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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.

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 [

For European currency option, Garman and Kohlhagen [

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 [

Back to the foreign exchange market, assume that the exchange rate follows an uncertain differential equation; Liu, Chen, and Ralescu [

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 [

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 [

The paper is organized as follows. In Section

Some notations and parameters.

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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.

Suppose that

Let

Uncertain differential equation

A spectrum of

Let

Let

Assume that the exchange rate

Vasicek model [

By solving stochastic differential equation (

The moment generating function of a random variable

If

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:

By using formula (

By Definition

When uncertain differential equation has no analytic solution, the solution can be calculated by some numerical methods. Interested readers can refer to [

In this section, we study the European currency option pricing problem and provide the pricing formula of European currency option under model (

In [

Let

Due to selling the contract, the bank can receive

To ensure the fairness of the contract, we have

Under model (

Under model (

By Theorem

By Theorem

According to formula (

Since the expectation of the stochastic integral is zero, the expected value of

Since the variance is determined from the diffusion term, namely, the stochastic integral, the two deterministic drift terms in

Since

Observe that

By use of Fubini theorem as well as that for the stochastic integral, we have

Since

Continuing with simple integration leads to

According to Remark

Thus

By the same way, we get

By Theorem

Hence,

According to Definition

Herein we discuss the properties of this pricing formula, and the result is shown by the following theorem.

Let

Theorem

Since

Since

Since

Since

Similarity to European call currency option, the definition, formula and property of European put currency option pricing with a strike price

Under model (

Under model (

By Theorem

By Theorem

From Definition

Let

By Theorem

Since

Since

Since

Since

In this section, some numerical examples are included to illustrate the pricing formulas under model (

Table

Parameters setting in model (

| 0.1 | | 0.0065 |

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| 0.15 | | 0.0001 |

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| 1.1 | | 1.2 |

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| 0.35 | | 0.32 |

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| 0.0002 | | 6.65 |

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| 0.0003 | | 6.5 |

European call currency option price

According to common sense, the European call currency option price is increasing with the maturity date. From Figure

In what follows, we discuss the influence of the parameters (

European call currency option price

European call currency option price

Figure

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

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest related to this work.

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.