Valuing currency swap contracts in uncertain financial market

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.

Appendix A: uncertainty theory

Uncertainty theory, founded by Liu (2007) and refined by Liu (2009), 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 (2017); Zhu (2010), and game theory (Gao et al. 2016; Yang and Gao 2016). In this section, we recall some basic results that are used in this work.

1.1 Uncertin variable

Suppose $\varGamma$ is a nonempty set and $\mathcal {L}$ is a $\sigma$-algebra over $\varGamma$. Each element $\varLambda$ in $\mathcal {L}$ is called an event. Let $\mathcal {M}$ be a set function defined from $\mathcal {L}$ to [0,1]. The concept of uncertain measure is defined as follows.

Definition A.1

(Liu 2007) The set function $\mathcal {M}$ is called an uncertain measure if it satisfies

Axiom 1. $\mathcal {M}\{\varGamma \}=1$ for the universal set $\varGamma$;

Axiom 2. $\mathcal {M}\{\varLambda \}+\mathcal {M}\{\varLambda ^c\}=1$ for any event $\varLambda$;

Axiom 3. For any countable sequence of events $\{\varLambda _i\},$ we have

$$\begin{aligned} \mathcal {M}\left\{ \mathop {\bigcup }\limits _{i=1}^{\infty } \varLambda _i \right\} \le \sum \limits _{i=1}^{\infty } \mathcal {M}\left\{ \varLambda _i \right\} . \end{aligned}$$

(A.1)

Besides, in order to provide the operational law, Liu (2009) defined the product uncertain measure on the product $\sigma$-algebre $\mathcal {L}$ as follows.

Axiom 4. Let $(\varGamma _k,\mathcal {L}_k,\mathcal {M}_k)$ be uncertainty spaces for k = 1, 2, $\cdots$ The product uncertain measure $\mathcal {M}$ is an uncertain measure satisfying

$$\begin{aligned} \mathcal {M}\left\{ \mathop {\prod }\limits _{k=1}^{\infty } \varLambda _k \right\} =\mathop {\bigwedge }\limits _{k=1}^{\infty } \mathcal {M}_k \left\{ \varLambda _k \right\} \end{aligned}$$

(A.2)

where $\varLambda _k$ are arbitrarily chosen events from $\mathcal {L}_k$ for $k = 1, 2 \ldots$, respectively.

Based on the concept of uncertain measure, we can define an uncertain variable.

Definition A.2

(Liu 2007) An uncertain variable is a function $\xi$ from an uncertainty space $(\varGamma ,\mathcal {L},\mathcal {M})$ to the set of real numbers such that $\{\xi \in B\}$ is an event for any Borel set B of real numbers.

Definition A.3

(Liu 2007) The uncertainty distribution $\varPhi$ of an uncertain variable $\xi$ is defined by

$$\begin{aligned} \varPhi (x)=\mathcal {M}\{ \xi \le x \} \end{aligned}$$

(A.3)

for any real number x.

Definition A.4

(Liu 2010) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\varPhi (x)$. Then the inverse function $\varPhi ^{-1}(\alpha )$ is called the inverse uncertainty distribution of $\xi$.

Definition A.5

(Liu 2007) Let $\xi$ be an uncertain variable. The expected value of $\xi$ is defined by

$$\begin{aligned} \text {E}[\xi ]={\int }_0^{+ \infty } \mathcal {M}\{\xi \ge x \}\text {d}x-{\int }_{-\infty }^{0}\mathcal {M}\{\xi \le x \}\text {d}x, \end{aligned}$$

(A.4)

provided that at least one of the above two integrals is finite.

Theorem A.1

(Liu 2010) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\varPhi$. If the expected value exists, then

$$\begin{aligned} \text {E}[\xi ]={\int }_0^1 \varPhi ^{-1}(\alpha ) \text {d} \alpha . \end{aligned}$$

(A.5)

1.2 Uncertain process

Definition A.6

(Liu 2008) Let $(\varGamma ,\mathcal {L},\mathcal {M})$ be an uncertain space and let T be a totally ordered set (e.g. time). An uncertain process is a function $X_t(\gamma )$ from $T \times (\varGamma ,\mathcal {L},\mathcal {M})$ to the set of real numbers such that $\{X_t \in B \}$ is an event for any Borel set B at each time t.

Definition A.7

(Liu 2014) Uncertain processes $X_{1t}$$, X_{2t}, \ldots , X_{nt}$ are said to be independent if for any positive integer k and any times $t_1, t_2, \ldots ,t_{k}$, the uncertain vectors

$$\begin{aligned} \xi _{i}=\left( X_{it_{1}}, X_{it_{2}}, \ldots , X_{it_{k}} \right) , i=1, 2, \ldots , n \end{aligned}$$

(A.6)

are independent, i.e., for any Borel sets $B_1, B_2, \ldots , B_n$ of k-dimensional real vectors, we have

$$\begin{aligned} \mathcal {M}\left\{ \bigcap \limits _{i=1}^{n} \left( \xi _{i} \in B_{i} \right) \right\} =\bigwedge \limits _{i=1}^{n} \mathcal {M}\left\{ \xi _{i} \in B_{i} \right\} . \end{aligned}$$

(A.7)

Definition A.8

(Liu 2009) An uncertain process $C_t$ is said to be a canonical Liu process if

• (i) $C_0$ = 0 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 0 and variance $t^2$.

It is easy to see that the uncertainty distribution of $C_t$ is

$$\begin{aligned} \varPhi _t(x) = \left( 1 + \text {exp}\left( -\displaystyle \frac{\pi x}{\sqrt{3}t} \right) \right) . \end{aligned}$$

Definition A.9

(Liu 2008) Let $\xi _1$, $\xi _2$, $\cdots$ be iid uncertain interarrival times. Define $S_0$ = 0 and $S_n$ = $\xi _1$ + $\xi _2$ + $\cdots$ + $\xi _n$ for $n \ge 1$. Then the uncertain process

$$\begin{aligned} N_t = \max \limits _{n \ge 0} \big \{n | S_n \le t \big \} \end{aligned}$$

(A.8)

is called an uncertain renewal process.

Definition A.10

(Liu 2009) Let $X_t$ be an uncertain process and let $C_t$ be a Liu process. For any partition of closed interval [a, b] with $a=t_1< t_2< \cdots < t_{k+1} = b$, the mesh is written as

$$\begin{aligned} \varDelta = \max \limits _{1\le i \le k } |t_{i+1} - t_i | . \end{aligned}$$

Then Liu integral of $X_t$ with respect to $C_t$ is defined as

$$\begin{aligned} {\int }_a^b X_t \text {d}C_t = \lim \limits _{\varDelta \rightarrow 0} \sum \limits _{i=1}^{k} X_{t_i}\left( C_{t_{i+1}} - C_{t_{i}}\right) , \end{aligned}$$

(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.

1.3 Uncertain differential equation

Definition A.11

(Liu (2008)) Suppose $C_t$ is a Liu process, and f and g are two functions. Then

$$\begin{aligned} \text {d}X_{t} = f(t,X_t)\text {d}t + g(t,X_t)\text {d}C_{t} \end{aligned}$$

(A.10)

is called an uncertain differential equation. A solution is a general Liu process $X_t$ that satisfies (A.10) identically in t.

Theorem A.2

(Liu 2010) Let $u_t$ and $v_t$ be two integrable uncertain processes. Then the uncertain differential equation

$$\begin{aligned} \text {d}X_t = u_t X_t \text {d}t + v_t X_t \text {d}C_t \end{aligned}$$

(A.11)

has a solution

$$\begin{aligned} X_t = X_{0}\text {exp}\left( \int _{0}^{t} u_s \text {d}s + \int _{0}^{t} v_s \text {d}C_s \right) . \end{aligned}$$

(A.12)

Definition A.12

(Yao and Chen 2013) Let $\alpha$ be a number with 0 < $\alpha$ < 1. An uncertain differential equation

$$\begin{aligned} \text {d}X_t=f(t,X_t)\text {d}t + g(t,X_t)\text {d}C_t \end{aligned}$$

(A.13)

is said to have an $\alpha$-path $X_t^{\alpha }$ if it solves the corresponding ordinary differential equation

$$\begin{aligned} \text {d}X_t^{\alpha } = f(t,X_t^{\alpha })\text {d}t + |g(t,X_t^{\alpha })|\varPhi ^{-1}(\alpha )\text {d}t \end{aligned}$$

(A.14)

where $\varPhi ^{-1}(\alpha )$ is the inverse standard normal uncertainty distribution, i.e.,

$$\begin{aligned} \varPhi ^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\text {ln}\frac{\alpha }{1-\alpha }. \end{aligned}$$

(A.15)

Theorem A.3

(Yao and Chen 2013) Let $X_t$ and $X_t^{\alpha }$ be the solution and the $\alpha$-path of the uncertain differential equation

$$\begin{aligned} \text {d}X_t = f(t,X_t)\text {d}t + g(t,X_t)\text {d}C_t , \end{aligned}$$

(A.16)

respectively. Then

$$\begin{aligned} \mathcal {M}\{X_t \le X_t^{\alpha }, \forall t \} = \alpha , \end{aligned}$$

(A.17)

$$\begin{aligned} \mathcal {M}\{X_t > X_t^{\alpha }, \forall t \} = 1- \alpha . \end{aligned}$$

(A.18)