On The Calibration of Short-Term Interest Rates Through a CIR Model

The aim of the present work is to provide a new numerical methodology for the CIR framework, which we call the CIR# model, that well fits the term structure of short interest rates as observed in a real market. Our approach is based on a proper translation of interest rates so that the market volatility structure is preserved as well as the analytical tractability of the original CIR model.

Cox, Ingersoll & Ross (1985) [11] proposed a term structure model, well known as the CIR model, to describe the price of discount zero-coupon bonds with variuous maturities under no-arbitrage condition. This model generalizes the Vasicek model [34] to the case of non constant volatility and assumes that the evolution of the underlying short term interest rate is a diffusion process, i.e. a continuous Markov process unique solution to the following stochastic differential equation (SDE)

with initial condition $r(0)=r_{0}>0$. $(W(t))_{t\geq 0}$ denotes a standard Brownian motion under the risk neutral probability measure, intended to model a random risk factor. The interest rate process $(r(t))_{t\geq 0}$ is usually known as the CIR process or square root process. The SDE (1) is classified as a one-factor time-homogeneous model, because the parameters $k,\theta$ and $\sigma,$ are time-independent and the short interest rate dynamic is driven only by the market price of risk $\lambda(t,r(t)):=\lambda r(t),$ where $\lambda$ is a constant. Therefore the SDE (1) is composed of two parts: the “mean reverting” drift component $k[\theta-r(t)],$ which ensures the rate $r(t)$ is elastically pulled towards a long-run mean value $\theta>0$ at a speed of adjustment $k>0,$ and the random component $W(t),$ which is scaled by the standard deviation $\sigma\sqrt{r(t)}$. The volatility of the instantaneous short rate is denoted by $\sigma>0$.

The paths of the CIR process never reach negative values and their behaviour depends on the relationship between the three constant positive parameters $k,\theta,\sigma$. Indeed, it can be shown (see, e.g., Jeanblanc, Yor & Chesney (2009) [19, Ch. 3]) that if the condition $2k\theta\geq\sigma^{2}$ is satisfied, then the interest rates $r(t)$ are strictly positive for all $t>0$, and, for small $r(t)$, the process rebounds as the random perturbation dampens with $r(t)\rightarrow 0$. Furthermore, the CIR process belongs to the class of processes satisfying the “affine property”, i.e., the logarithm of the characteristic function of the transition distribution of such processes is an affine (linear plus constant) function with respect to their initial state (for more details the reader can refer to Duffie, Filipović & Schachermayer (2003) [13, Section 2]. As a consequence, the non-arbitrage price of a discount zero coupon bond with maturity $T>0$ and underlying interest rate dynamics described by a CIR process, is given by

where $A(\cdot)$ and $B(\cdot)$ are deterministic functions (see Cox, Ingersoll & Ross (1985) [11]). The final condition is $P(T,r(T))=1,$ which corresponds to the nominal value of the bond conventionally set equal to 1 (monetary unit). Since bonds are commonly quoted in terms of yields rather than prices, the formula (2) allows derivation of the yield-to-maturity curve

which tends to the asymptotic value $\varUpsilon=2k\theta/(\gamma+k+\lambda)$ as $T\to\infty,$ where $\gamma:=\sqrt{(k+\lambda)^{2}+2\sigma^{2}}.$

Thus the CIR process is characterized by the following properties: 1) short interest rates never become negative; 2) if the interest rate reaches the zero value (it may occur under the condition $2k\theta<\sigma^{2}$), it is immediately reflected in positive values; 3) the diffusion term in (1) increases when $r(t)$ increases; 4) the transition density of short rates is a noncentral chi-squared distribution, which converges to a gamma distribution as time grows towards infinity; 5) the trajectories describing the short rate dynamics cannot be explicitly derived, but exact simulation is still possible and bond prices are explicitly computable from it.

The remainder of the paper is organized as follows. Section 2 summarizes the existing literature on the CIR model and the related extensions. Section 3 describes the principal steps of the proposed CIR# model; Section 4 presents in more detail the numerical procedure and tests the goodness-of-fit of the new methodology to market data. Finally, Section 5 concludes.

The CIR model became very popular in finance among practitioners because it was perceived as an improvement on the Vasicek model, not allowing for negative rates and introducing a rate dependent volatility, as well as for its relatively handy implementation and analytical tractability. Other applications include stochastic volatility modelling in option pricing problems (see Orlando and Taglialatela (2017) [32], Heston (1993) [16]), or default intensities in credit risk (see Duffie (2005) [12]).

Despite the previously listed properties, the CIR model fails as a satisfactory calibration to market data since it depends on a small number of constant parameters, $k,\theta$ and $\sigma.$ As proved by Keller-Ressel and Steiner (2008) [22, Theorem 3.9 and Section 4.2] the yield-to-maturity curve of any time-homogenous, affine one-factor model is either normal (i.e., a strictly increasing function of $T-t$), humped (i.e., with one local maximum and no minimum on $]0,\infty[$) or inverse (i.e., a strictly decreasing function of $T-t$) (see Figure 1). For the CIR process, the yield is normal when $r(t)\leq k\theta/(\gamma-2(k+\lambda))$, while it is inverse when $r(t)\geq k\theta/(k+\lambda).$ For intermediate values the yield curve is humped.

However, Carmona and Tehranchi (2006), [9, Section 2.3.5] explained that: “Tweaking the parameters can produce yield curves with one hump or one dip (a local minimum), but it is very difficult (if not impossible) to calibrate the parameters so that the hump/dip sits where desired. There are not enough parameters to calibrate the models to account for observed features contained in the prices quoted on the markets” (see Figure 2).

Thus the need for more sophisticated models for an exact fit to the currently-observed yield curve, which could take into account multiple correlated sources of risk as well as shocks and/or structural changes of the market, led some years later to the development of extensions of the CIR model. Among the best known we mention: the Hull-White (1990) [17] model based on the idea of considering time-dependent coefficients; the Chen (1996) [10] three-factor model; the CIR++ model by Brigo & Mercurio (2001) [7] that considers short rates shifted by a deterministic function chosen to fit exactly the initial term structure of interest rates; the jump diffusion JCIR model (see Brigo & Mercurio (2006) [8]) and JCIR++ by Brigo & El-Bachir (2006)[6] where jumps are described by a time-homogeneous Poisson process; the CIR2 and CIR2++ two-factor models (see Brigo & Mercurio (2006) [8]). Very recently, Zhu (2014) [35], in order to incorporate the default clustering effects, proposed a CIR process with jumps modelled by a Hawkes process (which is a point process that has self-exciting property and the desired clustering effect), Moreno et al. (2015) [27] presented a cyclical square-root model, and Najafi et al. (2017) ([28], [29]) proposed some extensions of the CIR model where a mixed fractional Brownian motion applies to display the random part of the model.

Note that all the above cited extensions to CIR model preserve the positivity of interest rates, in some cases through reasonable restrictions on the parameters. But the financial crisis of 2008 and the ensuing quantitative easing policies brought down interest rates, as a consequence of reduced growth of developed economies, and accustomed markets to unprecedented negative interest regimes under the so called “new normal”. As observed in Engelen (2015) [14] and BIS (2015) [5]:

Therefore, the need for adjusting short term interest rate models for negative rates has become an additional characteristic that a “good” model should possess. It is worth noting that the main drawback of the Vasicek model [34], which allows for negative interest rates, is that the conditional volatility of changes in the interest rate is constant, independent on the level of it, and this may unrealistically affect the prices of bonds that can grow exponentially (see Rogers [33]). For this reason, the Vasicek model is unused by practitioners.

In the following we will illustrate our original approach, but first let us recap the main issues of the CIR model:

1. i.

   Negative interest rates are precluded;

2. ii.

   The diffusion term in (1) goes to zero when $r(t)$ is small (in contrast with market data);

3. iii.

   The instantaneous volatility $\sigma$ is constant (in real life $\sigma$ is calibrated continuously from market data);

4. iv.

   There are no jumps (e.g. caused by government fiscal and monetary policies, by release of corporate financial results, etc.);

5. v.

   Risk premia are linear with interest rates (false if credit worthiness of a counterparty and market volatility are considered);

6. vi.

   The change in interest rates depends only on the market risk.

The aim of the present work is to provide a new methodology that gives an answer to points i.- iv. by preserving the structure of the original CIR model (1) to describe the dynamics of spot interest rates observed in financial markets. For this purpose the first step is partitioning the available market data sample into sub-samples - not necessarily of the same size - in order to capture all the statistically significant changes of variance in real spot rates and consequently, to give an account of jumps (see Section 4.2). This should allow to overcome the critical issue pointed out in iv.. After that, to overcome challenges i.- ii., the real spot rates are properly translated to shift them away from zero or negative values and such that the diffusion term in (1) is not dampened by the proximity to zero but fully reflects the same level of volatility present on the market (see Section 4.3).

The second step consists in fitting an “optimal” - as explained in Section 4.5 - ARIMA model to each sub-sample of market data. To ensure that the residuals of the chosen “optimal” ARIMA model in each sub-sample look like Gaussian white noise, the Johnson’s transformation (Johnson (1949)[20]) is applied to the standardized residuals (see Section 4.4).

As a third step, the parameters $k,\theta,\sigma$ in (1) are calibrated to the (eventually) shifted market interest rates by estimating them for each sub-sample of available data, as explained in Section 4.4.1 (which allows to overcome the issue iii.). For this purpose, trajectories of the CIR process are simulated by a strong convergent discretization scheme, using the standardized residuals of the “optimal” ARIMA model selected for each sub-sample in place of realizations of a standard Brownian motion. As a result, exact CIR fitted values to real data are calculated and the computational cost of the numerical procedure is considerably reduced. Finally, the short-term interest rates estimated by the CIR model are shifted back and compared to real data. As a measure of goodness-of-fit to the available market data, we compute:

• 1.

  the statistics $R^{2}$ given by the following expression [23]

  $$R^{2}=1-\frac{\sum\limits_{h=1}^{m}(e_{h}-\overline{e})^{2}}{\sum\limits_{h=1}^{m}(r_{h}-\overline{r})^{2}},$$

  (3)

  where $e_{h}=r_{h}-\widehat{r}_{h}$ denotes the residual between the observed market interest rate $r_{h}$ and the corresponding fitted value $\widehat{r}_{h},$ evaluated on a data sample of size $m\geq 2.$ Furthermore, $\overline{e}$ and $\overline{r}$ denote the sample mean of $e_{h}$ and $r_{h}$, respectively;

• 2.

  the square root of the mean square error (RMSE)

  $$\varepsilon=\sqrt{\frac{1}{m}\sum_{h=1}^{m}{e^{2}_{h}}}.$$

  (4)

In this section we will address briefly the CIR# model’s progress on future interest rate forecasts from a window of observed market data. Ex-ante forecasts require a thorough analysis and will be extensively treated in a forthcoming research. It is worth noting that in this work we decided to impose the most challenging conditions by modelling the shortest part of the yield curve (e.g. the overnight rate) and using only a handful of number of observations. For instance, with monthly data we have found that $m=8$ observations are sufficient for a good calibration. Thus we start to consider a fixed size window of 8 real interest rates that is rolled through time, each month adding the new rate and taking off the oldest rate. The length of this window (8 months) is the historical period over which we forecast the next-month spot rate value. The numerical procedure described in Sections 4.3–4.5 has been applied to forecast future next-month interest rates based on monthly EUR interest rates with overnight maturity. The predicted curve is shown in Figure 9, compared with the real observed term structure.

It is evident that the predicted next-month spot rates computed by the CIR# model follow the market trend. Moreover, the values of $R^{2}_{CIR}$ and RMSE $\varepsilon,$ computed to measure the goodness-of-fit of forecast interest rates to real data, are respectively $0.8045$ and $0.1392$.

Several different extensions of the original model have been proposed to date, with the aim of overcoming the limitations of the CIR model: from one-factor models including time-varying coefficients or jump diffusions to multi-factor models. All these extensions preserve the positivity of interest rates but, in some cases, the analytical tractability of the basic model is violated. Our approach, instead, is based on a proper translation of interest rates such that the market volatility structure is maintained as well as the analytical tractability of the original CIR model. Thus the suggested CIR# model is quite powerful for the following reasons. First, all the improvements are obtained within the CIR framework in order to preserve the single-factor property and the analytical tractability of the original model. Second, market interest rates are properly translated away from zero and/or negative values. The market data sample is partitioned into sub-groups in order to capture all the statistically significant changes of variance in real spot rates and therefore gives an account of jumps. Third, we have introduced a new way of calibration of the CIR model parameters to actual data. The standard Brownian motion process in the random part of the model is replaced with normally distributed standardized residuals of “optimal” ARIMA models suitably chosen. As a result, exact CIR fitted values to real data are calculated and the computational cost of the numerical procedure is considerably reduced. Fourth, we have shown that the CIR# model is efficient and able to follow very closely the structure of market short-term interest rates (especially for short maturities that, notoriously, are very difficult to handle) and to predict future interest rates better then the original CIR model. As a measure of goodness-of-fit, we obtained high values of the statistics $R^{2}$ and small values of the square error $\varepsilon$ for each sub-group and the entire data sample. Future research will show the predictive power of the model by extending the dataset in terms of frequency and size.