Interest rates and bonds

We use

$$e^{R(t,\tau)\cdot\tau} = \frac{1}{p(t,\tau)}$$ (36)

as the defining equation of the relation between the price $p(t,\tau)$ of a zero-coupon bond with maturity $\tau$ at time $t$, i.e. the price at time $t$ of the guaranteed payoff $1$ at time $t+\tau$, and the corresponding spot (interest) rate $R(t,\tau)$. Hence, $R(t,\tau)$ is the at $t$ guaranteed continuous interest rate during the time interval $[t,t+\tau]$. For future points of time ($t>0$), $p(t,\tau)$, respectively $R(t,\tau)$, are assumed to be random variables. We now turn to the considered interest rate model of Chen and Scott (1992) with two stochastic factors.

The model is usually called Cox-Ingersoll-Ross-2 (CIR-2) as it relies heavily on the work of Cox, Ingersoll and Ross (1985) which is a so-called short rate model with only one (economically interpretable) stochastic factor (modelled by a square-root process). However, the authors also formulated the main ideas for a theory with multiple stochastic factors. In our description of the model, we closely follow Fischer, May and Walther (2003), which also includes comments on the model choice which we want to adopt for our purposes (see also Subsection 4.3).

The concrete model setup is given by the two stochastic factors $x = (x_1, x_2)$ fulfilling the stochastic differential equations

$$dx_i = (b_i - a_i\cdot x_i)dt + \sigma_i\sqrt{x_i}dW_{i} \quad (i=1,2)$$ (37)

where $b_i$, $a_i$ and $\sigma_i$ are positive constants. One has $x_i>0$ if $2b_i > \sigma_i^2$. $W_{i}(t)$ is the $i$-th brownian motion at time $t$, $W_{1}$ and $W_{2}$ are independent (not correlated). Equation (37) defines a so-called mean reversion process. The parameter $a$ is called the strength of the mean reversion and $b/a$ the mean reversion level, i.e. the long-term mean of the process $x_i$. The implied spot interest rate at time $t$ for a maturity $\tau$ is

$$R(t,\tau) = \sum_{i=1}^{2} \left( -\frac{\log A_i(\tau)}{\tau} + \frac{B_i(\tau)}{\tau}\,x_i(t)\right) ,$$ (38)

the implied zero-coupon bond price at time $t$ for the maturity $\tau$

$$p(t,\tau,x(t)) = \prod\limits_{i=1}^{2}A_i(\tau)e^{-B_{i}(\tau)x_i(t)} .$$ (39)

The respective functions $A_i$ and $B_i$ are given by

$$A_i(\tau) = \left[\frac{2h_ie^{(a_i+\lambda_i+h_i)\tau/2}} {2h_i+(a_i+\lambda_i+h_i)(e^{\tau h_i}-1)}\right]^{2b_i/\sigma_i^2}$$ (40)

and

$$B_i(\tau) = \left[\frac{2(e^{\tau h_i}-1)} {2h_i+(a_i+\lambda_i+h_i)(e^{\tau h_i}-1)}\right] ,$$ (41)

with

$$h_i = \sqrt{(a_i+\lambda_i)^2 + 2\sigma_i^2} .$$ (42)

The parameter $\lambda_i$ concerns the change of measure (physical to martingale measure) and can together with all other parameters be estimated from historical interest rates. In the one-factor case, a particular function of $\lambda$ is interpreted as the so-called market price of risk (Cox, Ingersoll and Ross (1985); see also Fischer, May and Walther (2002)). For more than one factor, an economic interpretation is not possible or at least not obvious.

It is clear that the price of any coupon bond can be computed as the sum of the prices of the respective set of zero-coupon bonds.