# Pricing of European Currency Options with Uncertain Exchange Rate and Stochastic Interest Rates

Authors: Xiao Wang
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

Nowadays, the currency option is one of the best investment tools for companies and individuals to hedge against adverse movements in exchange rates. It can be divided into European currency option, American currency option, Asian currency option, and so forth, where European currency option is a contract giving the owner the right to buy or sell one unit of foreign currency with a specified price at a maturity date [1]. The theoretical models of currency option pricing have been a hot issue in mathematical finance, and the key point of this topic is how to get an appropriate pricing formula.For European currency option, Garman and Kohlhagen [2] first proposed G-K model, where both domestic and foreign interest rates are assumed to be constant and the exchange rate is governed by a geometric Brownian motion. However, it is unrealistic for the exchange rate to obey the geometric Brownian motion in the subsequent literatures. By modifying the G-K model, more and more methodologies for the currency option pricing have been proposed, such as Bollen and Rasieland [3], Carr and Wu [4], Sun [5], Swishchuk et al. [6], Xiao et al. [7], and Wang, Zhou, and Yang [8].In the above-mentioned literature, the exchange rate follows a stochastic differential equation under the framework of probability theory. When we use probability theory, the available probability distribution needs to be close to the true frequency. However, some emergencies coming from wars, political policies, or natural disasters may affect the exchange rate. In this case, it is difficult to obtain available statistical data about exchange rate, and the assumption that the exchange rate follows a stochastic differential equation may be out of work. At this time, belief degrees given by some domain experts are used to estimate values or distributions. To model the belief degree, uncertainty theory was established by Liu [9]. In addition, to describe the evolution of an uncertain phenomenon, uncertain process [10] and a Liu process [11] were subsequently proposed. To further express uncertain dynamic systems, uncertain differential equation was proposed [10] and has been widely applied to control and financial market.Back to the foreign exchange market, assume that the exchange rate follows an uncertain differential equation; Liu, Chen, and Ralescu [1] proposed Liu-Chen-Ralescu currency model. In addition, Shen and Yao [12] proposed a mean-reverting currency model under uncertain environment. Recently, Ji and Wu [13] provided an uncertain currency model with jumps. Besides that, Sheng and Shi [14] proposed the mean-reverting currency model under Asian currency option. In these currency models listed above, the exchange rate is governed by an uncertain process instead of stochastic process, and the interest rates are taken as constant.However, considering the fluctuation of interest rate market from time to time, it is unreasonable to regard the interest rates as constant. Up to now, interest rate is mainly studied under the framework of uncertainty theory or probability theory. When the sample size of interest rate is too small (even no-sample) to estimate a probability distribution, interest rate is usually described by an uncertain process under the framework of uncertainty theory. Chen and Gao [15] started from an assumption that the short interest rate follows uncertain process and proposed three equilibrium models. Sun, Yao, and Fu [16] proposed another interest rate model on the basis of exponential Ornstein-Uhlenbeck equation under the uncertain environment. Suppose that both domestic and foreign interest rates follow uncertain differential equations, Wang and Ning [17] proposed an uncertain currency model, where the exchange rate also follows an uncertain differential equation.While there is a large amount of historical data about interest rate, the short interest rate is usually described by a stochastic process. Under the framework of probability theory, Morton [18] proposed the first interest rate model and Ho and Lee [19] then extended it. In addition, many other economists have built other models, such as Hull and White [20] and Vasicek [21]. Taking into account two factors, randomness and uncertainty, we propose a new currency model in this paper. In detail, both domestic and foreign interest rates follow stochastic differential equations, while the exchange rate follows an uncertain differential equation.The paper is organized as follows. In Section2, we mainly introduce uncertain differential equation, Liu-Chen-Ralescu model, Vasicek model, and moment generating function. In Section 3, we propose a new currency model with uncertain exchange rate and stochastic. The pricing formula of European currency option under the proposed model is derived in Section 4. Some numerical examples are carried out in Section 5. Finally, Section 6 makes a brief conclusion. For convenience, some notations and parameters employed in the later sections are shown in Table 1.Table 1
Some notations and parameters.
r t: domestic interest rate at t K: strike price  f t: foreign interest rate at t T: maturity date  Z t: spot domestic currency price of a unit of foreign exchange at t C t: a Liu process  μ: log-drift of the spot currency price Zt W t: a Wiener process  σ: log-diffusion of the spot currency price Zt M: uncertain measure  C E: European call currency option price E [ ξ ]: expected value of ξ  P E: European put currency option price v a r [ ξ ]: variance of ξ  Φ - 1 ( α ) = 3 π ln ⁡ α 1 - α N ( μ , σ 2 ): normal distribution


## 2. Preliminaries

In this section, we first introduce uncertain differential equation. Then, an uncertain currency model and Vasicek model are recalled. Finally, we introduce the moment generating function.


### 2.1. Uncertain Differential Equation

Definition 1 (see Liu [10]).
Suppose thatCt is a Liu process, and f and g are two functions. Given an initial value X0,(1)dXt=ft,Xtdt+gt,XtdCtis called anuncertain differential equation with an initial value X0.Definition 2 (see Yao and Chen [22]).
Letα be a number with 0<α<1. Uncertain differential equation (1) is said to have an α-path Xtα if it solves the corresponding ordinary differential equation(2)dXtα=ft,Xtαdt+gt,XtαΦ-1αdtwhere Φ-1α=3/πln⁡α/(1-α).Example 3.
Uncertain differential equationdXt=Xtdt+XtdCt with X0=1 has an α-path(3)Xtα=exp⁡t+Φ-1αtand its spectrum is shown in Figure 1.Figure 1
A spectrum ofα-paths of dXt=Xtdt+XtdCt.Theorem 4 (see Liu [23]).
Letη and τ be an uncertain variable and a random variable, respectively. Then(4)Eητ=EηEτ.Theorem 5 (see Yao and Chen [22]).
LetXt and Xtα be the solution and α-path of uncertain differential equation (1), respectively. Then, for any monotone function J, we have(5)EJXt=∫01JXtαdα.


### 2.2. Uncertain Currency Model with Fixed Interest Rates

Assume that the exchange rateZt follows an uncertain differential equation and the domestic and foreign interest rates are constant; Liu, Chen, and Ralescu [1] proposed Liu-Chen-Ralescu model:(6)dXt=rXtdtdYt=fYtdtdZt=μZtdt+σZtdCtwhere Xt represents the riskless domestic currency with the fixed domestic interest rate r and Yt represents the riskless foreign currency with the fixed foreign interest rate f. The meaning of remaining parameters in model (6) can be seen in Table 1.


### 2.3. Vasicek Model

Vasicek model [21] is a classical stochastic interest rate model and is defined by a stochastic differential equation of the form(7)drt=ab-rtdt+σdWtdescribing the interest rate process rt, where σ determines the volatility of the interest rate, a represents the rate of adjustment, and b is the long run average value.By solving stochastic differential equation (7), we have(8)rt=r0exp⁡-at+b1-exp⁡-at+σexp⁡-at∫0texp⁡asdWs,and(9)rt~Nr0exp⁡-at+b1-exp⁡-at,σ22a1-exp⁡-2at.


### 2.4. Moment Generating Function

Definition 6 (see Shaked and Shanthikumar [24]).
The moment generating function of a random variableξ is Mξ[t]=E[exp⁡(tξ)],t∈R wherever this expectation exists.Remark 7.
Ifξ~N(μ,σ2), then Eexp⁡(tξ)=exp⁡μt+1/2σ2t2,t∈R.


## 2.1. Uncertain Differential Equation

Definition 1 (see Liu [10]).
Suppose thatCt is a Liu process, and f and g are two functions. Given an initial value X0,(1)dXt=ft,Xtdt+gt,XtdCtis called anuncertain differential equation with an initial value X0.Definition 2 (see Yao and Chen [22]).
Letα be a number with 0<α<1. Uncertain differential equation (1) is said to have an α-path Xtα if it solves the corresponding ordinary differential equation(2)dXtα=ft,Xtαdt+gt,XtαΦ-1αdtwhere Φ-1α=3/πln⁡α/(1-α).Example 3.
Uncertain differential equationdXt=Xtdt+XtdCt with X0=1 has an α-path(3)Xtα=exp⁡t+Φ-1αtand its spectrum is shown in Figure 1.Figure 1
A spectrum ofα-paths of dXt=Xtdt+XtdCt.Theorem 4 (see Liu [23]).
Letη and τ be an uncertain variable and a random variable, respectively. Then(4)Eητ=EηEτ.Theorem 5 (see Yao and Chen [22]).
LetXt and Xtα be the solution and α-path of uncertain differential equation (1), respectively. Then, for any monotone function J, we have(5)EJXt=∫01JXtαdα.


## 2.2. Uncertain Currency Model with Fixed Interest Rates

Assume that the exchange rateZt follows an uncertain differential equation and the domestic and foreign interest rates are constant; Liu, Chen, and Ralescu [1] proposed Liu-Chen-Ralescu model:(6)dXt=rXtdtdYt=fYtdtdZt=μZtdt+σZtdCtwhere Xt represents the riskless domestic currency with the fixed domestic interest rate r and Yt represents the riskless foreign currency with the fixed foreign interest rate f. The meaning of remaining parameters in model (6) can be seen in Table 1.


## 2.3. Vasicek Model

Vasicek model [21] is a classical stochastic interest rate model and is defined by a stochastic differential equation of the form(7)drt=ab-rtdt+σdWtdescribing the interest rate process rt, where σ determines the volatility of the interest rate, a represents the rate of adjustment, and b is the long run average value.By solving stochastic differential equation (7), we have(8)rt=r0exp⁡-at+b1-exp⁡-at+σexp⁡-at∫0texp⁡asdWs,and(9)rt~Nr0exp⁡-at+b1-exp⁡-at,σ22a1-exp⁡-2at.


## 2.4. Moment Generating Function

Definition 6 (see Shaked and Shanthikumar [24]).
The moment generating function of a random variableξ is Mξ[t]=E[exp⁡(tξ)],t∈R wherever this expectation exists.Remark 7.
Ifξ~N(μ,σ2), then Eexp⁡(tξ)=exp⁡μt+1/2σ2t2,t∈R.


## 3. Model Establishment

In this part, we generalize the Liu-Chen-Ralescu model through the use of stochastic interest rates. We employ Vasicek model for the domestic and foreign interest rates. Besides, assume that the exchange rate follows an uncertain differential equation; a new currency model is then proposed as follows:(10)drt=a1b1-rtdt+σ1dW1tdft=a2b2-ftdt+σ2dW2tdZt=μZtdt+σZtdCtwhere σ1 is the diffusion of rt; σ2 is the diffusion of ft; a1,a2,b1, and b2 are the constant parameters; W1t and W2t are independent Wiener processes and Wit and Ct are independent for i=1,2.By using formula (9), we have(11)rt~Nr0exp⁡-a1t+b11-exp⁡-a1t,σ122a1-exp⁡-2a1tand(12)ft~Nf0exp⁡-a2t+b21-exp⁡-a2t,σ222a21-exp⁡-2a2t.By Definition2, we obtain that the α-path of the exchange rate Zt is(13)Ztα=Z0·exp⁡μt+σtΦ-1α.Remark 8.
When uncertain differential equation has no analytic solution, the solution can be calculated by some numerical methods. Interested readers can refer to [22, 25, 26].


## 4. European Currency Option Pricing

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


### 4.1. Pricing Formula of European Call Currency Option

In [1], we can see that European call currency option is a contract endowing the holder the right to buy one unit of foreign currency at a maturity date T for K units of domestic currency, where K is commonly called a strike price.LetCE represent this contract price in domestic currency. The investor needs to pay CE to buy this contract at time 0. The payoff of the investor is (ZT-K)+ in domestic currency at the maturity date T. So the expected profit of the investor at time 0 is(14)-CE+EZT-K+exp⁡-∫0Trsds.Due to selling the contract, the bank can receiveCE at time 0. At the maturity date T, the bank also pays (1-K/ZT)+ in foreign currency. The expected profit of the bank at time 0 is(15)CE-Z0E1-KZT+exp⁡-∫0Tfsds.To ensure the fairness of the contract, we have(16)-CE+EZT-K+exp⁡-∫0Trsds=CE-Z0E1-KZT+exp⁡-∫0Tfsds.In this way, the investor and the bank have an identical expected profit.Definition 9.
Under model (10), given a strike price K and a maturity date T, the European currency call option price CE is(17)CE=12EZT-K+exp⁡-∫0Trsds+Z02E1-KZT+exp⁡-∫0Tfsds.Theorem 10.
Under model (10), given a strike price K and a maturity date T, the European currency call option price is(18)CE=12∫01ZTα-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫011-KZTα+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2where(19)ZTα-K+=Z0exp⁡μT+σTΦ-1α-K+and(20)1-KZTα+=1-KZ0exp⁡μT+σTΦ-1α+.Proof.
By Theorem4, we have(21)EZT-K+exp⁡-∫0Trsds=EZT-K+Eexp⁡-∫0Trsds.
By Theorem5 and formula (13), we have(22)EZT-K+=∫01ZTα-K+dα,where(23)ZTα-K+=Z0exp⁡μT+σTΦ-1α-K+.
According to formula (8), we get(24)∫0Trtdt=∫0Tr0exp⁡-a1tdt+b1∫0T1-exp⁡-a1tdt+∫0Tσ1exp⁡-a1t∫0texp⁡a1sdWsdt=b1T+1-exp⁡-a1Tr0-b1a1+σ1∫0T∫0texp⁡as-tdWsdt.
Since the expectation of the stochastic integral is zero, the expected value of∫0Trtdt is provided by(25)E∫0Trtdt=b1T+1-exp⁡-a1Tr0-b1a1.
Since the variance is determined from the diffusion term, namely, the stochastic integral, the two deterministic drift terms in∫0Trtdt make no contribution to the variance. This gives(26)var∫0Trtdt=varσ1∫0T∫0texp⁡a1s-tdWsdt.
Sincevar[ξ]=E[ξ2]-E[ξ]2, we have(27)var∫0Trtdt=Eσ1∫0T∫0texp⁡a1s-tdWsdt2-Eσ1∫0T∫0texp⁡a1s-tdWsdt2.
Observe thatEσ1∫0T∫0texpa1(s-t)dWsdt=0, and this implies that(28)var∫0Trtdt=Eσ1∫0T∫0texp⁡a1s-tdWsdt2.
By use of Fubini theorem as well as that for the stochastic integral, we have(29)Eσ1∫0T∫0texp⁡a1s-tdWsdt2=Eσ1∫0T∫sTexp⁡a1s-tdtdWs2=Eσ12∫0T∫sTexp⁡a1s-tdt2ds.
SincedWs2=ds, the square term eliminates the source of randomness, and(30)Eσ12∫0T∫sTexp⁡a1s-tdt2ds=σ12∫0T∫sTexp⁡a1s-tdt2ds.
Continuing with simple integration leads to(31)var∫0Trtdtdt=σ12∫0T1-exp⁡a1s-Ta12ds=σ12a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1,and(32)-∫0Trtdt~N-b1T-1-exp⁡-a1Tr0-b1a1,σ12a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
According to Remark7, we have(33)Eexp⁡-∫0Trtdt=exp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
Thus(34)EZT-K+exp⁡-∫0Trsds=∫01ZTα-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
By the same way, we get(35)Eexp⁡-∫0Tfsds=exp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.
By Theorem5 and formula (13), we have(36)E1-KZT+=∫011-KZTα+dα,where(37)1-KZTα+=1-KZ0exp⁡μT+σTΦ-1α+.
Hence,(38)E1-KZT+exp⁡-∫0Tfsds=∫011-KZTα+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.
According to Definition9 and formulas (34) and (38), the pricing formula of European call currency option is immediately obtained.Herein we discuss the properties of this pricing formula, and the result is shown by the following theorem.Theorem 11.
LetCE be the European call currency option price under model (10). Then
( 1 )   C E is a decreasing function of b1, b2, r0, f0, and K.
( 2 )   C E is an increasing function of μ and Z0.Proof.
Theorem10 tells us that CE can be expressed as(39)CE=12∫01Z0exp⁡μT+σTΦ-1α-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫011-KZ0exp⁡μT+σTΦ-1α+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.(1)  Since a1T+exp⁡(-a1T)-1>0(a1,T>0),(40)exp⁡-b1T-b1a1exp⁡-a1T-1=exp⁡-b1a1a1T+exp⁡-a1T-1is decreasing with b1 and CE is decreasing with b1.
Sincea2T+exp⁡(-a2T)-1>0(a2,T>0),(41)exp⁡-b2T-b2a2exp⁡-a2T-1=exp⁡-b2a2a2T+exp⁡-a2T-1is decreasing with b2 and CE is decreasing with b2.
Sinceexp(-r0) is decreasing with r0, CE is decreasing with r0.
Sinceexp(-f0) is decreasing with f0, CE is decreasing with f0.
Since(42)Z0exp⁡μT+σTΦ-1α-Kand(43)Z0-Kexp⁡μT+σTΦ-1αare decreasing with K, CE is decreasing with K.
( 2 )  Since(44)Z0exp⁡μT+σTΦ-1α-Kand(45)Z0-Kexp⁡μT+σTΦ-1αK,Z0,T>0are increasing with μ and Z0, CE is increasing with μ and Z0.


### 4.2. Pricing Formula of European Put Currency Option

Similarity to European call currency option, the definition, formula and property of European put currency option pricing with a strike priceK, and a maturity date T are handy to get and are shown as below.Definition 12.
Under model (10), given a strike price K and a maturity date T, the European put currency option price PE is(46)PE=12EK-ZT+exp⁡-∫0Trsds+Z02EKZT-1+exp⁡-∫0Tfsds.Theorem 13.
Under model (10), given a strike price K and a maturity date T, the European put currency option price is(47)PE=12∫01K-ZTα+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫01KZTα-1+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2where(48)K-ZTα+=K-Z0exp⁡μT+σTΦ-1α+and(49)KZTα-1+=KZ0exp⁡μT+σTΦ-1α-1+.Proof.
By Theorem5 and formula (13), we have(50)EK-ZT+=∫01K-ZTα+dαand(51)EKZT-1+=∫01KZTα-1+dα,where(52)K-ZTα+=K-Z0exp⁡μT+σTΦ-1α+and(53)KZTα-1+=KZ0exp⁡μT+σTΦ-1α-1+.
By Theorem4, we have(54)EK-ZT+exp⁡-∫0Trsds=EK-ZT+Eexp⁡-∫0Trsdsand(55)EKZT-1+exp⁡-∫0Tfsds=EKZT-1+Eexp⁡-∫0Tfsds.
From Definition12 and formulas (33), (35), (54), and (55), the pricing formula of the European put currency option is immediately obtained.Theorem 14.
LetPE be the European put currency option price under model (10). Then
( 1 )   P E is a decreasing function of b1, b2, μ, r0, f0, and Z0,
( 2 )   P E is an increasing function of K.Proof.
By Theorem13, PE can be written as(56)PE=12∫01K-Z0exp⁡μT+σTΦ-1α+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫01KZ0exp⁡μT+σTΦ-1α-1+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.(1)  Since a1T+exp⁡(-a1T)-1>0(a1,T>0),(57)exp⁡-b1T-b1a1exp⁡-a1T-1=exp⁡-b1a1a1T+exp⁡-a1T-1is decreasing with b1 and PE is decreasing with b1.
Sincea2T+exp(-a2T)-1>0(a2,T>0),(58)exp⁡-b2T-b2a2exp⁡-a2T-1=exp⁡-b2a2a2T+exp⁡-a2T-1is decreasing with b2 and PE is decreasing with b2.
Since(59)K-Z0exp⁡μT+σTΦ-1αand(60)Kexp⁡μT+σTΦ-1α-Z0K,Z0,T>0are decreasing with μ and Z0, PE is decreasing with μ and Z0.
Sinceexp(-r0) is decreasing with r0, PE is decreasing with r0.
Sinceexp(-f0) is decreasing with f0, PE is decreasing with f0.
( 2 )  Since(61)K-Z0exp⁡μT+σTΦ-1αand(62)Kexp⁡μT+σTΦ-1α-Z0are increasing with K, PE is decreasing with K.


## 4.1. Pricing Formula of European Call Currency Option

In [1], we can see that European call currency option is a contract endowing the holder the right to buy one unit of foreign currency at a maturity date T for K units of domestic currency, where K is commonly called a strike price.LetCE represent this contract price in domestic currency. The investor needs to pay CE to buy this contract at time 0. The payoff of the investor is (ZT-K)+ in domestic currency at the maturity date T. So the expected profit of the investor at time 0 is(14)-CE+EZT-K+exp⁡-∫0Trsds.Due to selling the contract, the bank can receiveCE at time 0. At the maturity date T, the bank also pays (1-K/ZT)+ in foreign currency. The expected profit of the bank at time 0 is(15)CE-Z0E1-KZT+exp⁡-∫0Tfsds.To ensure the fairness of the contract, we have(16)-CE+EZT-K+exp⁡-∫0Trsds=CE-Z0E1-KZT+exp⁡-∫0Tfsds.In this way, the investor and the bank have an identical expected profit.Definition 9.
Under model (10), given a strike price K and a maturity date T, the European currency call option price CE is(17)CE=12EZT-K+exp⁡-∫0Trsds+Z02E1-KZT+exp⁡-∫0Tfsds.Theorem 10.
Under model (10), given a strike price K and a maturity date T, the European currency call option price is(18)CE=12∫01ZTα-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫011-KZTα+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2where(19)ZTα-K+=Z0exp⁡μT+σTΦ-1α-K+and(20)1-KZTα+=1-KZ0exp⁡μT+σTΦ-1α+.Proof.
By Theorem4, we have(21)EZT-K+exp⁡-∫0Trsds=EZT-K+Eexp⁡-∫0Trsds.
By Theorem5 and formula (13), we have(22)EZT-K+=∫01ZTα-K+dα,where(23)ZTα-K+=Z0exp⁡μT+σTΦ-1α-K+.
According to formula (8), we get(24)∫0Trtdt=∫0Tr0exp⁡-a1tdt+b1∫0T1-exp⁡-a1tdt+∫0Tσ1exp⁡-a1t∫0texp⁡a1sdWsdt=b1T+1-exp⁡-a1Tr0-b1a1+σ1∫0T∫0texp⁡as-tdWsdt.
Since the expectation of the stochastic integral is zero, the expected value of∫0Trtdt is provided by(25)E∫0Trtdt=b1T+1-exp⁡-a1Tr0-b1a1.
Since the variance is determined from the diffusion term, namely, the stochastic integral, the two deterministic drift terms in∫0Trtdt make no contribution to the variance. This gives(26)var∫0Trtdt=varσ1∫0T∫0texp⁡a1s-tdWsdt.
Sincevar[ξ]=E[ξ2]-E[ξ]2, we have(27)var∫0Trtdt=Eσ1∫0T∫0texp⁡a1s-tdWsdt2-Eσ1∫0T∫0texp⁡a1s-tdWsdt2.
Observe thatEσ1∫0T∫0texpa1(s-t)dWsdt=0, and this implies that(28)var∫0Trtdt=Eσ1∫0T∫0texp⁡a1s-tdWsdt2.
By use of Fubini theorem as well as that for the stochastic integral, we have(29)Eσ1∫0T∫0texp⁡a1s-tdWsdt2=Eσ1∫0T∫sTexp⁡a1s-tdtdWs2=Eσ12∫0T∫sTexp⁡a1s-tdt2ds.
SincedWs2=ds, the square term eliminates the source of randomness, and(30)Eσ12∫0T∫sTexp⁡a1s-tdt2ds=σ12∫0T∫sTexp⁡a1s-tdt2ds.
Continuing with simple integration leads to(31)var∫0Trtdtdt=σ12∫0T1-exp⁡a1s-Ta12ds=σ12a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1,and(32)-∫0Trtdt~N-b1T-1-exp⁡-a1Tr0-b1a1,σ12a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
According to Remark7, we have(33)Eexp⁡-∫0Trtdt=exp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
Thus(34)EZT-K+exp⁡-∫0Trsds=∫01ZTα-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1.
By the same way, we get(35)Eexp⁡-∫0Tfsds=exp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.
By Theorem5 and formula (13), we have(36)E1-KZT+=∫011-KZTα+dα,where(37)1-KZTα+=1-KZ0exp⁡μT+σTΦ-1α+.
Hence,(38)E1-KZT+exp⁡-∫0Tfsds=∫011-KZTα+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.
According to Definition9 and formulas (34) and (38), the pricing formula of European call currency option is immediately obtained.Herein we discuss the properties of this pricing formula, and the result is shown by the following theorem.Theorem 11.
LetCE be the European call currency option price under model (10). Then
( 1 )   C E is a decreasing function of b1, b2, r0, f0, and K.
( 2 )   C E is an increasing function of μ and Z0.Proof.
Theorem10 tells us that CE can be expressed as(39)CE=12∫01Z0exp⁡μT+σTΦ-1α-K+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫011-KZ0exp⁡μT+σTΦ-1α+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.(1)  Since a1T+exp⁡(-a1T)-1>0(a1,T>0),(40)exp⁡-b1T-b1a1exp⁡-a1T-1=exp⁡-b1a1a1T+exp⁡-a1T-1is decreasing with b1 and CE is decreasing with b1.
Sincea2T+exp⁡(-a2T)-1>0(a2,T>0),(41)exp⁡-b2T-b2a2exp⁡-a2T-1=exp⁡-b2a2a2T+exp⁡-a2T-1is decreasing with b2 and CE is decreasing with b2.
Sinceexp(-r0) is decreasing with r0, CE is decreasing with r0.
Sinceexp(-f0) is decreasing with f0, CE is decreasing with f0.
Since(42)Z0exp⁡μT+σTΦ-1α-Kand(43)Z0-Kexp⁡μT+σTΦ-1αare decreasing with K, CE is decreasing with K.
( 2 )  Since(44)Z0exp⁡μT+σTΦ-1α-Kand(45)Z0-Kexp⁡μT+σTΦ-1αK,Z0,T>0are increasing with μ and Z0, CE is increasing with μ and Z0.


## 4.2. Pricing Formula of European Put Currency Option

Similarity to European call currency option, the definition, formula and property of European put currency option pricing with a strike priceK, and a maturity date T are handy to get and are shown as below.Definition 12.
Under model (10), given a strike price K and a maturity date T, the European put currency option price PE is(46)PE=12EK-ZT+exp⁡-∫0Trsds+Z02EKZT-1+exp⁡-∫0Tfsds.Theorem 13.
Under model (10), given a strike price K and a maturity date T, the European put currency option price is(47)PE=12∫01K-ZTα+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫01KZTα-1+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2where(48)K-ZTα+=K-Z0exp⁡μT+σTΦ-1α+and(49)KZTα-1+=KZ0exp⁡μT+σTΦ-1α-1+.Proof.
By Theorem5 and formula (13), we have(50)EK-ZT+=∫01K-ZTα+dαand(51)EKZT-1+=∫01KZTα-1+dα,where(52)K-ZTα+=K-Z0exp⁡μT+σTΦ-1α+and(53)KZTα-1+=KZ0exp⁡μT+σTΦ-1α-1+.
By Theorem4, we have(54)EK-ZT+exp⁡-∫0Trsds=EK-ZT+Eexp⁡-∫0Trsdsand(55)EKZT-1+exp⁡-∫0Tfsds=EKZT-1+Eexp⁡-∫0Tfsds.
From Definition12 and formulas (33), (35), (54), and (55), the pricing formula of the European put currency option is immediately obtained.Theorem 14.
LetPE be the European put currency option price under model (10). Then
( 1 )   P E is a decreasing function of b1, b2, μ, r0, f0, and Z0,
( 2 )   P E is an increasing function of K.Proof.
By Theorem13, PE can be written as(56)PE=12∫01K-Z0exp⁡μT+σTΦ-1α+dαexp⁡-b1T-1-exp⁡-a1Tr0-b1a1+σ122a12T-21-exp⁡-a1Ta1+1-exp⁡-2a1T2a1+Z02∫01KZ0exp⁡μT+σTΦ-1α-1+dαexp⁡-b2T-1-exp⁡-a2Tf0-b2a2+σ222a22T-21-exp⁡-a2Ta2+1-exp⁡-2a2T2a2.(1)  Since a1T+exp⁡(-a1T)-1>0(a1,T>0),(57)exp⁡-b1T-b1a1exp⁡-a1T-1=exp⁡-b1a1a1T+exp⁡-a1T-1is decreasing with b1 and PE is decreasing with b1.
Sincea2T+exp(-a2T)-1>0(a2,T>0),(58)exp⁡-b2T-b2a2exp⁡-a2T-1=exp⁡-b2a2a2T+exp⁡-a2T-1is decreasing with b2 and PE is decreasing with b2.
Since(59)K-Z0exp⁡μT+σTΦ-1αand(60)Kexp⁡μT+σTΦ-1α-Z0K,Z0,T>0are decreasing with μ and Z0, PE is decreasing with μ and Z0.
Sinceexp(-r0) is decreasing with r0, PE is decreasing with r0.
Sinceexp(-f0) is decreasing with f0, PE is decreasing with f0.
( 2 )  Since(61)K-Z0exp⁡μT+σTΦ-1αand(62)Kexp⁡μT+σTΦ-1α-Z0are increasing with K, PE is decreasing with K.


## 5. Numerical Examples

In this section, some numerical examples are included to illustrate the pricing formulas under model (10). In addition, we mine the influence of other parameters (T,a1,a2,σ1,σ2,σ) on the pricing formulas. For the sake of simplicity, this part just considers the case of European call currency option.Table2 presents the required parameters. Under model (10), we calculate the European call currency option prices and depict them in Figure 2, where T is the maturity date.Table 2
Parameters setting in model (10).
a1 0.1 μ 0.0065  a2 0.15 σ 0.0001  b1(%) 1.1 r 0 ( % ) 1.2  b2(%) 0.35 f 0 ( % ) 0.32  σ1 0.0002 Z 0 6.65  σ2 0.0003 K 6.5Figure 2
European call currency option priceCE under model (10).According to common sense, the European call currency option price is increasing with the maturity date. From Figure2, we can see that the price computed by model (10) suits this principle. If the maturity date T is taken as 100, the European call currency option price is CE=2.1782.In what follows, we discuss the influence of the parameters (a1,a2,σ1,σ2,σ) on CE and show the results by a series of experiments. Consider the first case, where the parameters a1,a2,T are fixed and the other parameters (σ1,σ2,σ) are changing. The second case is considering CE under different parameters a1 and a2, where the other parameters are not changing. Figures 3 and 4 show CE versus its parameters σ1,σ2,σ,a1, and a2. The default parameters can be referred to Table 2 and the maturity date T=100.Figure 3
European call currency option priceCE versus parameters σ1, σ2, and σ, where (1)  CE versus σ1;(2)CE versus σ2;(3)CE versus σ.Figure 4
European call currency option priceCE versus parameters a1 and a2, where (1)CE versus a1;(2)CE versus a2.Figure3 shows that the European call currency option price under model (10) is increasing with σ1,σ2, and σ. From Figure 4, we can obtain the following conclusion: the European call currency option price is increasing with a1 while it is decreasing with respect to a2.


## 6. Conclusion

This paper proposed a new currency model, in which the exchange rate follows an uncertain differential equation, while the domestic and foreign interest rates follow stochastic differential equations. Subsequently, we provided the pricing formulas of European currency option under this currency model. Meanwhile, we find that the European call currency option price under the proposed model is increasing with the initial exchange rate, the log-drift, the diffusion, the log-diffusion, the parametera1, and the maturity date, while it is decreasing with the initial domestic interest rate, the initial foreign interest rate, the strike price, and the parameters a2,b1,b2. At last, some numerical examples were included to illustrate that the pricing formulas under the proposed currency model are reasonable.



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