Valuing currency swap contracts in uncertain financial market

Transcription

1 Fuzzy Optim Decis Making Valuing currency swap contracts in uncertain financial market Yi Zhang 1 Jinwu Gao 1,2 Zongfei Fu 1 Springer Science+Business Media, LLC, part of Springer Nature 218 Abstract Swap is a financial contract between two counterparties who agree to exchange one cash flow stream with the other according to some predetermined rules. When the cash flows are interest payments of different currencies, the swap is called a currency swap. In this paper, it is assumed that the exchange rate follows some uncertain differential equations, and the currency swap contracts in uncertain financial market are discussed. For dealing with long-term, short-term and super-short circumstances, three currency swap models are proposed, respectively. Their explicit solutions are developed through Yao Chen formula. Moreover, a numerical method is designed for simplifying calculation. Finally, examples are given to show the effectiveness of the theory developed in this paper. Keywords Currency swap Exchange rate Uncertain currency model Uncertain process Yao Chen formula This work was supported in part by National Natural Science Foundation of China under Grant and China Scholarship Council under Grant B Jinwu Gao jgao@ruc.edu.cn Yi Zhang ethanzhang@ruc.edu.cn Zongfei Fu fuzf@ruc.edu.cn 1 Uncertain Systems Lab, School of Information, Renmin University of China, Beijing 1872, China 2 Present Address: Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC , USA

2 Y. Zhang et al. 1 Introduction A domestic multinational corporation usually owns a loan from a foreign corporation. The foreign corporation will pay it back in foreign currency when the loan is due. Thus, the domestic multinational corporation faces a foreign exchange risk that the domestic price of foreign currency or the exchange rate may decrease in the future. In order to hedge against the foreign exchange risk, the domestic multinational corporation may sign a currency swap contract with a foreign multinational corporation that is in the opposite situation. A currency swap is a foreign exchange derivative between two counterparties to exchange the principal and interest payments of a loan in one currency for equivalent amounts of the other. Countries and large corporations prefer to use currency swap contract to hedge against the future exchange rate risk. For example, as the world s second-largest economy, China holds a scale of about 3 billion RMB s currency swap contracts. The valuation of a currency swap is important for the participating corporations. But, the valuation is affected by the complex exchange rate processes. In 1962, Mundell (1963 and Fleming (1962 proposed a static exchange rate model (Mundell Fleming model. This model portrayed the short-run relationship between an economy s nominal exchange rate and interest rate. However, the future exchange rate is variable and dynamic since it is not only an onward trend from the past, but also affected by the current information, like the timely policies and news. For example, in the latest American presidential election, when Donald Trump took the lead, the exchange rate between RMB and US dollar began a fast decreasing trend. Traditionally, such noises were assumed as random variables, and the exchange rate, like stock price and interest rate, was described by stochastic differential equations. In literature, Biger and Hull (1993 discussed the valuation of currency option under the assumption that the exchange rate followed a geometric Brownian motion, and the interest rate was constant. Amin and Jarrow (1991 incorporated stochastic interest rate. Furthermore, Bates (1996 used a stochastic differential equation with jumps to describe the exchange rate. For more detailed expositions to stochastic exchange rate models, the readers may refer to Björk (29. It is shown that stochastic models behave well in financial market. But are stochastic models universal? Certainly not. The exchange rate is sensitive to the timely news and policies. For example, on June 24, 216, when the Prime Minister of UK Cameron announced that the United Kingdom would exit the European Union, the exchange rate between English pound and US dollar slumped. Another case in point is the recent American president election. When Donald Trump took the lead, the exchange rate between RMB and US dollar decreased sharply. Such factors will cause the exchange rate deviating from the previous tendencies. The exchange rate may follow a new pattern with little historical records. Even with the historical data or the sophisticated mathematical formulas, we don t knows how the exchange rate evolves in the future. Liu (27 showed that it will lead to counter-intuitive results if we ignore this fact and impose on a distribution deviated from its original one moderately. In such situation, the domain experts intuitions and experiences provide us a powerful alternative. We

3 Valuing currency swap contracts in uncertain can expect their personal belief degrees to give hints and predictions of the future exchange rate. In order to model belief degrees of the domain experts mathematically, uncertainty theory was founded by Liu (27 in 27, and refined by Liu (29 in 29. Uncertainty theory is a branch of mathematics based on normality, duality, subadditivity, and product axioms, in which an uncertain quantity is casted into an uncertain variable with some uncertainty distribution. In order to describe the evolution of an uncertain phenomenon, Liu (28 proposed the concept of uncertain process as a sequence of uncertain variables indexed by time. The study of uncertain differential equation was also initiated by Liu (28. Chen and Liu (21 proved the existence and uniqueness theorem of uncertain differential equations. Yao and Chen (213 found a relationship between an uncertain differential equation and a family of ordinary differential equations, which is called Yao Chen formula. This formula is an important tool for solving uncertain different equations. In order to model the discontinuous uncertain dynamic systems, Yao (215 combined uncertain differential equation with uncertain renewal process, and proposed uncertain differential equation with jumps. For more details about uncertain differential equation, the readers may refer to Liu (21 and Yao (216. Uncertain differential equation was then applied to the area of finance. Liu (29 initiated an uncertain stock model as well as its European option pricing formulas. Chen and Gao (213 introduced three short interest rate models and derived an uncertain term structure equation for valuing zero-coupon bond. We mainly consider three uncertain currency models that are developed for different situations. The Liu Chen Dan model by Liu et al. (215 assumes that the exchange rate follows a geometric Liu process, which fits in the short term circumstance. The Shen Yao model by Shen and Yao (216 assumes that the exchange rate follows a mean-reversion uncertain process, and is applicable to the long term circumstance. The Ji Wu model by Ji and Wu (216 deals with the super-short term circumstance by assuming that the exchange rate follows an uncertain differential equation with jumps. In this paper, we consider the currency swap contracts in the uncertain financial market, where the exchange rate follows different uncertain processes. Firstly, assuming that the exchange rate follows a mean-reversion process, we investigate the long term currency swap contract, and illustrate the valuation formula as well as its properties by both theorems and numerical experiments. Secondly, under the assumption that the exchange rate follows the geometric Liu process, we discuss the short term currency swap contract. Theorems and numerical experiments are proposed to price the short term currency swap contract. Finally, suppose that the exchange rate is an uncertain process with jumps, we give an explicit valuation formula to price the supershort term currency swap contract. A numerical method is given due to the difficulty to calculate the value via the formula. The properties of the formula are demonstrated by both theorems and numerical examples. The rest of this paper is organized as follows. In Sects. 2, 3, 4, we present three valuation formulas for long-term, short term and super short term currency swap contracts, respectively. Theorems and numerical examples are also given to show the properties of the formulas. In order to keep the readability of this paper, some preliminaries are provided in the Appendix at the end of this paper.

4 Y. Zhang et al. 2 Valuing the long term uncertain currency swap contract A currency swap is a foreign exchange derivative between a domestic corporation and a foreign corporation, both of which aim to hedge against the potential exchange risk by exchanging the principal and interest payments of a loan from one currency to another. The value of the swap for the domestic corporation is the present value of domestic currency minus the present value of foreign currency, and vice versa. Since the exchange rate is uncertain and changing with time, it is natural for us to formulate the exchange rate as an uncertain process. In the long run, the exchange rate cannot keep at extremely high or extremely low level. For example, if the exchange rate for the domestic corporations is in a high level, then there is a trade surplus. If the trade surplus lasts for a long time, then the foreign corporations will change the partners. This is a disaster for the export trade. So the exchange rate will fall back to a reasonable level. Similarly, if the exchange rate goes extremely lower, then the exchange rate will also rise to a reasonable level due to the pressure of the domestic corporations. Thus, the exchange rate will revert to a reasonable level in the long run. Shen and Yao (216 proposed an uncertain currency model with a mean-reversion exchange rate as follows, dx t = ux t dt dy t = vy t dt (2.1 dz t = (m az t dt + dc t. The first equation depicts the value of the domestic currency at time t, X t, and the parameter u denotes the domestic interest rate. The second equation gives the value of the foreign currency at time t, Y t, and the parameter v represents the foreign interest rate. In the third equation, Z t is the exchange rate (the domestic price of one unit of foreign currency at time t that follows a mean-reversion uncertain differential equation, where C t is a Liu process that captures the small disturbance of the market, and its coefficients ( > measures the volatility of the exchange rate. Figure 1 gives the exchange rate between EUR and USD from January 21, 215 to February 8, 217. It shows that there is a long run equilibrium exchange rate m/a, where a determines the speed of reversions. Let V d (measured in domestic currency and V f (measured in foreign currency denote the values of the currency swap contract for the domestic corporation and foreign corporation, respectively. Accordingly, S d and S f denote their own principles that are measured in foreign currency and domestic currency, respectively. Suppose that Z is the initial exchange rate. It is easy to see that S d Z = S f.attimet,the contract regulates that the domestic corporation should pay the foreign principle and its interest. At the same time, it receives the domestic principle and its interest. So the payoff of the domestic corporation at time T is S f exp(ut S d exp(vt Z T. (2.2 Considering the time value of the money, the present value of the payoff is

5 Valuing currency swap contracts in uncertain Z t T Fig. 1 EUR/USD from 215/1/21 to 217/2/8 (S f exp(ut S d exp(vt Z T exp( ut ( = S f S d exp (v u T Z T. In order to own the right to exchange the currency at time T, the domestic corporation must pay a V d at the initial time. So the net return of the domestic corporation at time zero is ( Rd = V d + S f S d exp (v u T Z T. On the other hand, the net return of the foreign corporation at time zero is (measured in domestic currency ( Rd = V d S f + S d exp (v u T Z T. Therefore, the fair value of the currency swap for the domestic corporation should ensure that the two parties have the same expected return (measured in domestic currency, i.e., E [ R d] = E [ R d ]. (2.3 At time T, the foreign corporation, however, should pay the domestic principle and its interest. Meanwhile, it receives the foreign principle and its interest. With a similar process, we obtain that the fair value of the currency swap for the foreign corporation follows

6 Y. Zhang et al. E [ R ] [ ] f = E R f (2.4 and ( / R f = V f + S d S f exp (u v T Z T. Definition 2.1 The currency swap contract regulates that the principles of the domestic corporation and foreign corporation are S d and S f, and they should exchange the principles as well as the interests at time T. If we assume the exchange rate follows an uncertain currency model, then the fair values of the currency swap for the domestic corporation and foreign corporation are ( V d = S f S d exp (v u T E[Z T ] (2.5 and ( / V f = S d S f exp (u v T E[Z T ], (2.6 respectively. Theorem 2.1 Assume the exchange rate follows the uncertain currency model (2.1, and the currency swap is described in Definition 2.1. Then the fair values of the domestic corporation and foreign corporation are ( 1 V d = S f S d exp (v u T ΨT 1 (αdα, (2.7 and ( / 1 V f = S d S f exp (u v T ΨT 1 (αdα, (2.8 respectively, where ΨT 1 (α = Z exp( at + m + ( Φ 1 (α 1 exp( at a (2.9 and Φ 1 (α = 3 π ln α 1 α.

7 Valuing currency swap contracts in uncertain Proof Solving the ordinary differential equation dz α t = ( m az α t + Φ 1 (α dt, we have Z α t = Z exp( at + m + ( Φ 1 (α 1 exp( at. a This means the uncertain differential equation dz t = (m az t dt + dc t has an α-path Z α t = Z exp( at + m + ( Φ 1 (α 1 exp( at. a It follows from Yao Chen formula Yao and Chen (213 that Z T uncertainty distribution has an inverse Ψ 1 T (α = Z α T. Then, by Theorem A.1, the theorem is verified. Theorem 2.2 Let V d and V f be the values of the currency swap contract for the domestic corporation and foreign corporation under model (2.1. Then, we can get the monotonicity of V d and V f with the parameters in (2.7 and (2.8, and the results are shown in the table below. In this table, the symbol + means monotone increasing and the symbol means monotone decreasing. Parameters Z m u v V d + V f Proof By Theorem 2.1, the valuation formula of the currency swap for the domestic corporation can be expressed as ( V d = S f S d exp (v u T 1 ( Z exp( at + m + ( Φ 1 (α 1 exp( at dα. a

8 Y. Zhang et al orign Z = 2.5 m = 2.5 u =.1 v = orign Z = 2.5 m = 2.5 u =.1 v = V d 3 V f Time 1.4 Time Fig. 2 V d and V f with respect to expiration date T (1 Since Z exp( at is an increasing function with respect to Z, V d is a decreasing function of the initial exchange rate. (2 It follows from a > and T > that 1 exp( at>. Since m + Φ 1 (α is an increasing function with respect to m, V d is a decreasing function with respect to m. (3 Since exp((v u T is a decreasing function with respect to u, V d is an increasing function with respect to the domestic interest rate. (4 Since exp((v u T is an increasing function with respect to v, V d is a decreasing function with respect to the foreign interest rate. (5 By comparing the valuation formulas of V d and V f in (2.5 and (2.6, this conclusion can be easily proved. Example 2.1 Assume that there are two multinational corporations with a currency swap contract. The contract regulates that S d = 1 million units and S f = 2 million units. At the beginning, Z = 2. Here we assume that the exchange rate follows model (2.1, where u =.4, v =.6, m = 4, a = 2, =.5, T = 1. By Theorem 2.1,we obtain that V f = million units and V d = million units. This means this contract is beneficial to the foreign corporation. The black line in Fig. 2 shows the variations of V d and V f with different expiration date T. Furthermore, setting Z = 2.5 with other parameters unchanged, we get V d and V f with respect to time T that are depicted in green line in Fig. 2. We see that the green line (value of V d /V f when Z = 2.5 is under/above the black line (value of V d /V f when Z = 2. This also verifies that V d decreases with Z, while V f increases with Z. Similarly, we depict V d /V f in red, yellow and blue lines in Fig. 2 when m = 2.5, u =.1 and v =.1, respectively. 3 Valuing the short term uncertain currency swap contract In a short term, the exchange rate does not go to extremely high or extremely low. It just fluctuates at a fixed level (see Fig. 3. In order to describe the short term exchange rate, Liu et al. (215 proposed the following currency model:

9 Valuing currency swap contracts in uncertain Z t T Fig. 3 GBP/USD from 215/7/14 to 215/8/14 dx t = ux t dt dy t = vy t dt dz t = ez t dt + Z t dc t, (3.1 where the parameter X t represents the domestic currency, u is the domestic interest rate, Y t represents the foreign currency, v is the foreign interest rate, Z t represents the exchange rate which means the domestic price of one unit of foreign currency at time t, C t is a Liu process which is used to capture the small disturbance of the market, and Z t follows a geometric Liu process with a log-drift e and a log-diffusion ( >. Theorem 3.1 Suppose that the exchange rate follows the uncertain currency model (3.1, and the currency swap is described in Definition 2.1. Then the fair values of the currency swap for the domestic corporation and foreign corporation are ( 1 V d = S f S d exp (v u T ΨT 1 (αdα, (3.2 and ( / 1 V f = S d S f exp (u v T ΨT 1 (αdα, (3.3 respectively, where ( 3 T ΨT 1 (α = Z exp et + π ln α. 1 α

10 Y. Zhang et al. Proof Using Theorem A.2, we know has a solution whose inverse uncertainty distribution is dz t = ez t dt + Z t dc t Z t = Z exp ( et + C t, ( 3 t Ψt 1 α (α = Z exp et + ln. π 1 α Thus, Z T has an inverse uncertainty distribution ( 3 T ΨT 1 (α = Z exp et + π ln α. 1 α Then, by Theorem A.1, the theorem is verified. Theorem 3.2 Let V d and V f be the values of the currency swap contract for the domestic corporation and foreign corporation under model (3.1. Then, we can get the monotonicity of V d and V f with the parameters in (3.2 and (3.3, and the results are shown in the table below. In this table, the symbol + means monotone increasing and the symbol means monotone decreasing. Parameters Z e u v V d + V f Proof By Theorem 3.1, the valuation formula of the currency swap for the domestic corporation can be expressed as ( ( 1 3 T α V d = S f S d exp (v u T Z exp et + ln dα. π 1 α (1 Since ( 3 T Z exp et + π ln α 1 α is an increasing function with respect to Z, V d is a decreasing function of the initial exchange rate.

11 Valuing currency swap contracts in uncertain orign Z = 2.5 e = 2.5 u =.2 v = orign Z = 2.5 e = 2.5 u =.2 v =.1 V d 5 V f Time 1.58 Time Fig. 4 V d and V f with respect to expiration date T (2 Since ( 3 T exp et + π ln α 1 α is an increasing function with respect to e, V d is a decreasing function with respect to the log-drift e. (3 Since exp((v u T is a decreasing function with respect to u, V d is an increasing function with respect to the domestic interest rate. (4 Since exp((v u T is an increasing function with respect to v, V d is a decreasing function with respect to the foreign interest rate. (5 By comparing the valuation formulas of V d and V f in (3.2 and (3.3, this conclusion can be easily proved. Example 3.1 Assume that there are two multinational corporations that have signed a currency swap contract. The contract regulates that S d = 1 million units and S f = 2 million units. At the beginning, Z = 2. Here we assume that the exchange rate follows model (3.1, where u =.4, v =.6, e = 2, =.5, T =.3. By Theorem 3.1, we get that V f = million units and V d = million units. This means this contract is beneficial to the foreign corporation. The black line in Fig. 4 shows the variations of V d and V f with different expiration date T. Furthermore, setting e = 2.5 with other parameters unchanged, we obtain V d and V f with respect to time T that are drawn as red line in Fig. 4. We see that the red line (value of V d /V f when e = 2.5 is under/above the black line (value of V d /V f when e = 2. This also verifies that V d decreases with e, while V f increases with e. Similarly, we depict V d /V f in green, yellow and blue lines in Fig. 4 when Z = 2.5, u =.2 and v =.1, respectively. 4 Valuing the super short term uncertain currency swap contract In a supper short term, the exchange rate is more fluctuating (See Fig. 5. Thus, the sample path in such situation cannot be seen as continuous anymore. To deal with such

12 Y. Zhang et al jump 8.26 Z t jump 8.24 T Fig. 5 GBP/CNY on 216/1/28 circumstance, Ji and Wu (216 proposed the following uncertain currency model with jumps: dx t = ux t dt dy t = vy t dt (4.1 dz t = ez t dt + Z t dc t + δz t dn t, where X t represents the domestic currency with a domestic interest rate u, Y t represents the domestic foreign currency with a foreign interest rate v, Z t represents the exchange rate with a log-drift e and a log-diffusion ( >, the parameter δ is a positive jump size, and N t is an uncertain renewal process. Theorem 4.1 Assume the exchange rate follows the uncertain currency model (4.1, and the currency swap is described in Definition 2.1. Then the fair values for the domestic corporation and foreign corporation are ( + V d = S f S d exp (v u T Υ T (xdx, (4.2 and ( / + V f = S d S f exp (u v T Υ T (xdx, (4.3 respectively, where Υ t (x = ( ( ( ( ln a et ln b sup 1 Φ t 1 Ψ t, ab x/z ln(1 + δ and Φ t (x, Ψ t (x are the uncertainty distributions of C t and N t, respectively.

13 Valuing currency swap contracts in uncertain Proof According to Yao (215, Z t has an analytic solution and Z t = Z exp ( et + C t ( 1 + δ Nt, Υ t (x = M{Z t x} = M{Z exp(et + C t (1 + δ N t x} = M{exp(et + C t (1 + δ N t x/z }. By using the independence of the uncertain variables, we have Υ t (x = sup M{(exp(et + C t a ((1 + δ N t b} ab x/z = sup M{et + C t ln a} M{N t ln(1 + δ ln b} ab x/z { ln a et } { ln b } = sup M C t M N t. ab x/z ln(1 + δ Since C t and N t have the uncertainty distributions Φ t (x and Ψ t (x, wehave { M C t ln a et } ( ln a et = 1 Φ t and Thus, Υ t (x = { ln b } ( ln b M N t = 1 Ψ t. ln(1 + δ ln(1 + δ ( ( ( ( ln a et ln b sup 1 Φ t 1 Ψ t. ab x/z ln(1 + δ Then it follows from Z t > that E[Z T ]= = = M{Z T x}dx Υ T (xdx sup ab x/z ( ( ( ( ln a et ln b 1 Φ T 1 Ψ T dx, ln(1 + δ where Φ t and Ψ t are the uncertainty distributions of C t and N t, respectively. The theorem is verified.

14 Y. Zhang et al. Theorem 4.2 Let V d and V f be the values of the currency swap contract for the domestic corporation and foreign corporation under model (4.1. Then, we obtain the monotonicity of V d and V f with the parameters in (4.2 and (4.3, and the results are shown in the table below. In the following table, the symbol + means monotone increasing and the symbol means monotone decreasing. Parameters Z e δ u v V d + V f Proof By Theorem 3.1, the valuation formula of the currency swap for the domestic corporation can be expressed as ( V d = S f S d exp (v u T E[Z T ] ( + = S f S d exp (v u T ( ( ln b 1 Ψ T dx. ln(1 + δ sup ab x/z ( ( ln a et 1 Φ T (1 Since ( V d = S f S d exp (v u T E[Z T ] ( = S f S d exp (v u T E [Z exp ( ] ( Nt et + C t 1 + δ holds, V d is a decreasing function of the initial exchange rate. (2 Since 1 Φ T ( ln a et is an increasing function with respect to e, V d is a decreasing function with respect to the log-drift e. (3 Since ( ln b 1 Ψ T ln(1 + δ is an increasing function with respect to δ, V d is a decreasing function with respect to the jump size.

15 Valuing currency swap contracts in uncertain (4 Since exp((v u T is a decreasing function with respect to u, V d is an increasing function with respect to the domestic interest rate. (5 Since exp((v u T is an increasing function with respect to v, V d is a decreasing function with respect to the foreign interest rate. (6 By comparing the valuation formulas of V d and V f in (4.2 and (4.3, this conclusion can be easily proved. Although we have obtained the explicit solution of the currency swap contract when the exchange rate follows (4.1, it is hard to compute the equation (4.2. Next, we provide a numerical method as an alternative. First, in order to get the explicit solution, we have to calculate infinite integral. After substituting 1/y for x by the method of changing variables, we get V d as follows, ( { 1 V d = S f + S d exp (v u T ( ( ] ln b 1 Ψ T dy ln(1 + δ [ 1 y 2 }. sup ab 1/(yZ ( ( ln a et 1 Φ T ln b Since Ψ T (x = Ψ T ( x holds, the optimal b satisfies = n for positive ln(1 + δ integers n. That is, the optimal b satisfies b = (1 + δ n. As a result, the optimal a 1 satisfies a = yz (1 + δ n. Then ( ( ( ( ln a et ln b sup 1 Φ T 1 Ψ T ab 1/(yZ ln(1 + δ ( ( 1 = max (1 Φ T ln 1 n 1 yz (1 + δ n et ( 1 Ψ T (n, and ( V d = S f + S d exp et { 1 (v u T ] ( 1 Ψ T (n dy 1 y 2 }. [ ( ( 1 max (1 Φ T ln n 1 1 yz (1 + δ n The computation procedure is summarized as follows, Step Set the values of the parameters S d, S f, u, v,, e, Z, T, N. Step 1 Partition the interval [, 1] into N parts. The partition points are denoted by < y < y 1 < < y N 1 < y N = 1. Step 2 Calculate max n 1 (1 Φ T ( 1 ln ( 1 y i Z (1 + δ n et ( 1 Ψ T (n, i = 1, 2,...,N.

16 Y. Zhang et al V.7154 d.7155 orign Z = 2.5 e = 2.5 δ =.6 u =.2 v =.1 V f orign Z = 2.5 e = 2.5 δ =.6 u =.2 v = Time.8339 Time Fig. 6 V d and V f with respect to expiration date T Step 3 Calculate ( { 1 V d = S f + S d exp (v u T N et ( ]} 1 Ψ T (n. N [ 1 i=1 y 2 i max n 1 ( ( 1 (1 Φ T ln 1 y i Z (1 + δ n Example 4.1 Assume that there are two multinational corporations who sign a currency swap contract. The contract regulates that S d = 1 million units and S f = 2 million units. At the beginning, Z = 2. Here we assume that the exchange rate follows model (4.1, where u =.4, v =.6, e = 2, δ =.5, =.5, T =.5. By using Theorem 4.1, we get that V f =.8395 million units and V d =.718 million units. This means this contract is beneficial to the foreign corporation. The black line in Fig. 6 shows the variations of V d and V f with different expiration date T. Furthermore, setting δ =.6 with other parameters unchanged, we get V d and V f with respect to time T that are depicted in yellow line in Fig. 4. We can see that the yellow line (value of V d /V f when δ =.6 is under/above the black line (value of V d /V f when δ =.5. This also verifies that V d decreases with δ, while V f increases with δ. Similarly, we depict V d /V f in green, red, blue and purple lines in Fig. 4 when Z = 2.5, e = 2.5, u =.2 and v =.1, respectively. 5 Conclusions In this paper, we investigated the currency swap contracts in the uncertain financial market. Three valuation models were proposed for the long term, short term, and super short term currency swap contracts, respectively. By using Yao Chen formula, we derived their explicit solutions. Moreover, the properties of the solutions were demonstrated by both theorems and numerical examples.

17 Valuing currency swap contracts in uncertain Appendix A: uncertainty theory Uncertainty theory, founded by Liu (27 and refined by Liu (29, is a branch of axiomatic mathematics for modeling human uncertainty. Except for its widely applications in uncertain optimization and uncertain finance, it has been used in many other key areas such as optimal control Guo and Gao (217; Zhu (21, and game theory (Gao et al. 216; Yang and Gao 216. In this section, we recall some basic results that are used in this work. Uncertin variable Suppose Γ is a nonempty set and L is a -algebra over Γ. Each element Λ in L is called an event. Let M be a set function defined from L to [,1]. The concept of uncertain measure is defined as follows. Definition A.1 (Liu 27 The set function M is called an uncertain measure if it satisfies Axiom 1. M{Γ }=1 for the universal set Γ ; Axiom 2. M{Λ}+M{Λ c }=1 for any event Λ; Axiom 3. For any countable sequence of events {Λ i }, we have { } M Λ i M {Λ i }. i=1 i=1 (A.1 Besides, in order to provide the operational law, Liu (29 defined the product uncertain measure on the product -algebre L as follows. Axiom 4. Let (Γ k, L k, M k be uncertainty spaces for k =1,2, The product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } k=1 k=1 (A.2 where Λ k are arbitrarily chosen events from L k for k = 1, 2..., respectively. Based on the concept of uncertain measure, we can define an uncertain variable. Definition A.2 (Liu 27 An uncertain variable is a function ξ from an uncertainty space (Γ, L, M to the set of real numbers such that {ξ B} is an event for any Borel set B of real numbers. Definition A.3 (Liu 27 The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x = M{ξ x} (A.3 for any real number x.

18 Y. Zhang et al. Definition A.4 (Liu 21 Let ξ be an uncertain variable with regular uncertainty distribution Φ(x. Then the inverse function Φ 1 (α is called the inverse uncertainty distribution of ξ. Definition A.5 (Liu 27Letξ be an uncertain variable. The expected value of ξ is defined by E[ξ] = + M{ξ x}dx M{ξ x}dx, (A.4 provided that at least one of the above two integrals is finite. Theorem A.1 (Liu 21 Let ξ be an uncertain variable with regular uncertainty distribution Φ. If the expected value exists, then E[ξ] = 1 Φ 1 (αdα. (A.5 Uncertain process Definition A.6 (Liu28Let(Γ,L, M be an uncertain space and let T be a totally ordered set (e.g. time. An uncertain process is a function X t (γ from T (Γ, L, M to the set of real numbers such that {X t B} is an event for any Borel set B at each time t. Definition A.7 (Liu 214 Uncertain processes X 1t, X 2t,...,X nt are said to be independent if for any positive integer k and any times t 1, t 2,...,t k, the uncertain vectors ξ i = ( X it1, X it2,...,x itk, i = 1, 2,...,n (A.6 are independent, i.e., for any Borel sets B 1, B 2,...,B n of k-dimensional real vectors, we have { n } n M (ξ i B i = M {ξ i B i }. (A.7 i=1 Definition A.8 (Liu 29 An uncertain process C t is said to be a canonical Liu process if (i C = and almost all sample path are Lipschitz continuous, (ii C t has stationary and independent increments, (iii every increment C s+t C s is a normal uncertain variable with expected value and variance t 2. It is easy to see that the uncertainty distribution of C t is Φ t (x = i=1 ( ( 1 + exp π x. 3t

19 Valuing currency swap contracts in uncertain Definition A.9 (Liu 28 Letξ 1, ξ 2, be iid uncertain interarrival times. Define S = and S n = ξ 1 + ξ ξ n for n 1. Then the uncertain process { N t = max n Sn t } (A.8 n is called an uncertain renewal process. Definition A.1 (Liu 29LetX t be an uncertain process and let C t be a Liu process. For any partition of closed interval [a, b] with a = t 1 < t 2 < < t k+1 = b, the mesh is written as Δ = max 1 i k t i+1 t i. Then Liu integral of X t with respect to C t is defined as b a X t dc t = lim Δ i=1 k ( X ti Cti+1 C ti, (A.9 provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be integrable. Uncertain differential equation Definition A.11 (Liu (28 Suppose C t is a Liu process, and f and g are two functions. Then dx t = f (t, X t dt + g(t, X t dc t (A.1 is called an uncertain differential equation. A solution is a general Liu process X t that satisfies (A.1 identically in t. Theorem A.2 (Liu 21 Let u t and v t be two integrable uncertain processes. Then the uncertain differential equation dx t = u t X t dt + v t X t dc t (A.11 has a solution ( t t X t = X exp u s ds + v s dc s. (A.12 Definition A.12 (Yao and Chen 213 Let α be a number with <α<1. An uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t (A.13

20 Y. Zhang et al. is said to have an α-path X α t if it solves the corresponding ordinary differential equation dx α t = f (t, X α t dt + g(t, X α t Φ 1 (αdt (A.14 where Φ 1 (α is the inverse standard normal uncertainty distribution, i.e., 3 Φ 1 (α = π ln α 1 α. (A.15 Theorem A.3 (Yao and Chen 213 Let X t and X α t be the solution and the α-path of the uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t, (A.16 respectively. Then M{X t Xt α, t} =α, (A.17 M{X t > Xt α, t} =1 α. (A.18 References Amin, K., & Jarrow, R. (1991. Pricing foreign currency options under stochastic interest rates. Journal of International Money and Finance, 1(3, Bates, D. (1996. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. The Review of Financial Studies, 9(1, Biger, N., & Hull, J. (1993. The valuation of currency options. Financial Management, 12(1, Björk, T. (29. Arbitrage theory in continuous time. Oxford: Oxford University Press. Chen, X., & Gao, J. (213. Uncertain term structure model of interest rate. Soft Computing, 17(4, Chen, X., & Liu, B. (21. Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1, Fleming, J. (1962. Domestic financial policies under fixed and under floating exchange rates. IMF Economic Review, 9(3, Gao, J., Yang, X., & Liu, D. (216. Uncertain Shapley value of coalitional game with application to supply chain alliance. Applied Soft Computing. Grabbe, J. (1983. The pricing of call and put options on foreign exchange. Journal of International Money and Finance, 2(3, Guo, C., & Gao, J. (217. Optimal dealer pricing under transaction uncertainty. Journal of Intelligent Manufacturing, 28(3, Ji, X., & Wu, H. (216. A currency exchange rate model with jumps in uncertain environment. Soft Computing. Liu, B. (27. Uncertainty theory (2nd ed.. Berlin: Springer. Liu, B. (28. Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1, Liu, B. (29. Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1, 3 1. Liu, B. (21. Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer. Liu, B. (214. Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13(3,

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