Two-factor term structure model with uncertain volatility risk

Abstract

This paper aims to study two-factor uncertain term structure model where the volatility of the uncertain interest rate is driven by another uncertain differential equation. In order to solve this model, the nested uncertain differential equation method is employed. This paper is also devoted to the study of the numerical solutions for the proposed nested uncertain differential equation using the $\alpha$-path methods. We also use the built two-factor term structure model to value the bond price with the help of proposed numerical method. Finally, we give a numerical example where the price of a zero-coupon bond is calculated based on the $\alpha$-path methods.

Appendix: Uncertainty theory

Based on the axioms of uncertain measure, an uncertainty theory was founded by Liu (2007) in 2007 and refined by Liu (2010) in 2010.

Definition 2

(Liu 2007) Let $\text{ L }$ be a $\sigma$-algebra on a nonempty set $\varGamma .$ A set function is called an uncertain measure if it satisfies the following axioms: Axiom 1: (normality axiom) for the universal set $\varGamma .$ Axiom 2: (duality axiom) for any event $\Lambda$. Axiom 3: (subadditivity axiom) For every countable sequence of events $\Lambda _1,\Lambda _2,\ldots ,$ we have

The triplet is called an uncertainty space. Besides, the product uncertain measure on the product $\sigma$-algebra $\text{ L }$ was defined by Liu (2009a, b) as follows: Axiom 4: (product axiom) Let be uncertainty spaces for $k=1,2,\ldots$ Then the product uncertain measure is an uncertain measure satisfying

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

An uncertain variable is a function from an uncertainty space to the set of real numbers. The uncertainty distribution $\varPhi$ of an uncertain variable $\xi$ is defined by for any real number x. An uncertainty distribution $\varPhi (x)$ is said to be regular if it is a continuous and strictly increasing function with respect to x at which $0<\varPhi (x)<1$, and

$$\begin{aligned} \lim \limits _{x\rightarrow -\infty } \varPhi (x) = 0, \lim \limits _{x\rightarrow +\infty } \varPhi (x) = 1. \end{aligned}$$

Definition 3

(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 $\xi$.

If the expected value of $\xi$ exists, it was proved by Liu (2010) that

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

An uncertain process is essentially a sequence of uncertain variables indexed by time. The study of uncertain process was started by Liu (2008) in 2008.

Definition 4

(Liu 2008) Let T be an index set and let be an uncertainty space. An uncertain process is a measurable function from to the set of real numbers, i.e., for each $t\in T$ and any Borel set B of real numbers, the set

$$\begin{aligned} \{X_t\in B\}=\{\gamma \in \varGamma |X_t(\gamma )\in B\} \end{aligned}$$

is an event.

Definition 5

(Liu 2009a, b) An uncertain process $C_t\ (t\ge 0)$ is said to be a canonical Liu process if

• (i) $C_0=0$ and almost all sample paths 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 $t^2$, whose uncertainty distribution is

  $$\begin{aligned} \varPhi (x)=\left( 1+\exp \left( \frac{-\pi x}{\sqrt{3}t}\right) \right) ^{-1},\ x\in \mathfrak {R}. \end{aligned}$$

Based on canonical Liu process, Liu calculus was then defined by Liu (2009a, b).

Definition 6

(Liu 2009a, b) Let $X_t$ be an uncertain process and $C_t$ be a canonical 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

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

Then Liu integral of $X_t$ is defined by

$$\begin{aligned} \int _a^bX_t\mathrm{d}C_t=\lim _{\Delta \rightarrow 0}\sum _{i=1}^kX_{t_{i}}\cdot (C_{t_{i+1}}-C_{t_{i}}) \end{aligned}$$

provided that the limit exists almost surely and is finite.

Definition 7

(Liu 2009a, b) Let $C_t$ be a canonical Liu process and $Z_t$ be an uncertain process. If there exist uncertain processes $\mu _s$ and $\sigma _s$ such that

$$\begin{aligned} Z_t=Z_0+\int _0^t\mu _s\mathrm{d}s+\int _0^t\sigma _s\mathrm{d}C_s \end{aligned}$$

for any $t\ge 0,$ then $Z_t$ is said to be differentiable and has an uncertain differential

$$\begin{aligned} \mathrm{d}Z_t=\mu _t\mathrm{d}t+\sigma _t\mathrm{d}C_t. \end{aligned}$$

Liu (2009a, b) verified the fundamental theorem of uncertain calculus, i.e., for a canonical Liu process $C_t$ and a continuous differentiable function h(t, c), the uncertain process $Z_t=h(t,C_t)$ has an uncertain differential

$$\begin{aligned} \mathrm{d}Z_t=\frac{\partial h}{\partial t}(t,C_t)\mathrm{d}t+ \frac{\partial h}{\partial c}(t,C_t)\mathrm{d}C_t. \end{aligned}$$

Uncertain calculus provides a theoretical foundation for constructing uncertain differential equations. Uncertain differential equation driven by canonical Liu process is defined as follows.

Definition 8

(Liu 2008) Suppose $C_t$ is a canonical Liu process, and f and g are some given functions. Given an initial value $X_0$, then

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

(19)

is called an uncertain differential equation with an initial value $X_0$. A solution is an uncertain process $X_t$ that satisfies Eq. (19) identically in t.

Definition 9

(Yao 2014) Let $C_t$ be an n-dimensional canonical Liu process. Suppose $f(t,{\varvec{x}})$ is a vector-valued function from $T\times \mathfrak {R}^m$ to $\mathfrak {R}^m$, and $g(t,{\varvec{x}})$ is a matrix-valued function from $T\times \mathfrak {R}^m$ to the set of $m\times n$ matrices. Then

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

(20)

is called an m-dimensional uncertain differential equation driven by an n-dimensional canonical Liu process. A solution is an m-dimensional uncertain process that satisfied (20) identically in each t.

The concept of $\alpha$-path is introduced as follows.

Definition 10

(Yao and Chen 2013) The $\alpha$-path $(0<\alpha <1)$ of an uncertain differential equation

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

with initial value $X_0$ is a deterministic function $X_t^{\alpha }$ with respect to t that solves the corresponding ordinary differential equation

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

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

$$\begin{aligned} \varPhi ^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha },\quad 0<\alpha <1. \end{aligned}$$

Theorem 3 (Yao and Chen 2013) Let $X_t$ and $X_t^{\alpha }$ be the solution and $\alpha$-path of the uncertain differential equation

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

respectively. Then

(21)

(22)

The proved formulas show that the solutions of an uncertain differential equation are related to a class of ordinary differential equations.