkowski [
In the seminal paper by Heston [
More recently, D’Ippoliti et al. [
Let us comment briefly on the existing literature in the same vein. van Haastrecht et al. [
Our goal is to derive semianalytical solutions for prices of plainvanilla FX options in a model in which the instantaneous volatility component is specified by the extended Heston model with lognormally and loguniformly distributed jump amplitudes for the exchange rate and the volatility process, respectively, whereas the shortterm 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 Section
Let
Processes
Processes
The process
The process
The Poisson processes
The model’s parameters satisfy the stability conditions:
Note that we postulate that the instantaneous squared volatility process
We will first establish the general representation for the value of the foreign exchange (i.e., currency) European call option with maturity
As a preliminary step towards the general valuation result presented in Section
The prices at date
The following result is also well known (see, e.g., Section 14.1.1 in Musiela and Rutkowski [
The forward exchange rate
Since manifestly
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 shortterm rates. Since the proof of Theorem
Let the foreign exchange model be given by SDEs (
The proof of Theorem
(i) Under Assumptions (A.3) and (A.5), the following equalities are valid:
The next result extends Lemma 6.1 in Ahlip and Rutkowski [
Let the dynamics of processes
For the reader’s convenience, we sketch the proof of the lemma. Let us set, for
Under the assumptions of Lemma
It should also be stressed that no closedform analytical expression for
Let us now introduce a convenient change of the underlying probability measure, from the domestic spot martingale measure
The
An application of the Girsanov theorem shows that the process
The following auxiliary result is easy to establish and thus its proof is omitted. Recall that
Under Assumptions (A.1)–(A.6), the dynamics of the forward exchange rate
It is easy to check that, under Assumptions (A.1)–(A.6), the process
The
Using Lemma
The price of the FX call option satisfies
To complete the proof Theorem
To obtain explicit formulae for the conditional probabilities above, it suffices to derive the corresponding conditional characteristic functions:
The following equality holds:
Straightforward computations show that
In view of the formula established in Lemma
Under Assumptions (A.1)–(A.4), the process
Using equality (
For the sake of conciseness, we denote
In view of Assumptions (A.1)–(A.6), we may use the following representation for the Brownian motion
By combining Proposition
Given the dynamics (
The first asserted formula is an immediate consequence of (
We split the proof of Theorem
We will first compute
Consequently, by conditioning first on the sample path of the process
In order to compute the conditional characteristic function
To complete the proof of Theorem
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 putcall parity relationship for FX options (see formula (
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 jumpdiffusion model, and we compare this approach with other related models that were proposed in Moretto et al. [
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 fixed
For a quoted volatility
Another relevant feature is that currency derivatives are based on the notion of
In the numerical results presented in Tables
Market volatility
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% 
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  1.5205 
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 



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% 
The dynamics of the exchange rate and volatility, as given by (
Values of parameters of the HCIR/LN/LU model (










0.1  0.2  0.02606  0.091  0.1000  0.00258  0.1000  0.9786  0.0644 
In Table
Prices of ATM USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity  Heston price  HCIR price  HCIRLNLU 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  0.102082 
Prices of 25% USD/EUR European exchange rate call options using data of June 13, 2005.
Maturity  Heston price  HCIR price  HCIRLNLU 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  0.0849682 
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 (
Maturity  Heston price  HCIR price  HCIRLNLU 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) 
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 (
Maturity  Heston price  HCIR price  HCIRLNLU 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) 
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 (
Maturity  Heston price  HCIR price  HCIRLNLU 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) 
Graphs for ATM options prices given in Table
Graphs for 25% options prices given in Table
Graphs for options prices given in Table
Graphs for options prices given in Table
Graph for option 15% prices given in Table
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to Enrico Moretto for consenting to use data reported in [