Research ArticlePricing of European Currency Options with UncertainExchange Rate and Stochastic Interest Rates

XiaoWang 12

1School of Economics and Management Beijing Institute of Petrochemical Technology Beijing 102617 China2Beijing Academy of Safety Engineering and Technology Beijing 102617 China

Correspondence should be addressed to Xiao Wang wangxiao19871125163com

Received 30 March 2018 Accepted 13 December 2018 Published 4 February 2019

Academic Editor Gian I Bischi

Copyright copy 2019 Xiao Wang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Suppose that the interest rates obey stochastic differential equations while the exchange rate follows an uncertain differentialequation this paper proposes a new currency model Under the proposed currency model the pricing formula of Europeancurrency options is then derived Some numerical examples recorded illustrate the quality of pricing formulas Meanwhile thispaper analyzes the relationship between the pricing formula and some parameters

1 Introduction

Nowadays the currency option is one of the best investmenttools for companies and individuals to hedge against adversemovements in exchange rates It can be divided into Europeancurrency option American currency option Asian currencyoption and so forth where European currency option is acontract giving the owner the right to buy or sell one unit offoreign currency with a specified price at a maturity date [1]The theoretical models of currency option pricing have beena hot issue in mathematical finance and the key point of thistopic is how to get an appropriate pricing formula

For European currency option Garman and Kohlhagen[2] first proposed G-K model where both domestic andforeign interest rates are assumed to be constant and theexchange rate is governed by a geometric Brownian motionHowever it is unrealistic for the exchange rate to obey thegeometric Brownian motion in the subsequent literatures Bymodifying the G-K model more and more methodologiesfor the currency option pricing have been proposed suchas Bollen and Rasieland [3] Carr and Wu [4] Sun [5]Swishchuk et al [6] Xiao et al [7] andWang Zhou andYang[8]

In the above-mentioned literature the exchange rate fol-lows a stochastic differential equation under the frameworkof probability theory When we use probability theory the

available probability distribution needs to be close to thetrue frequency However some emergencies coming fromwars political policies or natural disasters may affect theexchange rate In this case it is difficult to obtain availablestatistical data about exchange rate and the assumption thatthe exchange rate follows a stochastic differential equationmay be out of work At this time belief degrees given bysome domain experts are used to estimate values or distri-butions To model the belief degree uncertainty theory wasestablished by Liu [9] In addition to describe the evolution ofan uncertain phenomenon uncertain process [10] and a Liuprocess [11] were subsequently proposed To further expressuncertain dynamic systems uncertain differential equationwas proposed [10] and has been widely applied to control andfinancial market

Back to the foreign exchange market assume that theexchange rate follows an uncertain differential equation LiuChen and Ralescu [1] proposed Liu-Chen-Ralescu currencymodel In addition Shen and Yao [12] proposed a mean-reverting currency model under uncertain environmentRecently Ji and Wu [13] provided an uncertain currencymodel with jumps Besides that Sheng and Shi [14] proposedthe mean-reverting currency model under Asian currencyoption In these currency models listed above the exchangerate is governed by an uncertain process instead of stochasticprocess and the interest rates are taken as constant

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 2548592 10 pageshttpsdoiorg10115520192548592

2 Discrete Dynamics in Nature and Society

Table 1 Some notations and parameters

119903119905 domestic interest rate at 119905 119870 strike price119891119905 foreign interest rate at 119905 119879 maturity date119885119905 spot domestic currency price of a unit offoreign exchange at 119905 119862119905 a Liu process

120583 log-drift of the spot currency price 119885119905 119882119905 a Wiener process120590 log-diffusion of the spot currency price 119885119905 M uncertain measure119862119864 European call currency option price 119864[120585] expected value of 120585119875119864 European put currency option price var[120585] variance of 120585Φminus1(120572) = radic3120587 ln 1205721 minus 120572 119873(120583 1205902) normal distribution

However considering the fluctuation of interest rate mar-ket from time to time it is unreasonable to regard the interestrates as constant Up to now interest rate is mainly studiedunder the framework of uncertainty theory or probabilitytheoryWhen the sample size of interest rate is too small (evenno-sample) to estimate a probability distribution interestrate is usually described by an uncertain process underthe framework of uncertainty theory Chen and Gao [15]started from an assumption that the short interest rate followsuncertain process and proposed three equilibrium modelsSun Yao and Fu [16] proposed another interest ratemodel onthe basis of exponential Ornstein-Uhlenbeck equation underthe uncertain environment Suppose that both domestic andforeign interest rates follow uncertain differential equationsWang and Ning [17] proposed an uncertain currency modelwhere the exchange rate also follows an uncertain differentialequation

While there is a large amount of historical data aboutinterest rate the short interest rate is usually described bya stochastic process Under the framework of probabilitytheory Morton [18] proposed the first interest rate modeland Ho and Lee [19] then extended it In addition manyother economists have built other models such as Hulland White [20] and Vasicek [21] Taking into account twofactors randomness and uncertainty we propose a newcurrency model in this paper In detail both domestic andforeign interest rates follow stochastic differential equationswhile the exchange rate follows an uncertain differentialequation

The paper is organized as follows In Section 2 wemainlyintroduce uncertain differential equation Liu-Chen-Ralescumodel Vasicek model and moment generating function InSection 3 we propose a new currency model with uncertainexchange rate and stochastic The pricing formula of Euro-pean currency option under the proposed model is derivedin Section 4 Some numerical examples are carried out inSection 5 Finally Section 6 makes a brief conclusion Forconvenience somenotations and parameters employed in thelater sections are shown in Table 1

2 Preliminaries

In this section we first introduce uncertain differential equa-tion Then an uncertain currency model and Vasicek model

09

1

11

12

13

14

15

16

t

X

t

002 004 006 008 01 012 014 016 018 020

=02

=01

=09

=03

Figure 1 A spectrum of 120572-paths of d119883119905 = 119883119905d119905 + 119883119905d119862119905

are recalled Finally we introduce the moment generatingfunction

21 Uncertain Differential Equation

Definition 1 (see Liu [10]) Suppose that 119862119905 is a Liu processand 119891 and 119892 are two functions Given an initial value 1198830

d119883119905 = 119891 (119905 119883119905) d119905 + 119892 (119905 119883119905) d119862119905 (1)

is called an uncertain differential equationwith an initial value1198830Definition 2 (see Yao and Chen [22]) Let 120572 be a numberwith 0 lt 120572 lt 1 Uncertain differential equation (1) is saidto have an 120572-path 119883120572119905 if it solves the corresponding ordinarydifferential equation

d119883120572119905 = 119891 (119905 119883120572119905 ) d119905 + 1003816100381610038161003816119892 (119905 119883120572119905 )1003816100381610038161003816 Φminus1 (120572) d119905 (2)

whereΦminus1(120572) = (radic3120587) ln(120572(1 minus 120572))Example 3 Uncertain differential equation d119883119905 = 119883119905d119905 +119883119905d119862119905 with 1198830 = 1 has an 120572-path

119883120572119905 = exp (119905 + Φminus1 (120572) 119905) (3)

and its spectrum is shown in Figure 1

Theorem4 (see Liu [23]) Let 120578 and 120591 be anuncertain variableand a random variable respectively en

119864 [120578120591] = 119864 [120578] 119864 [120591] (4)

Discrete Dynamics in Nature and Society 3

Theorem 5 (see Yao and Chen [22]) Let 119883119905 and 119883120572119905 bethe solution and 120572-path of uncertain differential equation (1)respectively en for any monotone function 119869 we have

119864 [119869 (119883119905)] = int10

119869 (119883120572119905 )d120572 (5)

22 Uncertain Currency Model with Fixed Interest RatesAssume that the exchange rate 119885119905 follows an uncertaindifferential equation and the domestic and foreign interestrates are constant Liu Chen and Ralescu [1] proposed Liu-Chen-Ralescu model

d119883119905 = 119903119883119905d119905d119884119905 = 119891119884119905d119905d119885119905 = 120583119885119905d119905 + 120590119885119905d119862119905

(6)

where 119883119905 represents the riskless domestic currency with thefixed domestic interest rate 119903 and 119884119905 represents the risklessforeign currency with the fixed foreign interest rate 119891 Themeaning of remaining parameters in model (6) can be seenin Table 1

23 VasicekModel Vasicekmodel [21] is a classical stochasticinterest rate model and is defined by a stochastic differentialequation of the form

d119903119905 = 119886 (119887 minus 119903119905) d119905 + 120590d119882119905 (7)

describing the interest rate process 119903119905 where 120590 determinesthe volatility of the interest rate 119886 represents the rate ofadjustment and 119887 is the long run average value

By solving stochastic differential equation (7) we have

119903119905 = 1199030 exp (minus119886119905) + 119887 (1 minus exp (minus119886119905))+ 120590 exp (minus119886119905) int119905

0

exp (119886119904) d119882119904 (8)

and

119903119905 sim 119873(1199030 exp (minus119886119905)+ 119887 (1 minus exp (minus119886119905)) 12059022119886 (1 minus exp (minus2119886119905)))

(9)

24 Moment Generating Function

Definition 6 (see Shaked and Shanthikumar [24]) Themoment generating function of a random variable 120585 is119872120585[119905] = 119864[exp(119905120585)] 119905 isin R wherever this expectation exists

Remark 7 If 120585 sim 119873(120583 1205902) then 119864[exp(119905120585)] = exp(120583119905 +(12)12059021199052) 119905 isin R3 Model Establishment

In this part we generalize the Liu-Chen-Ralescu modelthrough the use of stochastic interest rates We employ

Vasicek model for the domestic and foreign interest ratesBesides assume that the exchange rate follows an uncertaindifferential equation a new currency model is then proposedas follows

d119903119905 = 1198861 (1198871 minus 119903119905) d119905 + 1205901d1198821119905d119891119905 = 1198862 (1198872 minus 119891119905) d119905 + 1205902d1198822119905d119885119905 = 120583119885119905d119905 + 120590119885119905d119862119905

(10)

where 1205901 is the diffusion of 119903119905 1205902 is the diffusion of 1198911199051198861 1198862 1198871 and 1198872 are the constant parameters 1198821119905 and 1198822119905are independent Wiener processes and 119882119894119905 and 119862119905 areindependent for 119894 = 1 2

By using formula (9) we have

119903119905 sim 119873(1199030 exp (minus1198861119905)

+ 1198871 (1 minus exp (minus1198861119905)) 120590212119886 (1 minus exp (minus21198861119905)))(11)

and

119891119905 sim 119873(1198910 exp (minus1198862119905)

+ 1198872 (1 minus exp (minus1198862119905)) 1205902221198862 (1 minus exp (minus21198862119905))) (12)

ByDefinition 2 we obtain that the 120572-path of the exchangerate 119885119905 is

119885120572119905 = 1198850 sdot exp (120583119905 + 120590119905Φminus1 (120572)) (13)

Remark 8 When uncertain differential equation has noanalytic solution the solution can be calculated by somenumerical methods Interested readers can refer to [22 2526]

4 European Currency Option Pricing

In this section we study the European currency optionpricing problem and provide the pricing formula of Europeancurrency option under model (10) For your convenience thecurrent time is set to 0

41 Pricing Formula of European Call Currency Option In[1] we can see that European call currency option is acontract endowing the holder the right to buy one unit offoreign currency at a maturity date 119879 for119870 units of domesticcurrency where119870 is commonly called a strike price

Let119862119864 represent this contract price in domestic currencyThe investor needs to pay 119862119864 to buy this contract at time 0The payoff of the investor is (119885119879 minus 119870)+ in domestic currencyat the maturity date 119879 So the expected profit of the investorat time 0 is

minus119862119864 + 119864[(119885119879 minus 119870)+ exp(minusint1198790

119903119904d119904)] (14)

4 Discrete Dynamics in Nature and Society

Due to selling the contract the bank can receive 119862119864 attime 0 At the maturity date119879 the bank also pays (1minus119870119885119879)+in foreign currency The expected profit of the bank at time 0is

119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] (15)

To ensure the fairness of the contract we have

minus 119862119864 + 119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)]

= 119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790119891119904d119904)]

(16)

In this way the investor and the bank have an identicalexpected profit

Definition 9 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price 119862119864is

119862119864 = 12119864 [(119885119879 minus 119870)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[(1 minus 119870119885119879)

+

exp(minusint1198790

119891119904ds)] (17)

Theorem 10 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price is

119862119864 = 12 int1

0

(119885120572119879 minus 119870)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10

(1 minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(18)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (19)

and

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (20)

Proof ByTheorem 4 we have

119864[(119885119879 minus 119870)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119885119879 minus 119870)+] 119864 [exp(minusint119879

0119903119904d119904)]

(21)

ByTheorem 5 and formula (13) we have

119864 [(119885119879 minus 119870)+] = int10(119885120572119879 minus 119870)+ d120572 (22)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (23)

According to formula (8) we get

int1198790

119903119905d119905 = int1198790

1199030 exp (minus1198861119905) d119905+ 1198871 int1198790

(1 minus exp (minus1198861119905)) d119905+ int1198790

(1205901 exp (minus1198861119905) int1199050

exp (1198861119904) d119882119904) d119905= 1198871119879 + (1 minus exp (minus1198861119879)) 1199030 minus 11988711198861+ 1205901 int119879

0int1199050exp (119886 (119904 minus 119905)) d119882119904d119905

(24)

Since the expectation of the stochastic integral is zero theexpected value of int119879

0119903119905d119905 is provided by

119864[int1198790

119903119905d119905] = 1198871119879 + (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 (25)

Since the variance is determined from the diffusion termnamely the stochastic integral the two deterministic driftterms in int119879

0119903119905d119905 make no contribution to the variance This

gives

var [int1198790

119903119905d119905]= var [1205901 int119879

0

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905] (26)

Since var[120585] = 119864[1205852] minus (119864[120585])2 we havevar [int119879

0119903119905d119905]

= 119864[(1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905))d119882119904d119905)2]

minus (119864[1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905])2

(27)

Discrete Dynamics in Nature and Society 5

Observe that119864[1205901 int1198790 int1199050 exp(1198861(119904minus119905))d119882119904d119905] = 0 and thisimplies that

var [int1198790119903119905d119905]

= 119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904dt)

2] (28)

By use of Fubini theorem as well as that for the stochasticintegral we have

119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904d119905)

2]

= 119864[(1205901 int1198790int119879119904exp (1198861 (119904 minus 119905)) d119905d119882119904)

2]

= 119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

(29)

Since (d119882119904)2 = d119904 the square term eliminates the sourceof randomness and

119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

= 12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904(30)

Continuing with simple integration leads to

var [int1198790

119903119905d119905] d119905 = 12059021 int1198790

(1 minus exp (1198861 (119904 minus 119879))1198861 )2 d119904

= 1205902111988621 (119879 minus2 (1 minus exp (minus1198861119879))1198861

+ 1 minus exp (minus21198861119879)21198861 )

(31)

and

minus int1198790

119903119905d119905 sim 119873(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 1205902111988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(32)

According to Remark 7 we have

119864[exp(minusint1198790

119903119905d119905)] = exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

(33)

Thus

119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)] = int1

0(119885120572119879 minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(34)

By the same way we get

119864[exp(minusint1198790119891119904d119904)] = exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(35)

ByTheorem 5 and formula (13) we have

119864[(1 minus 119870119885119879)+] = int1

0

(1 minus 119870119885120572119879)+

d120572 (36)

where

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (37)

Hence

119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] = int10

(1

minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(38)

6 Discrete Dynamics in Nature and Society

According to Definition 9 and formulas (34) and (38)the pricing formula of European call currency option isimmediately obtained

Herein we discuss the properties of this pricing formulaand the result is shown by the following theorem

Theorem 11 Let119862119864 be the European call currency option priceunder model (10) en(1) 119862119864 is a decreasing function of 1198871 1198872 1199030 1198910 and 119870(2) 119862119864 is an increasing function of 120583 and 1198850Proof Theorem 10 tells us that 119862119864 can be expressed as

119862119864 = 12 int1

0(1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502 int1

0

(1

minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(39)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(40)

is decreasing with 1198871 and 119862119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(41)

is decreasing with 1198872 and 119862119864 is decreasing with 1198872Since exp(minus1199030) is decreasing with 1199030119862119864 is decreasing with1199030Since exp(minus1198910) is decreasing with 1198910 119862119864 is decreasing

with 1198910

Since

1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (42)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (43)

are decreasing with 119870 119862119864 is decreasing with 119870(2) Since1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (44)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (119870 1198850 119879 gt 0) (45)

are increasing with 120583 and 1198850 119862119864 is increasing with 120583 and 1198850

42 Pricing Formula of European Put Currency OptionSimilarity to European call currency option the definitionformula and property of European put currency optionpricing with a strike price119870 and a maturity date 119879 are handyto get and are shown as below

Definition 12 Under model (10) given a strike price119870 and amaturity date 119879 the European put currency option price 119875119864is

119875119864 = 12119864 [(119870 minus 119885119879)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[( 119870119885119879 minus 1)

+

exp(minusint1198790

119891119904d119904)] (46)

Theorem 13 Under model (10) given a strike price 119870 and amaturity date 119879 the European put currency option price is

119875119864 = 12 int1

0

(119870 minus 119885120572119879)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

+ 11988502 int10

( 119870119885120572119879 minus 1)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(47)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (48)

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 3

Theorem 5 (see Yao and Chen [22]) Let 119883119905 and 119883120572119905 bethe solution and 120572-path of uncertain differential equation (1)respectively en for any monotone function 119869 we have

119864 [119869 (119883119905)] = int10

119869 (119883120572119905 )d120572 (5)

22 Uncertain Currency Model with Fixed Interest RatesAssume that the exchange rate 119885119905 follows an uncertaindifferential equation and the domestic and foreign interestrates are constant Liu Chen and Ralescu [1] proposed Liu-Chen-Ralescu model

d119883119905 = 119903119883119905d119905d119884119905 = 119891119884119905d119905d119885119905 = 120583119885119905d119905 + 120590119885119905d119862119905

(6)

where 119883119905 represents the riskless domestic currency with thefixed domestic interest rate 119903 and 119884119905 represents the risklessforeign currency with the fixed foreign interest rate 119891 Themeaning of remaining parameters in model (6) can be seenin Table 1

23 VasicekModel Vasicekmodel [21] is a classical stochasticinterest rate model and is defined by a stochastic differentialequation of the form

d119903119905 = 119886 (119887 minus 119903119905) d119905 + 120590d119882119905 (7)

describing the interest rate process 119903119905 where 120590 determinesthe volatility of the interest rate 119886 represents the rate ofadjustment and 119887 is the long run average value

By solving stochastic differential equation (7) we have

119903119905 = 1199030 exp (minus119886119905) + 119887 (1 minus exp (minus119886119905))+ 120590 exp (minus119886119905) int119905

0

exp (119886119904) d119882119904 (8)

and

119903119905 sim 119873(1199030 exp (minus119886119905)+ 119887 (1 minus exp (minus119886119905)) 12059022119886 (1 minus exp (minus2119886119905)))

(9)

24 Moment Generating Function

Definition 6 (see Shaked and Shanthikumar [24]) Themoment generating function of a random variable 120585 is119872120585[119905] = 119864[exp(119905120585)] 119905 isin R wherever this expectation exists

Remark 7 If 120585 sim 119873(120583 1205902) then 119864[exp(119905120585)] = exp(120583119905 +(12)12059021199052) 119905 isin R3 Model Establishment

In this part we generalize the Liu-Chen-Ralescu modelthrough the use of stochastic interest rates We employ

Vasicek model for the domestic and foreign interest ratesBesides assume that the exchange rate follows an uncertaindifferential equation a new currency model is then proposedas follows

d119903119905 = 1198861 (1198871 minus 119903119905) d119905 + 1205901d1198821119905d119891119905 = 1198862 (1198872 minus 119891119905) d119905 + 1205902d1198822119905d119885119905 = 120583119885119905d119905 + 120590119885119905d119862119905

(10)

where 1205901 is the diffusion of 119903119905 1205902 is the diffusion of 1198911199051198861 1198862 1198871 and 1198872 are the constant parameters 1198821119905 and 1198822119905are independent Wiener processes and 119882119894119905 and 119862119905 areindependent for 119894 = 1 2

By using formula (9) we have

119903119905 sim 119873(1199030 exp (minus1198861119905)

+ 1198871 (1 minus exp (minus1198861119905)) 120590212119886 (1 minus exp (minus21198861119905)))(11)

and

119891119905 sim 119873(1198910 exp (minus1198862119905)

+ 1198872 (1 minus exp (minus1198862119905)) 1205902221198862 (1 minus exp (minus21198862119905))) (12)

ByDefinition 2 we obtain that the 120572-path of the exchangerate 119885119905 is

119885120572119905 = 1198850 sdot exp (120583119905 + 120590119905Φminus1 (120572)) (13)

Remark 8 When uncertain differential equation has noanalytic solution the solution can be calculated by somenumerical methods Interested readers can refer to [22 2526]

4 European Currency Option Pricing

In this section we study the European currency optionpricing problem and provide the pricing formula of Europeancurrency option under model (10) For your convenience thecurrent time is set to 0

41 Pricing Formula of European Call Currency Option In[1] we can see that European call currency option is acontract endowing the holder the right to buy one unit offoreign currency at a maturity date 119879 for119870 units of domesticcurrency where119870 is commonly called a strike price

Let119862119864 represent this contract price in domestic currencyThe investor needs to pay 119862119864 to buy this contract at time 0The payoff of the investor is (119885119879 minus 119870)+ in domestic currencyat the maturity date 119879 So the expected profit of the investorat time 0 is

minus119862119864 + 119864[(119885119879 minus 119870)+ exp(minusint1198790

119903119904d119904)] (14)

4 Discrete Dynamics in Nature and Society

Due to selling the contract the bank can receive 119862119864 attime 0 At the maturity date119879 the bank also pays (1minus119870119885119879)+in foreign currency The expected profit of the bank at time 0is

119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] (15)

To ensure the fairness of the contract we have

minus 119862119864 + 119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)]

= 119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790119891119904d119904)]

(16)

In this way the investor and the bank have an identicalexpected profit

Definition 9 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price 119862119864is

119862119864 = 12119864 [(119885119879 minus 119870)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[(1 minus 119870119885119879)

+

exp(minusint1198790

119891119904ds)] (17)

Theorem 10 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price is

119862119864 = 12 int1

0

(119885120572119879 minus 119870)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10

(1 minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(18)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (19)

and

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (20)

Proof ByTheorem 4 we have

119864[(119885119879 minus 119870)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119885119879 minus 119870)+] 119864 [exp(minusint119879

0119903119904d119904)]

(21)

ByTheorem 5 and formula (13) we have

119864 [(119885119879 minus 119870)+] = int10(119885120572119879 minus 119870)+ d120572 (22)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (23)

According to formula (8) we get

int1198790

119903119905d119905 = int1198790

1199030 exp (minus1198861119905) d119905+ 1198871 int1198790

(1 minus exp (minus1198861119905)) d119905+ int1198790

(1205901 exp (minus1198861119905) int1199050

exp (1198861119904) d119882119904) d119905= 1198871119879 + (1 minus exp (minus1198861119879)) 1199030 minus 11988711198861+ 1205901 int119879

0int1199050exp (119886 (119904 minus 119905)) d119882119904d119905

(24)

Since the expectation of the stochastic integral is zero theexpected value of int119879

0119903119905d119905 is provided by

119864[int1198790

119903119905d119905] = 1198871119879 + (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 (25)

Since the variance is determined from the diffusion termnamely the stochastic integral the two deterministic driftterms in int119879

0119903119905d119905 make no contribution to the variance This

gives

var [int1198790

119903119905d119905]= var [1205901 int119879

0

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905] (26)

Since var[120585] = 119864[1205852] minus (119864[120585])2 we havevar [int119879

0119903119905d119905]

= 119864[(1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905))d119882119904d119905)2]

minus (119864[1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905])2

(27)

Discrete Dynamics in Nature and Society 5

Observe that119864[1205901 int1198790 int1199050 exp(1198861(119904minus119905))d119882119904d119905] = 0 and thisimplies that

var [int1198790119903119905d119905]

= 119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904dt)

2] (28)

By use of Fubini theorem as well as that for the stochasticintegral we have

119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904d119905)

2]

= 119864[(1205901 int1198790int119879119904exp (1198861 (119904 minus 119905)) d119905d119882119904)

2]

= 119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

(29)

Since (d119882119904)2 = d119904 the square term eliminates the sourceof randomness and

119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

= 12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904(30)

Continuing with simple integration leads to

var [int1198790

119903119905d119905] d119905 = 12059021 int1198790

(1 minus exp (1198861 (119904 minus 119879))1198861 )2 d119904

= 1205902111988621 (119879 minus2 (1 minus exp (minus1198861119879))1198861

+ 1 minus exp (minus21198861119879)21198861 )

(31)

and

minus int1198790

119903119905d119905 sim 119873(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 1205902111988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(32)

According to Remark 7 we have

119864[exp(minusint1198790

119903119905d119905)] = exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

(33)

Thus

119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)] = int1

0(119885120572119879 minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(34)

By the same way we get

119864[exp(minusint1198790119891119904d119904)] = exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(35)

ByTheorem 5 and formula (13) we have

119864[(1 minus 119870119885119879)+] = int1

0

(1 minus 119870119885120572119879)+

d120572 (36)

where

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (37)

Hence

119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] = int10

(1

minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(38)

6 Discrete Dynamics in Nature and Society

According to Definition 9 and formulas (34) and (38)the pricing formula of European call currency option isimmediately obtained

Herein we discuss the properties of this pricing formulaand the result is shown by the following theorem

Theorem 11 Let119862119864 be the European call currency option priceunder model (10) en(1) 119862119864 is a decreasing function of 1198871 1198872 1199030 1198910 and 119870(2) 119862119864 is an increasing function of 120583 and 1198850Proof Theorem 10 tells us that 119862119864 can be expressed as

119862119864 = 12 int1

0(1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502 int1

0

(1

minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(39)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(40)

is decreasing with 1198871 and 119862119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(41)

is decreasing with 1198872 and 119862119864 is decreasing with 1198872Since exp(minus1199030) is decreasing with 1199030119862119864 is decreasing with1199030Since exp(minus1198910) is decreasing with 1198910 119862119864 is decreasing

with 1198910

Since

1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (42)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (43)

are decreasing with 119870 119862119864 is decreasing with 119870(2) Since1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (44)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (119870 1198850 119879 gt 0) (45)

are increasing with 120583 and 1198850 119862119864 is increasing with 120583 and 1198850

42 Pricing Formula of European Put Currency OptionSimilarity to European call currency option the definitionformula and property of European put currency optionpricing with a strike price119870 and a maturity date 119879 are handyto get and are shown as below

Definition 12 Under model (10) given a strike price119870 and amaturity date 119879 the European put currency option price 119875119864is

119875119864 = 12119864 [(119870 minus 119885119879)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[( 119870119885119879 minus 1)

+

exp(minusint1198790

119891119904d119904)] (46)

Theorem 13 Under model (10) given a strike price 119870 and amaturity date 119879 the European put currency option price is

119875119864 = 12 int1

0

(119870 minus 119885120572119879)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

+ 11988502 int10

( 119870119885120572119879 minus 1)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(47)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (48)

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

4 Discrete Dynamics in Nature and Society

Due to selling the contract the bank can receive 119862119864 attime 0 At the maturity date119879 the bank also pays (1minus119870119885119879)+in foreign currency The expected profit of the bank at time 0is

119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] (15)

To ensure the fairness of the contract we have

minus 119862119864 + 119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)]

= 119862119864 minus 1198850119864[(1 minus 119870119885119879)+

exp(minusint1198790119891119904d119904)]

(16)

In this way the investor and the bank have an identicalexpected profit

Definition 9 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price 119862119864is

119862119864 = 12119864 [(119885119879 minus 119870)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[(1 minus 119870119885119879)

+

exp(minusint1198790

119891119904ds)] (17)

Theorem 10 Under model (10) given a strike price 119870 and amaturity date 119879 the European currency call option price is

119862119864 = 12 int1

0

(119885120572119879 minus 119870)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10

(1 minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(18)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (19)

and

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (20)

Proof ByTheorem 4 we have

119864[(119885119879 minus 119870)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119885119879 minus 119870)+] 119864 [exp(minusint119879

0119903119904d119904)]

(21)

ByTheorem 5 and formula (13) we have

119864 [(119885119879 minus 119870)+] = int10(119885120572119879 minus 119870)+ d120572 (22)

where

(119885120572119879 minus 119870)+ = (1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ (23)

According to formula (8) we get

int1198790

119903119905d119905 = int1198790

1199030 exp (minus1198861119905) d119905+ 1198871 int1198790

(1 minus exp (minus1198861119905)) d119905+ int1198790

(1205901 exp (minus1198861119905) int1199050

exp (1198861119904) d119882119904) d119905= 1198871119879 + (1 minus exp (minus1198861119879)) 1199030 minus 11988711198861+ 1205901 int119879

0int1199050exp (119886 (119904 minus 119905)) d119882119904d119905

(24)

Since the expectation of the stochastic integral is zero theexpected value of int119879

0119903119905d119905 is provided by

119864[int1198790

119903119905d119905] = 1198871119879 + (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 (25)

Since the variance is determined from the diffusion termnamely the stochastic integral the two deterministic driftterms in int119879

0119903119905d119905 make no contribution to the variance This

gives

var [int1198790

119903119905d119905]= var [1205901 int119879

0

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905] (26)

Since var[120585] = 119864[1205852] minus (119864[120585])2 we havevar [int119879

0119903119905d119905]

= 119864[(1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905))d119882119904d119905)2]

minus (119864[1205901 int1198790

int1199050

exp (1198861 (119904 minus 119905)) d119882119904d119905])2

(27)

Discrete Dynamics in Nature and Society 5

Observe that119864[1205901 int1198790 int1199050 exp(1198861(119904minus119905))d119882119904d119905] = 0 and thisimplies that

var [int1198790119903119905d119905]

= 119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904dt)

2] (28)

By use of Fubini theorem as well as that for the stochasticintegral we have

119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904d119905)

2]

= 119864[(1205901 int1198790int119879119904exp (1198861 (119904 minus 119905)) d119905d119882119904)

2]

= 119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

(29)

Since (d119882119904)2 = d119904 the square term eliminates the sourceof randomness and

119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

= 12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904(30)

Continuing with simple integration leads to

var [int1198790

119903119905d119905] d119905 = 12059021 int1198790

(1 minus exp (1198861 (119904 minus 119879))1198861 )2 d119904

= 1205902111988621 (119879 minus2 (1 minus exp (minus1198861119879))1198861

+ 1 minus exp (minus21198861119879)21198861 )

(31)

and

minus int1198790

119903119905d119905 sim 119873(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 1205902111988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(32)

According to Remark 7 we have

119864[exp(minusint1198790

119903119905d119905)] = exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

(33)

Thus

119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)] = int1

0(119885120572119879 minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(34)

By the same way we get

119864[exp(minusint1198790119891119904d119904)] = exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(35)

ByTheorem 5 and formula (13) we have

119864[(1 minus 119870119885119879)+] = int1

0

(1 minus 119870119885120572119879)+

d120572 (36)

where

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (37)

Hence

119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] = int10

(1

minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(38)

6 Discrete Dynamics in Nature and Society

According to Definition 9 and formulas (34) and (38)the pricing formula of European call currency option isimmediately obtained

Herein we discuss the properties of this pricing formulaand the result is shown by the following theorem

Theorem 11 Let119862119864 be the European call currency option priceunder model (10) en(1) 119862119864 is a decreasing function of 1198871 1198872 1199030 1198910 and 119870(2) 119862119864 is an increasing function of 120583 and 1198850Proof Theorem 10 tells us that 119862119864 can be expressed as

119862119864 = 12 int1

0(1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502 int1

0

(1

minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(39)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(40)

is decreasing with 1198871 and 119862119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(41)

is decreasing with 1198872 and 119862119864 is decreasing with 1198872Since exp(minus1199030) is decreasing with 1199030119862119864 is decreasing with1199030Since exp(minus1198910) is decreasing with 1198910 119862119864 is decreasing

with 1198910

Since

1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (42)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (43)

are decreasing with 119870 119862119864 is decreasing with 119870(2) Since1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (44)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (119870 1198850 119879 gt 0) (45)

are increasing with 120583 and 1198850 119862119864 is increasing with 120583 and 1198850

42 Pricing Formula of European Put Currency OptionSimilarity to European call currency option the definitionformula and property of European put currency optionpricing with a strike price119870 and a maturity date 119879 are handyto get and are shown as below

Definition 12 Under model (10) given a strike price119870 and amaturity date 119879 the European put currency option price 119875119864is

119875119864 = 12119864 [(119870 minus 119885119879)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[( 119870119885119879 minus 1)

+

exp(minusint1198790

119891119904d119904)] (46)

Theorem 13 Under model (10) given a strike price 119870 and amaturity date 119879 the European put currency option price is

119875119864 = 12 int1

0

(119870 minus 119885120572119879)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

+ 11988502 int10

( 119870119885120572119879 minus 1)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(47)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (48)

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

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Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 5

Observe that119864[1205901 int1198790 int1199050 exp(1198861(119904minus119905))d119882119904d119905] = 0 and thisimplies that

var [int1198790119903119905d119905]

= 119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904dt)

2] (28)

By use of Fubini theorem as well as that for the stochasticintegral we have

119864[(1205901 int1198790int1199050exp (1198861 (119904 minus 119905)) d119882119904d119905)

2]

= 119864[(1205901 int1198790int119879119904exp (1198861 (119904 minus 119905)) d119905d119882119904)

2]

= 119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

(29)

Since (d119882119904)2 = d119904 the square term eliminates the sourceof randomness and

119864[12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904]

= 12059021 int1198790

(int119879119904

exp (1198861 (119904 minus 119905)) d119905)2

d119904(30)

Continuing with simple integration leads to

var [int1198790

119903119905d119905] d119905 = 12059021 int1198790

(1 minus exp (1198861 (119904 minus 119879))1198861 )2 d119904

= 1205902111988621 (119879 minus2 (1 minus exp (minus1198861119879))1198861

+ 1 minus exp (minus21198861119879)21198861 )

(31)

and

minus int1198790

119903119905d119905 sim 119873(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 1205902111988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(32)

According to Remark 7 we have

119864[exp(minusint1198790

119903119905d119905)] = exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

(33)

Thus

119864[(119885119879 minus 119870)+ exp(minusint1198790119903119904d119904)] = int1

0(119885120572119879 minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 ))

(34)

By the same way we get

119864[exp(minusint1198790119891119904d119904)] = exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(35)

ByTheorem 5 and formula (13) we have

119864[(1 minus 119870119885119879)+] = int1

0

(1 minus 119870119885120572119879)+

d120572 (36)

where

(1 minus 119870119885120572119879)+ = (1 minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))

+ (37)

Hence

119864[(1 minus 119870119885119879)+

exp(minusint1198790

119891119904d119904)] = int10

(1

minus 119870119885120572119879)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(38)

6 Discrete Dynamics in Nature and Society

According to Definition 9 and formulas (34) and (38)the pricing formula of European call currency option isimmediately obtained

Herein we discuss the properties of this pricing formulaand the result is shown by the following theorem

Theorem 11 Let119862119864 be the European call currency option priceunder model (10) en(1) 119862119864 is a decreasing function of 1198871 1198872 1199030 1198910 and 119870(2) 119862119864 is an increasing function of 120583 and 1198850Proof Theorem 10 tells us that 119862119864 can be expressed as

119862119864 = 12 int1

0(1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502 int1

0

(1

minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(39)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(40)

is decreasing with 1198871 and 119862119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(41)

is decreasing with 1198872 and 119862119864 is decreasing with 1198872Since exp(minus1199030) is decreasing with 1199030119862119864 is decreasing with1199030Since exp(minus1198910) is decreasing with 1198910 119862119864 is decreasing

with 1198910

Since

1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (42)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (43)

are decreasing with 119870 119862119864 is decreasing with 119870(2) Since1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (44)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (119870 1198850 119879 gt 0) (45)

are increasing with 120583 and 1198850 119862119864 is increasing with 120583 and 1198850

42 Pricing Formula of European Put Currency OptionSimilarity to European call currency option the definitionformula and property of European put currency optionpricing with a strike price119870 and a maturity date 119879 are handyto get and are shown as below

Definition 12 Under model (10) given a strike price119870 and amaturity date 119879 the European put currency option price 119875119864is

119875119864 = 12119864 [(119870 minus 119885119879)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[( 119870119885119879 minus 1)

+

exp(minusint1198790

119891119904d119904)] (46)

Theorem 13 Under model (10) given a strike price 119870 and amaturity date 119879 the European put currency option price is

119875119864 = 12 int1

0

(119870 minus 119885120572119879)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

+ 11988502 int10

( 119870119885120572119879 minus 1)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(47)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (48)

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

According to Definition 9 and formulas (34) and (38)the pricing formula of European call currency option isimmediately obtained

Herein we discuss the properties of this pricing formulaand the result is shown by the following theorem

Theorem 11 Let119862119864 be the European call currency option priceunder model (10) en(1) 119862119864 is a decreasing function of 1198871 1198872 1199030 1198910 and 119870(2) 119862119864 is an increasing function of 120583 and 1198850Proof Theorem 10 tells us that 119862119864 can be expressed as

119862119864 = 12 int1

0(1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870)+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502 int1

0

(1

minus 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)))+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(39)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(40)

is decreasing with 1198871 and 119862119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(41)

is decreasing with 1198872 and 119862119864 is decreasing with 1198872Since exp(minus1199030) is decreasing with 1199030119862119864 is decreasing with1199030Since exp(minus1198910) is decreasing with 1198910 119862119864 is decreasing

with 1198910

Since

1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (42)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (43)

are decreasing with 119870 119862119864 is decreasing with 119870(2) Since1198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 119870 (44)

and

1198850 minus 119870exp (120583119879 + 120590119879Φminus1 (120572)) (119870 1198850 119879 gt 0) (45)

are increasing with 120583 and 1198850 119862119864 is increasing with 120583 and 1198850

42 Pricing Formula of European Put Currency OptionSimilarity to European call currency option the definitionformula and property of European put currency optionpricing with a strike price119870 and a maturity date 119879 are handyto get and are shown as below

Definition 12 Under model (10) given a strike price119870 and amaturity date 119879 the European put currency option price 119875119864is

119875119864 = 12119864 [(119870 minus 119885119879)+ exp(minusint119879

0

119903119904d119904)]+ 11988502 119864[( 119870119885119879 minus 1)

+

exp(minusint1198790

119891119904d119904)] (46)

Theorem 13 Under model (10) given a strike price 119870 and amaturity date 119879 the European put currency option price is

119875119864 = 12 int1

0

(119870 minus 119885120572119879)+ d120572 exp(minus1198871119879

minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861 + 12059021211988621 (119879

minus 2 (1 minus exp (minus1198861119879))1198861 + 1 minus exp (minus21198861119879)21198861 ))

+ 11988502 int10

( 119870119885120572119879 minus 1)+

d120572 exp(minus1198872119879

minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862 + 12059022211988622 (119879

minus 2 (1 minus exp (minus1198862119879))1198862 + 1 minus exp (minus21198862119879)21198862 ))

(47)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (48)

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (49)

Proof ByTheorem 5 and formula (13) we have

119864 [(119870 minus 119885119879)+] = int10

(119870 minus 119885120572119879)+ d120572 (50)

and

119864[( 119870119885119879 minus 1)+] = int1

0

( 119870119885120572119879 minus 1)+

d120572 (51)

where

(119870 minus 119885120572119879)+ = (119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ (52)

and

( 119870119885120572119879 minus 1)+ = ( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+ (53)

ByTheorem 4 we have

119864[(119870 minus 119885119879)+ exp(minusint1198790

119903119904d119904)]= 119864 [(119870 minus 119885119879)+] 119864 [exp(minusint119879

0

119903119904d119904)](54)

and

119864[( 119870119885119879 minus 1)+

exp(minusint1198790

119891119904d119904)]= 119864[( 119870119885119879 minus 1)

+]119864[exp(minusint1198790

119891119904d119904)] (55)

FromDefinition 12 and formulas (33) (35) (54) and (55)the pricing formula of the European put currency option isimmediately obtained

Theorem 14 Let 119875119864 be the European put currency option priceunder model (10) en(1) 119875119864 is a decreasing function of 1198871 1198872 120583 1199030 1198910 and 1198850(2) 119875119864 is an increasing function of119870

Proof ByTheorem 13 119875119864 can be written as

119875119864 = 12 int1

0(119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)))+ d120572

sdot exp(minus1198871119879 minus (1 minus exp (minus1198861119879)) (1199030 minus 1198871)1198861+ 12059021211988621 (119879 minus

2 (1 minus exp (minus1198861119879))1198861+ 1 minus exp (minus21198861119879)21198861 )) + 11988502sdot int10( 1198701198850 exp (120583119879 + 120590119879Φminus1 (120572)) minus 1)

+

d120572

sdot exp(minus1198872119879 minus (1 minus exp (minus1198862119879)) (1198910 minus 1198872)1198862+ 12059022211988622 (119879 minus

2 (1 minus exp (minus1198862119879))1198862+ 1 minus exp (minus21198862119879)21198862 ))

(56)

(1) Since 1198861119879 + exp(minus1198861119879) minus 1 gt 0 (1198861 119879 gt 0)exp(minus1198871119879 minus 11988711198861 (exp (minus1198861119879) minus 1))= exp(minus11988711198861 (1198861119879 + exp (minus1198861119879) minus 1))

(57)

is decreasing with 1198871 and 119875119864 is decreasing with 1198871Since 1198862119879 + exp(minus1198862119879) minus 1 gt 0 (1198862 119879 gt 0)

exp(minus1198872119879 minus 11988721198862 (exp (minus1198862119879) minus 1))= exp(minus11988721198862 (1198862119879 + exp (minus1198862119879) minus 1))

(58)

is decreasing with 1198872 and 119875119864 is decreasing with 1198872Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (59)

and119870

exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (1198701198850 119879 gt 0) (60)

are decreasing with 120583 and1198850 119875119864 is decreasing with 120583 and1198850Since exp(minus1199030) is decreasing with 1199030 119875119864 is decreasing with1199030Since exp(minus1198910) is decreasingwith1198910 119875119864 is decreasingwith1198910 (2) Since

119870 minus 1198850 exp (120583119879 + 120590119879Φminus1 (120572)) (61)

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Discrete Dynamics in Nature and Society

Table 2 Parameters setting in model (10)

1198861 01 120583 000651198862 015 120590 000011198871() 11 1199030() 121198872() 035 1198910() 0321205901 00002 1198850 6651205902 00003 119870 65

1501005000

05

1

15

2

25

T

CE

Figure 2 European call currency option price 119862119864 under model (10)

and

119870exp (120583119879 + 120590119879Φminus1 (120572)) minus 1198850 (62)

are increasing with 119870 119875119864 is decreasing with 1198705 Numerical Examples

In this section some numerical examples are included toillustrate the pricing formulas under model (10) In additionwemine the influence of other parameters (119879 1198861 1198862 1205901 1205902 120590)on the pricing formulas For the sake of simplicity thispart just considers the case of European call currencyoption

Table 2 presents the required parameters Under model(10) we calculate the European call currency option pricesand depict them in Figure 2 where 119879 is the maturitydate

According to common sense the European call currencyoption price is increasing with the maturity date FromFigure 2 we can see that the price computed by model(10) suits this principle If the maturity date 119879 is takenas 100 the European call currency option price is 119862119864 =21782

In what follows we discuss the influence of the parame-ters (1198861 1198862 1205901 1205902 120590) on 119862119864 and show the results by a seriesof experiments Consider the first case where the parametersa1 1198862 119879 are fixed and the other parameters (1205901 1205902 120590) arechanging The second case is considering 119862119864 under differentparameters 1198861 and 1198862 where the other parameters are notchanging Figures 3 and 4 show 119862119864 versus its parameters1205901 1205902 120590 1198861 and 1198862 The default parameters can be referredto Table 2 and the maturity date 119879 = 100

Figure 3 shows that the European call currency optionprice under model (10) is increasing with 1205901 1205902 and 120590From Figure 4 we can obtain the following conclusion theEuropean call currency option price is increasing with 1198861while it is decreasing with respect to 11988626 Conclusion

This paper proposed a new currency model in whichthe exchange rate follows an uncertain differential equa-tion while the domestic and foreign interest rates followstochastic differential equations Subsequently we providedthe pricing formulas of European currency option underthis currency model Meanwhile we find that the Euro-pean call currency option price under the proposed modelis increasing with the initial exchange rate the log-driftthe diffusion the log-diffusion the parameter 1198861 and thematurity date while it is decreasing with the initial domes-tic interest rate the initial foreign interest rate the strikeprice and the parameters 1198862 1198871 1198872 At last some numer-ical examples were included to illustrate that the pricingformulas under the proposed currency model are reason-able

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interestrelated to this work

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 9

21779

2178

21781

21782

21783

21784

21785

21786

21787

21779

217795

2178

217805

21781

217815

21782

217825

21783

217835

21781

21782

21783

21784

21785

21786

21787

21788

2 40(3)

2 40(2)

2 40(1)

1 2

times10minus4

times10minus4

times10minus4

CE CE CE

Figure 3 European call currency option price 119862119864 versus parameters 1205901 1205902 and 120590 where (1) 119862119864 versus 1205901 (2) 119862119864 versus 1205902 (3) 119862119864 versus120590

209

21

211

212

213

214

215

216

217

218

219

22

221

222

223

a1a2

001 002 003 004 0050(1)

001 002 003 004 0050(2)

CE

CE

Figure 4 European call currency option price 119862119864 versus parameters 1198861 and 1198862 where (1) 119862119864 versus 1198861 (2) 119862119864 versus 1198862

Acknowledgments

This study was funded by the National Natural ScienceFoundation of China (11701338) and a Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J17KB124) The author would like to thank ProfessorZhen Peng for helpful comments

References

[1] Y Liu X Chen and D A Ralescu ldquoUncertain currency modeland currency option pricingrdquo International Journal of IntelligentSystems vol 30 no 1 pp 40ndash51 2014

[2] M B Garman and S W Kohlhagen ldquoForeign currency optionvaluesrdquo Journal of International Money and Finance vol 2 no3 pp 231ndash237 1983

[3] N P B Bollen and E Rasiel ldquoThe performance of alternativevaluationmodels in theOTC currency options marketrdquo Journalof International Money and Finance vol 22 no 1 pp 33ndash642003

[4] P Carr and L Wu ldquoStochastic skew in currency optionsrdquoJournal of Financial Economics vol 86 no 1 pp 213ndash247 2007

[5] L Sun ldquoPricing currency options in the mixed fractionalBrownian motionrdquo Physica A Statistical Mechanics and itsApplications vol 392 no 16 pp 3441ndash3458 2013

[6] A Swishchuk M Tertychnyi and R Elliott ldquoPricing currencyderivatives with Markov-modulated Lersquovy dynamicsrdquo Insur-ance Mathematics amp Economics vol 57 pp 67ndash76 2014

[7] W-L XiaoW-G Zhang X-L Zhang andY-LWang ldquoPricingcurrency options in a fractional Brownian motion with jumpsrdquoEconomic Modelling vol 27 no 5 pp 935ndash942 2010

[8] C Wang S W Zhou and J Y Yang ldquoThe pricing of vulnerableoptions in a fractional brownian motion environmentrdquoDiscrete

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Discrete Dynamics in Nature and Society

Dynamics in Nature and Society vol 2015 Article ID 579213 10pages 2015

[9] B Liu Uncertainty eory vol 2nd Springer Berlin Germany2007

[10] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[11] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[12] Y Shen and K Yao ldquoA mean-reverting currency model in anuncertain environmentrdquo So Computing vol 20 no 10 pp4131ndash4138 2016

[13] X Ji and H Wu ldquoA currency exchange rate model with jumpsin uncertain environmentrdquo So Computing vol 21 no 18 pp5507ndash5514 2017

[14] Y Sheng and G Shi ldquoMean-reverting stock model and pricingrules for Asian currency optionrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 34 no6 pp 4261ndash4268 2018

[15] X W Chen and J Gao ldquoUncertain term structure model ofinterest raterdquo So Computing vol 17 no 4 pp 597ndash604 2013

[16] Y Sun K Yao and Z Fu ldquoInterest ratemodel in uncertain envi-ronment based on exponential OrnsteinndashUhlenbeck equationrdquoSo Computing vol 22 no 2 pp 465ndash475 2018

[17] X Wang and Y Ning ldquoAn uncertain currency model withfloating interest ratesrdquo So Computing vol 21 no 22 pp 6739ndash6754 2017

[18] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics andManagement Science vol 4 no 1 pp 141ndash1831973

[19] T S Y Ho and S Lee ldquoTerm structure movements and pricinginterest rate contingent claimsrdquo e Journal of Finance vol 41no 5 pp 1011ndash1029 1986

[20] J Hull and AWhite ldquoPricing interest rate derivative securitiesrdquoReview of Financial Studies vol 3 no 4 pp 573ndash592 1990

[21] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[22] K Yao and X W Chen ldquoA numerical method for solvinguncertain differential equationsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 25 no3 pp 825ndash832 2013

[23] Y Liu ldquoUncertain random programming with applicationsrdquoFuzzy Optimization and DecisionMaking vol 12 no 2 pp 153ndash169 2013

[24] M Shaked and J G Shanthikumar Stochastic Orders andeirApplications Academic Press San Diego Calif USA 1994

[25] R Gao ldquoMilne method for solving uncertain differential equa-tionsrdquoAppliedMathematics andComputation vol 274 pp 774ndash785 2016

[26] X Wang Y Ning T A Moughal and X Chen ldquoAdams-Simpson method for solving uncertain differential equationrdquoApplied Mathematics and Computation vol 271 pp 209ndash2192015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom