Discrete Dynamics in Nature and Society

Volume 2019, Article ID 2548592, 10 pages

https://doi.org/10.1155/2019/2548592

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

^{1}School of Economics and Management, Beijing Institute of Petrochemical Technology, Beijing 102617, China^{2}Beijing Academy of Safety Engineering and Technology, Beijing 102617, China

Correspondence should be addressed to Xiao Wang; moc.361@52117891oaixgnaw

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

Academic Editor: Gian I. Bischi

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

#### 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.