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.
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 Section 2, 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.
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 that is a Liu process, and and are two functions. Given an initial value ,is called an uncertain differential equation with an initial value .
Definition 2 (see Yao and Chen [22]). Let be a number with . Uncertain differential equation (1) is said to have an -path if it solves the corresponding ordinary differential equationwhere .
Example 3. Uncertain differential equation with has an -pathand its spectrum is shown in Figure 1.
Theorem 4 (see Liu [23]). Let and be an uncertain variable and a random variable, respectively. Then
Theorem 5 (see Yao and Chen [22]). Let and be the solution and -path of uncertain differential equation (1), respectively. Then, for any monotone function , we have
2.2. Uncertain Currency Model with Fixed Interest Rates
Assume that the exchange rate follows an uncertain differential equation and the domestic and foreign interest rates are constant; Liu, Chen, and Ralescu [1] proposed Liu-Chen-Ralescu model:where represents the riskless domestic currency with the fixed domestic interest rate and represents the riskless foreign currency with the fixed foreign interest rate . 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 formdescribing the interest rate process , where determines the volatility of the interest rate, represents the rate of adjustment, and is the long run average value.
By solving stochastic differential equation (7), we haveand
2.4. Moment Generating Function
Definition 6 (see Shaked and Shanthikumar [24]). The moment generating function of a random variable is wherever this expectation exists.
Remark 7. If , then
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:where is the diffusion of ; is the diffusion of ; , and are the constant parameters; and are independent Wiener processes and and are independent for .
By using formula (9), we haveand
By Definition 2, we obtain that the -path of the exchange rate is
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 for units of domestic currency, where is commonly called a strike price.
Let represent this contract price in domestic currency. The investor needs to pay to buy this contract at time 0. The payoff of the investor is in domestic currency at the maturity date . So the expected profit of the investor at time 0 is
Due to selling the contract, the bank can receive at time 0. At the maturity date , the bank also pays in foreign currency. The expected profit of the bank at time 0 is
To ensure the fairness of the contract, we haveIn this way, the investor and the bank have an identical expected profit.
Definition 9. Under model (10), given a strike price and a maturity date , the European currency call option price is
Theorem 10. Under model (10), given a strike price and a maturity date , the European currency call option price iswhereand
Proof. By Theorem 4, we haveBy Theorem 5 and formula (13), we havewhereAccording to formula (8), we getSince the expectation of the stochastic integral is zero, the expected value of is provided bySince the variance is determined from the diffusion term, namely, the stochastic integral, the two deterministic drift terms in make no contribution to the variance. This givesSince , we haveObserve that , and this implies thatBy use of Fubini theorem as well as that for the stochastic integral, we haveSince , the square term eliminates the source of randomness, andContinuing with simple integration leads toandAccording to Remark 7, we haveThusBy the same way, we getBy Theorem 5 and formula (13), we havewhereHence,According to Definition 9 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. Let be the European call currency option price under model (10). Then
is a decreasing function of , , , , and .
is an increasing function of and .
Proof. Theorem 10 tells us that can be expressed as Since ,is decreasing with and is decreasing with .
Since ,is decreasing with and is decreasing with .
Since is decreasing with , is decreasing with .
Since is decreasing with , is decreasing with .
Sinceandare decreasing with , is decreasing with .
Sinceandare increasing with and , is increasing with and .
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 price , and a maturity date are handy to get and are shown as below.
Definition 12. Under model (10), given a strike price and a maturity date , the European put currency option price is
Theorem 13. Under model (10), given a strike price and a maturity date , the European put currency option price iswhereand
Proof. By Theorem 5 and formula (13), we haveandwhereandBy Theorem 4, we haveandFrom Definition 12 and formulas (33), (35), (54), and (55), the pricing formula of the European put currency option is immediately obtained.
Theorem 14. Let be the European put currency option price under model (10). Then
is a decreasing function of , , , , , and ,
is an increasing function of .
Proof. By Theorem 13, can be written as Since ,is decreasing with and is decreasing with .
Since ,is decreasing with and is decreasing with .
Sinceandare decreasing with and , is decreasing with and .
Since is decreasing with , is decreasing with .
Since is decreasing with , is decreasing with .
Sinceandare increasing with , is decreasing with .
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 () on the pricing formulas. For the sake of simplicity, this part just considers the case of European call currency option.
Table 2 presents the required parameters. Under model (10), we calculate the European call currency option prices and depict them in Figure 2, where is the maturity date.
According to common sense, the European call currency option price is increasing with the maturity date. From Figure 2, we can see that the price computed by model (10) suits this principle. If the maturity date is taken as 100, the European call currency option price is .
In what follows, we discuss the influence of the parameters () on and show the results by a series of experiments. Consider the first case, where the parameters are fixed and the other parameters () are changing. The second case is considering under different parameters and , where the other parameters are not changing. Figures 3 and 4 show versus its parameters , and . The default parameters can be referred to Table 2 and the maturity date .
Figure 3 shows that the European call currency option price under model (10) is increasing with , and . From Figure 4, we can obtain the following conclusion: the European call currency option price is increasing with while it is decreasing with respect to .
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 parameter , 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 . At last, some numerical examples were included to illustrate that the pricing formulas under the proposed currency model are reasonable.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest related to this work.
Acknowledgments
This study was funded by the National Natural Science Foundation of China (11701338) and a Project of Shandong Province Higher Educational Science and Technology Program (J17KB124). The author would like to thank Professor Zhen Peng for helpful comments.