1. Introduction
Options originated from the earliest stock trading. Until the 1980s, due to the increasingly fierce exchange rate fluctuations in the international financial market and the development of international trade, foreign exchange options gradually developed. The first foreign exchange options were sterling and deutschemark options undertaken by the Philadelphia Stock Exchange in 1982. From the perspective of trading volume, the foreign exchange options market has become the deepest and largest global option market with strong liquidity [
1]. Therefore, establishing mathematical models for foreign exchange options is significant in modern mathematical finance [
2].
With the development of foreign exchange options, pricing has become a hot issue in the financial market. Scholars first use probability theory to solve the pricing problem of financial markets. However, many real and reliable data are needed in the application of probability theory, and there are certain obstacles during data collection in real life and regarding the reliability of data, which means that probability theory may cause loopholes in practical applications. To solve the above problems, scholars replace probability with the belief degree of experts in the respective professional field. However, subsequent studies found that the degree of belief is also not accurate and can be easily influenced by personal perspectives [
3], which can lead to huge deviations in the results.
It was not until 2007 that Liu [
4] first put forward the uncertainty theory that the problem of the irrationality of expert belief degree was solved. Uncertainty theory is a new branch of mathematics based on four axioms: normalization, duality, subadditivity, and product axiom. Due to its ability to deal with imprecise information such as subjective judgment, uncertainty theory boasts a series of major achievements. For more information, please refer to [
5,
6]. Liu [
5,
7] optimized uncertainty theory by using an uncertainty measure, and gave the definition of uncertain process. In order to study the uncertain calculus of uncertain processes, Liu proposed the Liu process [
6] in 2009, which is an uncertain process with stationary independent increment that can more accurately simulate the uncertain dynamic system.
On this basis, Liu [
7] established a differential equation that is driven by an uncertain process in 2008. Subsequently, many studies have been conducted. Chen and Liu [
8] put forward the existence and uniqueness theorem of solutions under the global Lipschitz condition. Gao [
9] proved that the theorem also holds under the local Lipschitz condition. Chen and Liu [
8] obtained the analytic solutions of linear uncertain differential equations. Yao and Chen [
10] proposed a numerical method to solve uncertain differential equations.
Since stock prices follow the Liu process, Liu [
7] introduced the uncertainty theory into a financial field for the first time and proposed an uncertain stock model. Soon after that, the European option price formula was derived from Liu’s work. Then, Chen [
11] further derived the American option pricing formula. Peng and Yao [
12] proposed another uncertain stock model and studied the pricing formula of the option. Gao [
13] gave the price formulas of the American barrier option. Chang and Sun [
14] proposed a nonlinear multiperiod portfolio selection model based on uncertainty theory to solve an uncertain multiperiod portfolio selection problem. Furthermore, in addition to the stock market, scholars applied the uncertainty model to the other financial fields. Chen and Gao [
15] inserted the uncertainty model into the interest rate market and proposed a pricing formula for zero coupon bonds. Liu [
16] introduced the uncertainty model into the insurance market field.
Scholars from different countries have performed many studies on using uncertain differential equations to solve financial problems. However, because the volatility of financial market is continuous, an uncertain differential equation cannot precisely reflect this property. Therefore, an uncertain fractional differential equation is more suitable for financial markets due to its continuity and memory.
Because the assumptions of fractional differential equations are more consistent with the facts and more reasonable to solve practical problems, many scholars have studied fractional differential equations. Lakshmikantham et al. [
17] expounded the basic theory of fractional differential equations. Lakshmikantham and Vasundhara Devi [
18] elaborated the details of the theory of fractional differential equations in Banach space. On the basis of the above theories, Belmekki et al. [
19] studied the existence of periodic solutions for nonlinear fractional differential equations. Kosmatov [
20] studied the integral equations and initial value problems of nonlinear fractional differential equations. Zhang [
21] explored the monotone iterative method for the initial value problem of fractional derivatives. In 2013, Zhu [
22] introduced uncertainty into fractional differential equations for the first time and gave two forms of UFDEs. Then, Zhu [
23] gave the Lipschitz condition and linear growth conditions, and obtained the existence and uniqueness of UFDE’s solution. Jin et al. [
24] studied the extremal problem of uncertain fractional differential equations. On the basis of the expected and optimistic values, Lu [
25] gave the pricing formulas of the Asian options, which are driven by an uncertain fractional differential equation.
Foreign exchange options are becoming increasingly important, but there are few scholars using uncertain fractional differential equations to price foreign exchange options. Therefore, this paper introduces an uncertain fractional option model and deduces its pricing formula.
5. Numerical Calculation
5.1. European Currency Option Pricing
Example 1. Assume that uncertain fractional currency Model (13) has , , , , and initial exchange rate , . Consider the European currency option with expiration date and a striking price . In this numerical example, the two currencies are RMB and SGD, and the initial exchange rate was changed a little. These two currencies and the financial markets in both countries were considered to be stable enough, so the parameters in the example are reasonable. In addition, represents the derivative of exchange rate at .
In order to simplify the calculation and reduce the computing time, a numerical calculation was adopted for the integral calculation in numerical examples. Numerical integration can solve most real-life problems, and its error acceptable. For this example, we used the compound trapezoid formula whose interval value was . Let , and we obtained the following European currency option price f.
As shown in
Table 1, the price of the European currency option increased with an increase in
p in
, and decreased with an increase in
p in
and
. The price jumped when
p changes from
to
because of the initial value of
. Such a jump is similar to the results obtained by Lu [
26]. Obviously, different initial values may cause different values of price
f.
With the setting of
,
,
,
,
,
, and
, we obtained
Table 1, which shows the variation in price
f with different
p, including its jump. The compound Simpson formula is another numerical method to calculate numerical integration, and its algebraic precision is better than that of the compound trapezoid formula. We then changed the numerical method, using the compound Simpson formula whose value was also
. Then, we obtained the following European currency option price
in
Table 2.
With the change in the method of numerical integration, we obtained another series of price
. From
Table 2, we can intuitively observe that the prices of option changed little, and some were the same in both tables. There existed a round-off error among computer calculations, the methodical errors of the two numerical methods were small enough to be ignored. Calculating
then generated
Figure 1.
As is shown in
Figure 1, most of the deviation values of prices were smaller than
and becoming smaller. The values of
f in the table were saved as four-digit decimals, so the change in the numerical integration formula made little difference.
Remark 3. Though the compound Simpson formula has better algebraic precision, the application of the compound trapezoid formula is more appropriate because of the shorter computing time and it being easy for code writing.
Now, we discuss the European currency option price with a constant
p and different
K, considering that
K in
and
. With the comparison of the two numerical methods above, we used the compound trapezoid formula to calculate numerical integration and obtain
f. Then, we could generate
Figure 2.
As is illustrated in
Figure 2, for different
p, price
f was a decreasing function of
K, and its decrease was nonlinear. Such a conclusion could also be obtained by observing the structure of Function (
17). While
K increased, the values of
and
decreased or were equal to 0.
Remark 4. The price of European currency option f in this paper was a decreasing function of K, while other parameters were constant, which is similar to the result in Liu et al. [28] for UDEs. Now, we discuss the European currency option price with
and
, considering the variation of the price while
u and
v changed in
. With the comparison of the two numerical methods above, we used the compound trapezoid formula to calculate numerical integration and obtain
f. Then, we could generate
Figure 3.
As is shown in
Figure 3, price
f was a decreasing function of both
u and
v. In Function (
17), taking derivatives in
u and
v, the decrease was nonlinear. In addition, as
u and
v increased simultaneously, the value of price
f decreased.
Remark 5. The price of European currency option f in this paper was a decreasing function of both u and v, while other parameters were constant, which is similar to the result in Liu et al. [28] for UDEs. 5.2. American Currency Option Pricing
Example 2. We assumed that uncertain fractional currency Model (13) had , , , , and the initial exchange rate , , and considered the American currency option with expiration date and striking price . In this numerical example, similar to the previous example, the two currencies were RMB and SGD, and the initial exchange rate was changed a little. These two currencies and the financial markets in both countries were considered to be stable enough, so the parameters in the example were reasonable. As in the previous example, represents the derivative of the exchange rate at .
Due to the particularity of the American currency option whose price is influenced by the maximal , we viewed as a discrete instead of a continuous case. It could also be reasonable to set the interval to be small enough and reduce the computing time. To solve the problem in this example, we divided the whole time T into 100 equal intervals. Hence, our work turned to calculating the maximum of these 100 values to obtain American currency option price f.
Similar to the previous example, we used the compound trapezoid and Simpson formulas whose interval values were both
to compute the integral of
over
. We first used the compound trapezoid formula to calculate American currency option price
f; results are shown in
Table 3.
As is shown in
Table 3, the price of the American currency option increased with the increase in
p in
, and decreased with the increase in
p in
and
. The price jumped when
p changed from
to
because of the initial value of
, similar to the price function of the European currency option. Such a jump is similar to that in the results obtained by Lu [
26]. Obviously, different initial values may cause different values of price
f.
Meanwhile, the value of prices were the same as that of the European currency option when . That is to say, with the setting of , , , , , and initial exchange rate , American currency option price f was an increasing function of time s.
Now, we change the method of numerical integration. With the use of the compound Simpson formula, we obtained the American currency option price
f in
Table 4.
With the change in the method of numerical integration, we obtain another series of price
. From
Table 4, we can intuitively observe that the prices of option changed little, and some were the same in the two tables. Similar to the previous part, we calculated
and generated
Figure 4.
As is shown in
Figure 4, most of the deviation values of prices were smaller than
and becoming smaller. For the same reason, we still chose the compound trapezoid formula instead of the compound Simpson formula.
Remark 6. Similar to Remark 3, though the compound Simpson formula has better algebraic precision, the application of the compound trapezoid formula is more appropriate because of the shorter computing time and it being easy for code writing.
Now, we discuss the American currency option price with a constant
p and different
K, considering that
K was
and
. With the comparison of two numerical methods above, we used the compound trapezoid formula to calculate the numerical integration and obtain
f. Then, we could generate
Figure 5.
As is illustrated in
Figure 5, for the different
p, price
f was a decreasing function of
K, and its decrease was nonlinear. Such a conclusion could also be obtained by observing the structure of Function (
20). While
K increased, the values of
and
decreased or were equal to 0.
Remark 7. Similar to Remark 4, the price of American currency option f in this paper was a decreasing function of K, while other parameters were constant, which is similar to the result in Liu et al. [28] for UDEs. Now, we discuss the American currency option price with
and
, considering variation in the price while
u and
v were changed in
. With the comparison of the two numerical methods above, we used the compound trapezoid formula to calculate the numerical integration and obtain
f. Then, we could generate
Figure 6.
As is shown in
Figure 6, price
f was a decreasing function of both
u and
v. In Function (
20), taking derivatives in
u and
v, the decrease was nonlinear. In addition, as
u and
v increased simultaneously, the value of price
f decreased.
Remark 8. The price of American currency option f in this paper was a decreasing function of both u and v, while other parameters were constant, which is similar to the result in Liu et al. [28] for UDEs.