International investing in uncertain financial market

Abstract

International investing is the strategy of selecting globally based investment instruments as a part of an investment portfolio. In order to diversify the portfolios and enhance growth opportunities, more and more firms choose to invest on foreign stocks and derivatives that bring not only stock price risk, but also the exchange rate risk. This paper considers foreign derivatives in an uncertain financial market. Under the assumption that both the exchange rate and the stock price follow uncertain differential equations, the domestic prices of foreign European options, American options and Asian options are developed by means of contour process, respectively. Some examples are finally provided to further demonstrate the properties of the pricing formulas.

Appendix A

Uncertainty theory, founded by Liu (2007) and refined by Liu (2009a), is a branch of axiomatic mathematics for modeling human uncertainty. Uncertainty theory has been applied in many key areas, for example Gao (2013) studied uncertain game, Liu (2009a) studied uncertain finance, Liu (2009b) studied uncertain programming. Let $\varGamma$ be a nonempty set, and $\text{ L }$ be a $\sigma$-algebra over $\varGamma$. Each element $\varLambda$ in $\text{ L }$ is called an event. Uncertain measure is defined as a function from $\text{ L }$ to [0,1]. In detail, Liu (2007) gave the concept of uncertain measure as follows.

Definition A.1

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

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

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

Axiom 3. For any countable sequence of events $\varLambda _1$, $\varLambda _2$, $\cdots$, we have

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

(A.1)

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

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

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

(A.2)

where $\varLambda _k$ are arbitrarily chosen events from $\text{ L }_k$ 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 $(\varGamma ,\text{ L },\text{ M })$ 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

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

(A.3)

for any real number x.

Moreover, if an uncertainty distribution $\varPhi (x)$ is continuous and strictly increasing function with respect to x at which $0<\varPhi (x)<1$, and $\lim \nolimits _{x \rightarrow - \infty } \varPhi (x) = 0$, $\lim \nolimits _{x \rightarrow + \infty } \varPhi (x) = 1$, then $\varPhi (x)$ is said to be regular.

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

(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

(A.6)

Definition A.6

(Liu 2007) Let $(\varGamma ,\text{ L },\text{ M })$ 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 $T \times (\varGamma ,\text{ L },\text{ M })$ 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 $t_1, t_2, \ldots ,t_{k}$, to be uncertain vectors

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

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

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

(A.8)

Definition A.8

(Liu 2009a) An uncertain process $C_t$ is said to be a 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 $t^2$.

Definition A.9

(Liu 2009a) 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

(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 is a type of differential equation driven by Liu process. Chen and Liu (2010) proved the existence and uniqueness theorem of uncertain differential equation. Yao and Chen (2013) found a relationship between an uncertain differential equation and a family of ordinary differential equations, which is called Yao-Chen formula. For more information about uncertain differential equations, the readers can refer to Yao’s book Yao (2016).

Definition A.10

(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.

Definition A.11

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

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

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.11)

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

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

Definition A.12

(Yao 2015) Let $X_t$ be an uncertain process. If for each $\alpha$ $\in$ (0,1), there exists a real function $X_t^{\alpha }$ such that

$$\begin{aligned} \text{ M }\{X_t \le X_t^{\alpha }, \forall t \}= & {} \alpha ,\\ \text{ M }\{X_t > X_t^{\alpha }, \forall t \}= & {} 1- \alpha , \end{aligned}$$

then $X_t$ is called a contour process. In this case, $X_t^{\alpha }$ is called the $\alpha$-path of the uncertain process.

For example, a solution of an uncertain differential equation is a contour process.

Theorem A.2

(Yao 2015) Let $X_t$ be a contour process with an $\alpha$-path $X_t^{\alpha }$. Then, its supremum process

$$\begin{aligned} Y_t = \sup \limits _{0 \le s \le t} X_s \end{aligned}$$

is a contour process with an $\alpha$-path

$$\begin{aligned} Y_t^{\alpha } = \sup \limits _{0 \le s \le t} X_s^{\alpha }. \end{aligned}$$

Theorem A.3

(Yao 2015) Let $X_t$ be a contour process with an $\alpha$-path $X_t^{\alpha }$. Then, its time integral process

is a contour process with an $\alpha$-path

Theorem A.4

(Yao 2015) Let $X_{1t}$, $X_{2t}$, $\ldots$, $X_{nt}$ be independent contour processes with $\alpha$-paths $X_{1t}^{\alpha }$, $X_{2t}^{\alpha }$, $\ldots$, $X_{nt}^{\alpha }$, respectively. If $f(x_1, x_2, \ldots , x_n)$ 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 the uncertain process

$$\begin{aligned} X_{t} =f(X_{1t}, X_{2t}, \ldots , X_{nt}) \end{aligned}$$

is a contour process with an $\alpha$-path

$$\begin{aligned} X_{t}^{\alpha } = f(X_{1t}^{\alpha }, \ldots ,X_{mt}^{\alpha }, X_{(m+1)t}^{1-\alpha }, \ldots , X_{nt}^{1-\alpha }). \end{aligned}$$