An uncertain currency model with floating interest rates

Abstract

Considering the uncertain fluctuations in the financial market from time to time, we propose a currency model with floating interest rates within the framework of uncertainty theory. Different from the classical stochastic currency models, this paper is assumed that the domestic interest rate, the foreign interest rate and the exchange rate follow uncertain differential equations. After that, the pricing formulas of European and American currency options are derived. The simulation experiments presented in this paper illustrate the performance of the proposed model, and the relationship between the option pricing formulas and all relevant parameters is analyzed.

Appendices

Appendix 1: Uncertain variable

Definition 7.1

(Liu (2007, 2009)) Let $\mathcal {L}$ be a $\sigma$-algebra on a nonempty set $\Gamma$. A set function $\mathcal {M}:\mathcal {L}\rightarrow [0,1]$ is called an uncertain measure if it satisfies the following axioms:

Axiom 1

(Normality Axiom) $\mathcal {M}\{\Gamma \}= 1$;

Axiom 2

(Duality Axiom) $\mathcal {M}\{\Lambda \}+\mathcal {M}\{\Lambda ^{c}\}=1$ for any event $\Lambda \in \mathcal {L}$;

Axiom 3

(Subadditivity Axiom) For every countable sequence of $\{\Lambda _i\}\in \mathcal {L}$, we have

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

Axiom 4

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

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

where $a \bigwedge b=\min (a,b)$ and $\Lambda _k \in {\mathcal {L}}_k$ for $k=1,2,\ldots$, respectively.

Definition 7.2

(Liu 2007) An uncertain variable $\xi$ is a measurable function from an uncertainty space $(\Gamma ,{\mathcal {L}},\mathcal {M})$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set

$$\begin{aligned} \{\xi \in B\}=\{\gamma \in \Gamma |\xi (\gamma )\in B\} \end{aligned}$$

is an event.

To describe an uncertain variable $\xi$ in practice, a concept of uncertainty distribution was defined by Liu (2007) as $\Phi (x)=\mathcal {M}\left\{ \xi \le x\right\}$ for any real number x. If $\Phi$ has an inverse function, $\Phi ^{-1}$ is called the inverse uncertainty distribution of $\xi$ Liu (2010).

Definition 7.3

(Liu 2009) Uncertain variables $\xi _1,\xi _2,\ldots ,\xi _n$ are said to be independent if

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

for any Borel sets $B_1,B_2,\ldots ,B_n$ of real numbers.

Definition 7.4

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

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

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

Theorem 7.1

(Liu 2010) Assume $\xi _1,\xi _2,\ldots ,\xi _n$ are independent uncertain variables with regular uncertainty distributions $\Phi _1,\Phi _2,\ldots ,\Phi _n$, respectively. If the function $f(x_1,x_2,\ldots ,x_n)$ is strictly increasing with respect to $x_1,x_2,\ldots$, $x_m$ and strictly decreasing with respect to $x_{m+1},x_{m+2},\ldots ,x_n$, then $\xi =f(\xi _1,\xi _2,\ldots ,\xi _n)$ has an inverse uncertainty distribution

$$\begin{aligned} \Psi ^{-1}(\alpha )= & {} f\left( \Phi _1^{-1}(\alpha ),\ldots , \Phi _m^{-1}(\alpha ),\right. \\&\left. \Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _n^{-1}(1-\alpha )\right) . \end{aligned}$$

In addition, Liu and Ha (2010) proved that the uncertain variable $\xi$ has an expected value

$$\begin{aligned} E[\xi ]= & {} \int _{0}^{1}f\left( \Phi _1^{-1}(\alpha ),\ldots , \Phi _m^{-1}(\alpha ),\right. \\&\left. \Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _n^{-1}(1-\alpha )\right) \mathrm{d}\alpha . \end{aligned}$$

Appendix 2: Uncertain process

An uncertain process is essentially a sequence of uncertain variables indexed by time or space and defined as follows:

Definition 7.5

(Liu 2008) Let $(\Gamma ,{\mathcal {L}},\mathcal {M})$ be an uncertainty space and T be a totally ordered set (e.g., time). An uncertain process is a function $X_{t}(\gamma )$ from $T\times (\Gamma ,{\mathcal {L}},\mathcal {M})$ to the set of real numbers such that

$$\begin{aligned} \{X_{t}\in B\}=\{\gamma \in \Gamma |X_{t}(\gamma )\in B\} \end{aligned}$$

is an event for any Borel set B at each t. For each $\gamma \in \Gamma$, the function $X_{t}(\gamma )$ is called a sample path of $X_{t}$.

Definition 7.6

(Liu 2008) An uncertain process $X_{t}$ is said to have an uncertainty distribution $\Phi _{t}(x)$ if at each t, the uncertain variable $X_{t}$ has the uncertainty distribution $\Phi _{t}(x)$. And the inverse function $\Phi _{t}^{-1}(\alpha )$ of $\Phi _{t}(x)$ is called the inverse uncertainty distribution of $X_{t}$.

Definition 7.7

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

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

are independent, i.e., for any k-dimensional Borel sets $\varvec{B}_{1}, \varvec{B}_{2}, \ldots , \varvec{B}_{n}$, we have

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

Theorem 7.2

(Liu 2014) Assume $X_{1t},X_{2t},\ldots ,X_{nt}$ are independent uncertain processes with regular uncertainty distributions $\Phi _{1t},\Phi _{2t},\ldots ,\Phi _{nt}$, respectively. If the function $f(x_1,x_2,\ldots ,x_n)$ is strictly increasing with respect to $x_1,x_2,\ldots$, $x_m$ and strictly decreasing with respect to $x_{m+1},x_{m+2},\ldots ,x_n$, uncertain process $X_{t}=f(X_{1t},X_{2t},\ldots ,X_{nt})$ has an inverse uncertainty distribution

$$\begin{aligned} \Psi ^{-1}_{t}(\alpha )= & {} f\left( \Phi _{1t}^{-1}(\alpha ),\ldots , \Phi _{mt}^{-1}(\alpha ),\right. \\&\left. \Phi _{m+1,t}^{-1}(1-\alpha ),\ldots ,\Phi _{nt}^{-1}(1-\alpha )\right) . \end{aligned}$$

Definition 7.8

(Liu (2009)) An uncertain process $C_{t}\ (t\ge 0)$ is called a canonical Liu process if

• (i) $C_{0} = 0$ and almost all sample paths are Lipschitz continuous,

• (ii) $C_{t}$ is a stationary independent increment process,

• (iii) every increment $C_{s+t}-C_{s}$ is a normal uncertain variable with expected value 0 and variance $t^{2}$.

Definition 7.9

(Liu 2009) Let $C_{t}$ be a canonical Liu process. Then, for any real numbers $\mu$ and $\sigma > 0$, the uncertain process

$$\begin{aligned} G_{t} = \exp (\mu t + \sigma C_{t}) \end{aligned}$$

is called a geometric Liu process, where $\mu$ is called the log-drift and $\sigma$ is called the log-diffusion.

Definition 7.10

(Liu 2009) Let $X_{t}$ be an uncertain process and let $C_{t}$ be a canonical Liu process. For any partition of the closed interval [a, b] with $a = t_{1}< t_{2}< \cdots < t_{k+1} = b$, the mesh is written as

$$\begin{aligned} \Delta =\sup _{1\le i\le k}|t_{i+1}-t_{i}|. \end{aligned}$$

Then, the Liu integral of $X_{t}$ with respect to $C_{t}$ is defined by

$$\begin{aligned} \int _{a}^{b}X_{t}\mathrm{d}C_{t}=\lim _{\Delta \rightarrow 0}\sum _{i=1}^{k}X_{t_{i}}\cdot (C_{t_{i+1}}-C_{t_{i}}) \end{aligned}$$

provided that the limit exists almost surely and is finite. In this case, the uncertain process $X_{t}$ is said to be integrable.