Keywords

1 Introduction

All the models considered belong to the class of diffusion models, that generate processes X(t) , described by the equation

$$\begin{aligned} dX(t) =\mu (X(t))dt+\sigma (X(t))dW(t), \; t>t_0, \; X(t_0)= X_0, \end{aligned}$$
(1)

where a specific determination of drift \( \mu (x) \) and volatility \( \sigma (x) \) defines one or another particular model. Some models, such as models: Vasicek, Cox-Ingersoll-Ross, geometric Brownian motion, Ahn-Gao, are well documented in the literature, but nevertheless their properties are listed here for convenience of comparison with other, less well-known or not investigated models. The analysis is the first part of the work devoted to the explanation of the most suitable short-term rate models to determine the term structure of a zero-coupon yield that is reproducing the actually observed yield, as far as possible, the best way.

2 The Vasicek Model [4]

For \(\mu (x)=k(\theta -x)\), \(\sigma ^2(x)=\sigma ^2\) the Eq. (1) generates the Ornstein-Uhlenbeck process that is known in finance literature as the Vasicek model. Probability density for this process is normal with the expectation \(E[X]=\theta \) and the variance \(\displaystyle {\textit{Var}}\;[X]=\frac{\sigma ^2}{2k}\):

$$\begin{aligned} f(x)=\sqrt{\frac{k}{\pi }} \frac{1}{\sigma } e^{-k\frac{(\scriptstyle x-\theta )^2}{\scriptstyle \sigma ^2}}. \end{aligned}$$
(2)

3 The CIR Model [5]

When the functions of drift and volatility are \( \mu (x) = k(\theta - x) \) and \( \sigma ^2 (x) = \sigma ^2 x \) from (1) for the short-term interest rate r(t) a nonnegative process is obtained. In financial literature such a process is named the Cox-Ingersoll-Ross model (the CIR model).

$$\begin{aligned} dr(t)=k(\theta -r(t))dt+\sqrt{2kD \frac{r(t)}{\theta }}dW(t), \end{aligned}$$

where \( \theta \) and D are the stationary expectation and variance respectively.

The CIR process has a gamma distribution with the scale parameter \(\displaystyle c=\frac{2k}{\sigma ^2} \) and the form parameter \(\displaystyle q = \frac{2k\theta }{\sigma ^2} \). So

$$\begin{aligned} f(x)=\frac{c^q x^{q-1}}{\varGamma (q)} e^{-cx}, \; q>0, \; x>0. \end{aligned}$$
(3)

The moments of this distribution are calculated by the formula

$$\begin{aligned} E[X^m]=\frac{\varGamma (m+q)}{c^m \varGamma (q)}, \end{aligned}$$

and important numerical characteristics are the expectation E[X],  the variance \( \textit{Var}\,[X], \) the skewness S and the kurtosis K:

4 The Duffie-Kan Model [6]

In the Duffie-Kan model the rate r(t) is generated by Eq. (1) with functions \( \mu (x)=k(\theta - x) \) and \(\displaystyle \sigma (x)=\sqrt{\gamma x +\delta } \equiv \sqrt{2kD\frac{x-r_0}{\theta -r_0}}\):

$$\begin{aligned} dr(t) = (\alpha r(t) + \beta )dt + \sqrt{\gamma r(t) +\delta } dW(t), \; \gamma r(0) + \delta > 0, \end{aligned}$$

where \( k = - \alpha > 0\), \(\displaystyle \theta = - \frac{\beta }{\alpha } > 0\), \(\displaystyle D = \frac{\beta \gamma - \alpha \delta }{2 \alpha ^2} > 0\), \(\displaystyle r_0 = - \frac{\delta }{\gamma } < \theta \).

The process r(t) has the stationary probability density f(x) which is a shifted gamma density with the shift parameter \( r_0 \), the scale parameter c and the form parameter q, i.e.

$$\begin{aligned} f(x)=\frac{c^q (x-r_0)^{q-1}}{\varGamma (q)} e^{-c(x-r_0)}, \; r_0< x < \infty , \end{aligned}$$
(4)

where \(\displaystyle q = \frac{(\theta - r_0)^2}{D}\), \(\displaystyle c = \frac{(\theta - r_0)}{D} >0 \), \( r_0 \) is the limit bottom value of interest rate r(t).

The important numerical characteristics of the stationary density

5 The Ahn-Gao Model [2]

In the Ahn-Gao model it is assumed that drift and volatility are nonlinear functions \( \mu (x) = k(\theta - x) x \) and \( \sigma ^2(x) = \sigma ^2 x^3. \) Such a process has the stationary probability density f(x) of form

$$\begin{aligned} f(x)=\frac{c^q}{\varGamma (q) x^{1+q}} e^{-\frac{\scriptstyle c}{\scriptstyle x}}, \; x>0, \end{aligned}$$
(5)

where the scale parameter \(\displaystyle c =\frac{2k\theta }{\sigma ^2} \) and the form parameter \(\displaystyle q = 2 + \frac{2k}{\sigma ^2}. \) The process of the Ahn-Gao model can be obtained from the CIR process by transformation \(\displaystyle X_{AG} = \frac{1}{X_{CIR}}. \) The important numerical characteristics of the stationary density of process are determined by formulae

6 The BDT Model [7]

The Black-Derman-Toy (BDT) model

$$\begin{aligned} dr(t) = [\alpha _1 r(t) - \alpha _2 r(t) \ln r(t)] dt + \beta r(t) dW(t), \; \alpha _2 > 0, \end{aligned}$$

by transformation \( Y(t) = \ln r(t) \) reduces to linear form

$$\begin{aligned} dY(t) = \left( \alpha _1 - \frac{\beta ^2}{2} - \alpha _2 Y(t)\right) dt + \beta dW(t). \end{aligned}$$

This equation allows a stationary solution and process Y(t) that is found in explicit form

$$\begin{aligned} Y(t) = \frac{1}{\alpha _2} \left( \alpha _1 - \frac{\beta ^2}{2}\right) + \xi (t), \; \xi (t) = \beta \int \limits _{-\infty }^{t}e^{-\alpha _2 s}dW(s), \end{aligned}$$

where \( \xi (t) \) is a stochastic Gaussian process with zero expectation, variance \(\displaystyle \textit{Var}\,[\xi (t)] = \frac{\beta ^2}{2 \alpha _2} \) and covariance \(\displaystyle \textit{Cov}\,[t_1, t_2] = \frac{\beta ^2}{2 \alpha _2}e^{-\alpha _2 |t_2 - t_1|}. \) Thus the BDT model generates a log-normal process and allows a stationary regime. The leading stationary moments of the interest rate are calculated by formulae

7 The Ait-Sahalia Model [8]

Ait-Sahalia has tested the based models of short interest rates (including those described here) by fitting them to the actually time series of rates. It was found that an acceptable level of goodness-of-fit of all these rates was rejected because of the drift and volatility properties. As a result he proposed the following functions of drift and diffusion

$$\begin{aligned} \mu (r) = \alpha _0 + \alpha _1 r + \alpha _2 r^2 + \alpha _{-1} \frac{1}{r}, \; \sigma ^2 (r) = \beta _0 + \beta _1 r + \beta _2 r^2. \end{aligned}$$

In this model, the non-linear functions of drift and diffusion allow a wide variety of forms. To \( \sigma ^2 (r) > 0 \) for any r, it is necessary that the diffusion function parameters ensure the fulfilment of inequalities

$$\begin{aligned} \beta _0> 0, \; \beta _2 > 0, \; \gamma ^2 \equiv 4 \beta _0 \beta _2 - \beta _1^2 \geqslant 0. \end{aligned}$$

Relevant to this function a probability density is given by the expression

$$\begin{aligned} f(x) = N x^B (\beta _0 + \beta _1 x + \beta _2 x^2)^{C - 1} e^{A x + G \text {arctg}\,(E + F x)}, \; x > 0, \end{aligned}$$

where N is the normalization constant,

$$\begin{aligned} A = \frac{2 \alpha _2}{\beta _2} < 0, \; B = \frac{2 \alpha _{-1}}{\beta _0} > 0, \; C = \frac{\alpha _1}{\beta _2} - \frac{\alpha _2 \beta _1}{\beta _2^2} - \frac{\alpha _{-1}}{\beta _0}, \end{aligned}$$
$$\begin{aligned} G = \frac{\displaystyle 2\left( 2 \alpha _0 + \frac{\alpha _2 \beta _1^2}{\beta _2^2} - \frac{\alpha _1 \beta _1}{\beta _2} - \frac{2 \alpha _2 \beta _0}{\beta _2} - \frac{\alpha _{-1} \beta _1}{\beta _0}\right) }{\gamma }, \; E = \frac{\beta _1}{\gamma }, \; F = \frac{\beta _2}{\gamma }. \end{aligned}$$

Since the density f(x) at \( x \rightarrow 0 \) has order \( O(x^B)\), \( B>0\), and at \(x\rightarrow \infty \) its order is \(O(x^{B + C} e^{Ax})\), \(A<0\), then for every finite m the moments \( E[X^m] \) exist, but their analytical expressions cannot be obtained, and they can be calculated only numerically.

8 The CKLS Model [1]

In the Chan-Karolyi-Longstaff-Sanders (CKLS) model it is assumed that \({\mu (x)=k(\theta -x)}\), \(\sigma ^2(x)=\sigma ^2 x^3. \) It turns out that a random process corresponding to this model has a stationary density

$$\begin{aligned} f(x) = \frac{n}{x^3} e^{-c\left( \left( \frac{\scriptstyle \theta }{\scriptstyle x}\right) ^2 - 2\frac{\scriptstyle \theta }{\scriptstyle x}\right) }, \; x>0, \end{aligned}$$
(6)

where \(\displaystyle c = \frac{k}{\theta \sigma ^2}\), n is normalization constant. Note that such random process has only the first stationary moment \( E[X] = \theta \).

9 The Unrestricted Model I [2]

In “unrestricted model I”

$$\begin{aligned} dr = (\alpha _1 + \alpha _2 r + \alpha _3 r^2) dt + \sqrt{\alpha _4 + \alpha _5 r + \alpha _6 r^3} dW \end{aligned}$$
(7)

all the preceding models are embedded, that is, at a certain setting parameters \( \{\alpha \} \) can get any of the previous models. The table in this case has the form

Restrictions of parameters

Model

Equation of processes

\( \alpha _3 = \alpha _5 = \alpha _6 = 0 \)

Vasicek

\( dr = k (\theta - r) dt + \sigma dW \)

\( \alpha _3 = \alpha _4 = \alpha _6 = 0 \)

CIR

\( dr = k (\theta - r) dt + \sigma \sqrt{r} dW \)

\( \alpha _3 = \alpha _6 = 0 \)

Duffie-Kan

\( dr = k (\theta - r) dt + \sqrt{\alpha + \beta r} dW \)

\( \alpha _1 = \alpha _4 = \alpha _5 = 0 \)

Ahn-Gao

\( dr = k (\theta - r) r dt + \sigma r^{1.5} dW \)

\( \alpha _3 = \alpha _4 = \alpha _5 = 0 \)

CKLS

\( dr = k (\theta - r) dt + \sigma r^{1.5} dW \)

Stationary probability density “unrestricted I” process has the form

$$\begin{aligned} f(x) = \frac{c(w)}{\sigma ^2(x)}e^{\textstyle \int \limits _{w}^{x}\frac{2\mu (u)}{\sigma ^2(u)}du} = \frac{c(w)}{\alpha _4 + \alpha _5 x + \alpha _6 x^3} e^{\textstyle \int \limits _{w}^{x}\frac{2\left( \alpha _1 + \alpha _2 u + \alpha _3 u^2\right) }{\alpha _4 + \alpha _5 u + \alpha _6 u^3}du}, \end{aligned}$$

where c(w) is the normalization constant, w is a fixed number from the set of possible values of a random process, the specific value of which does not play some role.

Getting the explicit form of expression for f(x) is possible, but it will be quite cumbersome in a general case, and we restrict ourselves to the case when the values of the parameters \( \{\alpha \} \) provide the performance properties of the probability density f(x). First, we note that the volatility of the real process needs to be a real function, so \( \sigma ^2(r) = \alpha _4 + \alpha _5 r + \alpha _6 r^3 \geqslant 0 \) for all values of r. At the same time analytic properties of the probability density depend on the type of the roots of equation \( \alpha _4 + \alpha _5 r + \alpha _6 r^3 = 0 \), \(\alpha _6 > 0 \). The sign of the discriminant \(\displaystyle \varDelta =\left( \frac{\alpha _5}{3 \alpha _6}\right) ^3 + \left( \frac{\alpha _4}{2 \alpha _6}\right) ^2 \) specifies the number of real and complex roots of the equation. When \( \varDelta > 0, \) there is one real and two complex conjugate roots. When \( \varDelta < 0, \) there are three different real roots. When \( \varDelta = 0, \) real roots are multiples.

Let \( \varDelta > 0 \) and the real root is \( r = r_0, \) then we can write

$$ \alpha _4 + \alpha _5 r + \alpha _6 r^3 = \alpha _6 (r - r_0)\left( r^2 + p r + q\right) , $$

where \( r_0\), p and q are a relatively sophisticated analytical expression and because of that are not listed here. However, if \( \alpha _4 = 0, \) then \( r_0 = 0\), \(p = 0\), \(\displaystyle q = \frac{\alpha _5}{\alpha _6}. \) In this case, the probability density is given by

(8)

where n is the normalization constant. For the existence of the probability density its parameters must satisfy the inequalities: \(\displaystyle \frac{\alpha _1}{\alpha _5} > 1\), \(\displaystyle \frac{\alpha _3}{\alpha _6} < 1. \) In order to at the same time there exist stationary moments it is necessary for the expectation \(\displaystyle \frac{\alpha _3}{\alpha _6} < 0.5\), for variance \(\displaystyle \frac{\alpha _3}{\alpha _6} < 0, \) for the third moment \(\displaystyle \frac{\alpha _3}{\alpha _6} < -0.5 \) and for the fourth moment \(\displaystyle \frac{\alpha _3}{\alpha _6} < - 1. \)

If \( \varDelta < 0, \) denote the roots of the equation \( r_0> r_1 > r_2 \) so

$$\begin{aligned} \alpha _4 + \alpha _5 r + \alpha _6 r^3 = \alpha _6 (r - r_0)(r - r_1)(r - r_2). \end{aligned}$$

Then the probability density is expressed in the form

$$\begin{aligned} f(x) = n (x - r_0)&^{\frac{\scriptstyle 2\left( \alpha _1 + \alpha _2 r_0 + \alpha _3 r_0^2\right) }{\scriptstyle \alpha _6 (r_0 - r_1)(r_0 - r_2)} - 1} \nonumber \\ \times&(x - r_1)^{- \frac{\scriptstyle 2\left( \alpha _1 + \alpha _2 r_1 + \alpha _3 r_1^2\right) }{\scriptstyle \alpha _6 (r_0 - r_1)(r_1 - r_2)} - 1} (x - r_2)^{\frac{\scriptstyle 2\left( \alpha _1 + \alpha _2 r_2 + \alpha _3 r_2^2\right) }{\scriptstyle \alpha _6 (r_0 - r_2)(r_1 - r_2)} - 1}. \end{aligned}$$
(9)

In this case the inequalities must be performed

$$\begin{aligned} 2(\alpha _1 + \alpha _2 r_0 + \alpha _3 r_0^2) > \alpha _6 (r_0 - r_1)(r_0 - r_2), \; \frac{\alpha _3}{\alpha _6} < 1. \end{aligned}$$

For the existence of the m-th moment other than that necessary to perform the conditions \(\displaystyle \frac{m}{2} + \frac{\alpha _3}{\alpha _6} < 1. \) Unfortunately, the analytical expression of the normalization constant n and moments \( E[r^m] \) is very cumbersome and they includes hypergeometric functions. Under these assumptions the process with such a density has a bottom line equal to the largest root, i.e. \( r(t) \geqslant r_0. \)

10 The Unrestricted Model II [1]

In the “unrestricted model II” process of short rate follows the equation

$$\begin{aligned} dr = k (\theta - r) dt + \sigma r^{\gamma } dW, \; \gamma > 0. \end{aligned}$$
(10)

Therefore \( \mu (x) = k (\theta - x)\), \(\sigma ^2(x) = \sigma ^2 x^{2 \gamma } \) and the stationary density f(x) has form

$$\begin{aligned} f(x) = \frac{n}{x^{2 \gamma }} e^{\scriptstyle \frac{1}{\scriptstyle x^{2 \gamma }}\left( \frac{\scriptstyle qx}{\scriptstyle 1 - 2 \gamma } - \frac{\scriptstyle cx^2}{\scriptstyle 2 - 2 \gamma }\right) }, \; x > 0, \end{aligned}$$
(11)

where \(\displaystyle q = \frac{2k\theta }{\sigma ^2}\), \(\displaystyle c = \frac{2k}{\sigma ^2}\), n is the normalization constant. The values of parameter \( \gamma , \) allowing the convergence of the integral of f(x) on the interval \( (0, \infty ), \) determined by the inequality \( \gamma > 0.5. \) At the same time, there are two critical points: \( \gamma = 0.5 \) (in this case, the model is transformed into a short-term rate model CIR) and \( \gamma = 1, \) when the probability density is reduced to a form that corresponds to process of the Brennan-Schwartz model [9]

$$\begin{aligned} f(x) = \frac{q^{1+c}}{x^{2+c} \varGamma (1+c)} e^{\scriptstyle -\frac{q}{\scriptstyle x}}, \; x > 0. \end{aligned}$$
(12)

When \( \gamma = 1.5, \) the “unrestricted model II” is known as the CKLS model. The Vasicek model is also a model embedded in the “unrestricted model II” at \( \gamma = 0. \) For existence of moments of order m,  it is necessary the fulfilment of inequality \( 2\gamma > m + 1. \) Unfortunately, the expression for the probability density in the general case does not allow the calculation of moments in analytical form, although for referred particular cases they are simply calculated. For the model CIR

$$\begin{aligned} E[X^m] = \frac{\varGamma (m + q)}{c^m \varGamma (q)}, \end{aligned}$$

for Brennan-Schwartz model

$$\begin{aligned} E[X^m] = q^m \frac{\varGamma (1 + c - m)}{\varGamma (1 + c)}, \end{aligned}$$

the moments of order m exist if the inequality \( m < 1 + c \) is fulfilled. So that

Model

\(\gamma \)

E[X]

\( \textit{Var}\,[X] \)

Skewness

Kurtosis

Vasicek

0

\( \theta \)

\(\displaystyle \frac{\sigma ^2}{2k} \)

0

3

CIR

0.5

\(\displaystyle \frac{q}{c} = \theta \)

\(\displaystyle \frac{q}{c^2} = \frac{\sigma ^2 \theta }{2k} \)

\(2 \sqrt{q} \)

\(\displaystyle 3 + \frac{6}{q} \)

Brennan-Schwartz

1.0

\(\displaystyle \frac{q}{c} = \theta \)

\(\displaystyle \frac{\theta ^2}{c - 1} \)

\(\displaystyle \frac{4\sqrt{c-1}}{c-2} \)

\(\displaystyle \frac{3(c-1)(c+6)}{(c-2)(c-3)} \)

CKLS

1.5

\(\displaystyle \frac{q}{c} = \theta \)

not exist

not exist

not exist

Even before the appearance of the “unrestricted model II” models were used, which then turned out to be special cases of this model. This is the model of the CIR (1980) [10], which is obtained from the Eq. (10), if we assume that \( \gamma = 1.5 \) and \( k = 0. \) Another particular version is the CEV model, i.e. the model of constant elasticity of variance that was proposed J. Cox and S. Ross (1976) [11], as in Eq. (10) made \( \theta = 0.\) The properties of the processes generated by these models can be understood by considering the limiting transition \( k \rightarrow 0 \) in the first model or \( \theta \rightarrow 0 \) in the second. When k and \( \theta \) are still finite the stationary regimes in the models exist and the probability density of processes for these models is expressed in the form (11). However, in the limiting case \( k = 0 \) or \( \theta = 0 \) stationary regimes of processes no longer exist, and the probability density cannot be expressed in the form (11), and can be obtained as solutions of partial differential equations

$$\begin{aligned} \frac{\partial f(x,t|y,s)}{\partial t} - \frac{1}{2} \frac{\partial ^2[\sigma ^2 x^3 f(x,t|y,s)]}{\partial x^2} = 0 \end{aligned}$$

for the CIR model (1980) and

$$\begin{aligned} \frac{\partial f(x,t|y,s)}{\partial t} + \beta \frac{\partial [x f(x,t|y,s)]}{\partial x} - \frac{\sigma ^2}{2} \frac{\partial ^2[ x^{2 \gamma } f(x,t|y,s)]}{\partial x^2} = 0 \end{aligned}$$

for the CEV model at the boundary condition for both equations

$$\begin{aligned} \lim \limits _{t \rightarrow s} f(x,t|y,s) = \delta (x - y). \end{aligned}$$

Unfortunately, these equations cannot be solved analytically, but we can say that for \( k = 0 \) or \( \theta = 0 \) the process generated by the Eq. (10) becomes unsteady for the CIR model (1980) with the constant expectation and increasing with time variance, and for the CEV model changing with time both the expectation and the variance.

The other non-stationary models are as following.

11 The Merton Model [12]

$$\begin{aligned} dr(t) = \alpha dt + \sigma dW(t) \end{aligned}$$

generates a nonstationary Gaussian process

$$\begin{aligned} r(t) = r(0) + \alpha t + \sigma W(t) \end{aligned}$$

with a linearly varying expectation and linearly increasing variance

$$\begin{aligned} E\left[ r | r(0)\right] = r(0) + \alpha t, \; \textit{Var}\,[r] = \sigma ^2 t. \end{aligned}$$

12 The Dothan Model [13]

The equation of the Dothan model

$$\begin{aligned} dr = \sigma r dW \end{aligned}$$

is solved in explicit form:

$$\begin{aligned} r(t) = r(0) e^{- 0.5 \sigma ^2 t + \sigma W(t)}, \end{aligned}$$

which implies that a random process generated by the model has a log-normal distribution and is non-stationary. The expectation is steady, but the variance increases exponentially with time

$$\begin{aligned} E[r|r(0)] = r(0), \; \textit{Var}\,[r|r(0)] = r(0)^2 \left( e^{\sigma ^2 t} - 1\right) . \end{aligned}$$

13 The GBM Model [14]

The GBM model is a model of process geometric Brownian motion

$$\begin{aligned} dr = \beta r dt + \sigma r dW \end{aligned}$$

was introduced into the modern financial analysis by P. Samuelson (1965). It generates a non-stationary process of geometric Brownian motion

$$\begin{aligned} r(t) = r(0) e^{(\beta - 0.5 \sigma ^2) t + \sigma W(t)}. \end{aligned}$$

In this case, the probability density of the interest rate is log-normal. Unlike BDT model, which also generates a log-normal process, moments of r(t) in the GBM model is not constant but increases exponentially with time, in particular,

Expressions for skewness and kurtosis formally coincide with the expressions of these characteristics of the BDT model, but parameter \( \lambda \) here is not constant and increases exponentially with time.

14 Conclusion

As mentioned above, the process of short-term rates is the basis for building a term structure of the yield of zero-coupon bonds. This explains the interest in the analysis of the processes of short-term rates. In the literature there are many articles that made empirical attempts to find a model of short-term rates, for which a term structure closest to the actual observed structure is obtained [1–3]. On the other hand there is also empirical evidence in the literature that the famous model of short-term rates do not provide an acceptable level of goodness-of-fit [8]. Therefore there is a need for analytical studies to determine the degree of risk in the use of a particular model of short-term rates of the yield. As a necessary basis for this information is needed about the probability properties of the short-term rate processes, expressed analytically. This is the subject of this paper that shall be considered as the first stage of this work.