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Introduction and General Theories of Interest Rates

Consider the case when a bond has been issued. In the transaction, one party has paid a principal amount/ initial price \(P\) to purchase the bond. The counterparty to the transaction is expected to pay it back according to some period of payment decided between the two parties, although the structure of repayments can take various forms. But the counterparty has to pay an additional overhead over \(P\) to account for factores like:

This overhead charged over the amount \(P\) is called the interest. This is pre-determined either by the parties involved or some governing agency(like a bank).

Our interest in this presentation is to give a description of the Cox-Ingersoll-Ross(CIR) model, it’s various characteristics, outline its usefulness and address some shortcomings for the same.

General Theories of Interst Rates

First we need to address some of the basic developments and concepts in the lead-up to the CIR model. My source for all this is Prof. Andrew Cairns excellent book on the subject, Interst Rate Models\(^1\). I will at this time give a brief description of the 4 theories mentioned in the beginning sections of the book. This might lay some important groundwork on which to place the CIR model as we come upon it.

  1. Expectations Theory

The expected return of a one-year bond, from time \(S\) is conjectured to be equal to the annualised one-year forward rate of interest from the period of \(S\) to \(S+1\). This can be mathematically represented as: \[e^{F(0,S,S+1)} = E[e^{R(S,S+1)}|\mathcal{F} _0]\]

Where \(\mathcal{F}_t\), in general, refers to the information available up until time \(t\).

Generally, this approach is characterized by the following propositions:

  1. the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or
  2. the expected rate of return over the next holding period is the same for bonds of all maturities.\(^0\)

But the above form is not ideal. The details are beyond the scope of this presentations. But what can be said is that, by the above equation, it would seem that consecutive returns would seem to be uncorrelated, which is highly unlikely\(^1\). And this is the assumption in the original paper by Cox,Ross & Ingersoll.

  1. Liquidity Preference Theory

This theory basically reveals the motives of potential investors. That is to say, investors would usually prefer short-term investments, without having to hold their capital in for too long.Also, the prices of shorter term bonds tend to be less volatile than long term bonds.

Also, worth mentioning is that investors will partake in higher volatility bonds if, the expected returns are just as likely to be high. This is often referred to as the risk premium. And it is a premium investors are likely to indulge in to offset higher risk, because of higher volatilities.

  1. Market Segmentation Theory

This theory asserts that bonds of different maturites trade in separate and distinct markets. That is markets do not tend to mix and that investors have clear demarcations for their investments.

  1. Arbitrage-Free Pricing Theory

And lastly, I turn my attention to Arbitrage-Free pricing theory which is the reason we can do such analyses consistently. The reason I wished to introduce this section of the General Theories of Interest, was because of an intuitive bit that Prof. Cairns put in his book; Interest Rate Models: An Introduction.

Under this approach of Arbitrage-Free Pricing we can usually decompose forward rates into 3 components:

  1. Expected future risk-free rate of interest, \(r(t)\)
  2. An adjustment for market price of risk \(^*\).
  3. Convexity adjustment to reflect \(E(e^X)\geqslant e^{E(X)}\)

With the forward rate being written as the sum of the three components:

\[f(0,T) = \mu + (r(0) - \mu)e^{-\alpha T}-\lambda\sigma(1-e^{-\alpha T})/\alpha\]

I thought that was quite interesting.

CIR Model & Process

A proper introduction into the CIR Model would be to define it as an Equilibrium model. The basic framework of such a model is to factor in how changes in the economy will affect the term structure of interest rates. This obviously entails building a stochatic equation for the evolution of the interest rate or the risk-free rate \(r_t\). The Fundamental theorm of Asset Pricing is then applied on the model to derive a set of theoretical bond prices. These bond prices are evoled to be arbitrage-free.

The CIR Model is given by \[dr_t=\alpha(\mu-r_t)dt + \sigma\sqrt{r_t}dW_t\] This was developed by J. C. Cox, J. E. Ingersoll and S. A. Ross is an equilibrium approach to interest rate estimation. This was the first improvement from the Vasicek model i.e. the first tractable model that kept rates of interest positive. The characteristics of this model are:

  1. An autoregressive process that pulls the rate to the long-term mean.
  2. Bounces back from negative i.e \(P(\forall{t}>0:X_t>0) = 1\), called the Feller condition.\(^5\)
  3. Absolute variance of \(r_t\) increases as \(r_t\) itself increases.
  4. There is a steady state distribution of \(r(t)\).
I’ve used a package in R called sde to model what CIR Interest rates will look like. These are three simulations using the funciton sde.sim in the aforementioned package. All properties mentioned above regarding the dispersion of the interest rate can be observed in the figure.
Three simulations of interst rates by the CIR process, using R and the *sde* package.

Three simulations of interst rates by the CIR process, using R and the sde package.

With these pre-requisites stated, lets look at the solution of the SDE.

For ease of comprehension, I’m going to change the parametrization of the above model to\(^3\): \[dX_t=(\theta_1-\theta_2X_t)dt + \theta_3\sqrt{X_t}dW_t\] This is with reference to the material in the textbook Simulation and Inference for Stochastic Differential Equations by Stefano M. Iacus. The nomenclature should help us follow the brief solution I am going to present.

The SDE has the explicit solution as: \[X_t=(X_0-\frac{\theta_1}{\theta_2})e^{-\theta_2t} + \theta_3e^{-\theta_2t}\int_{o}^te^{\theta_2u}\sqrt{X_u}dW_u\] The condition for positive rate or bouncing from the negative is accomplished once the condition \(2\theta_1>\theta_3^2\) is satisfied. This process is strictly positive, else rates can reach zero which is not an ideal scenario in finance and is called the Feller condition.

Next looking at the distribution of the \(X_t\)’s the evaluaiton can be made that it follows a \(\chi^2\) distribution, subject to some important assumptions in the parameters. The explicit form of the probabilty density is given by: \[p_\theta(t,y|x_0) = ce^{-(u+v)}(\frac{v}{u})^{q/2} I_q(2\sqrt{uv})\] where \(u = cx_0e^{-\theta_2t}, v = cy, q = 2\theta_1/\theta^3-1\) and \(I_q(.)\) is the modified first kind order for \(q\) Bessel function.

Though the \(\chi^2\) distribution is well known, in this instance, the behavior of the distribution shows some peculiar properties. Let’s take a closer look at these:

  1. As \(t\rightarrow0\) the distribution skews to the left and leaves fat tails on the right.
  2. Conversely, as \(t\rightarrow\infty\) the skewness reverses and we get fat tails on the left.
  3. With referrence to the initial model equation if \(\alpha\rightarrow\infty\), the mean goes to \(\mu\) and variance goes to \(0\)
  4. And as \(\alpha\rightarrow0\), the mean goes to the current interest rate \(r_t\) and variance goes to \(\sigma^2r_t(s-t)\) for time \(s>t\).

Simulated here is a \(\chi^2\) distribution using the sde package and the rcCIR function for a 10000 samples from the probability distribution adjusted to the \(\chi^2\) distribution. Also, on the right is the is the distribution from running a 1000 simulations of the diffusion of the interest rate and plotting its distribution. The 2 shapes seem to coincide confirming the nature of distribution of interest rates.

Chi-squared Distribution for CIR model

Chi-squared Distribution for CIR model

The next two graphs show the difference in the shape of the denisites when the time increment is modified. This characterisitc was previously mentioned, but a pictorial representation of it would help us in grasping exactly what is going on.

Now would be a good to mention that, the shape of the distribution will reflect a Gamma distribution for large time steps, which is what we observe on the left. And the regular \(\chi^2\) dsitribution ofr smaller time steps, which is also confirmed in the original paper by Cox-Ingersoll-Ross\(^6\).

Numerical Simulation of the CIR Process

In this section we will be looking at the Euler-Maruyama(EM) scheme for the solution of the CIR process. Cutting right to the chase, the scheme looks like so:

\[\hat{X}^n_{t_{i+1}}=\hat{X}^n_{t_i}+\frac{T}{n}(\theta_1-\theta_2\hat{X^n_{t_i}})+\sigma\sqrt{\hat{X^n_{t_i}}}(W_{t_{i+1}}-W_{t_i})\] The EM scheme has one drawback. For example, when the values of \(\hat{X}{_{t_i}}\) move down to zero, at the \((i+1)^{th}\), the algorithm will follow along the tangent of the curve according to the deterministic part of the equation, but because the random component has an equal probablity of being positive or negative, if the step-size is large enough, the approximation becomes negative while the the actual solution remains positive\(^7\). Or in more rigorous terms, the diffusion coeffiecient is not locally Lipschitz\(^7\).

CIR Model Limitations

The CIR Model which is an extension of the Vasicek model for modelling interest rates, while it has its fair share of advantages, where it fails is in its calibration to actual market data\(^2\). This can be observed in the distribution of the interest rates. Observing its density plot, one observes fat tails, and its obvious skewness. This left-hand constraint means that the CIR is less likely to produce small \(r_t\). In comparison, as an after thought, the Vasicek model produces a symmetric distribution, and tends to overestimates prices.

As discussed in the previous section, we’ve seen that the CIR process is not locally Lipschitz.

But, it was these shortcomings that later produced the extensions of the CIR model. Namely,

  1. The Hull-White model in 1990, considering time-dependent coefficients.
  2. The Chen three-factor model(1996).
  3. The HJM model.
  4. The Reciprocal CIR.

I’d like to briefly mention the Reciprocal model\(^8\), because it is a type of time-changed CIR model, which is just as popular as the Heston model, because it has certain advantages over the CIR model.

Assuming that the Feller condition (\(0<2k\theta<\sigma^2\)) is satisfied and taking \(k(x)=1/x\) i.e. we apply Ito’s formula over \(Y_t=1/X_t\), we obtain the SDE\(^6\) \[Y_t=y + \int_0^t\tilde{\kappa}(\tilde{\theta}-Y_s)Y_sds - \int^t_0\sigma Y_s^{3/2}dB_s\] where \(Y_0=y=1/x, \tilde{\kappa}=\kappa/(\kappa\theta-\sigma^2)\) and \(\tilde{\theta}= \kappa\theta-\sigma^2\). This SDE exhibits the usual quadratic drift and the “3/2 volatility”. Claimed advantages of this particular extension is. The nomenclature of the parameters is changed to keep consistency with the undelying literature:

  1. It behaves as a Heston model under different parameters, incorporating the essence of drift volatility.
  2. Provides more accurate results than the standard CIR model because of its non-affine nature.
  3. Non-affine models seem to be more consistent with real world data.

Referrences

[1] Interest Rate Models: An Introduciton, Andrew J.G. Cairns

[2] A Fast and Exact Simulation for CIR Process, Anqi Shao, UNIVERSITY OF FLORIDA, 2012

[3] Simulation and Inference for Stochastic Differential Equations[pg 64], Stefano M. Iacus, 2008 Springer Science+Business Media, LLC

[4] CIR Modeling of Interest Rates, Zan Miao, Linnaeus University, Sweden, 2018

[5] A Theory of the Term Structure of Interest Rates, John C. Cox, Jonathan E. Ingersoll, Jr., Stephen A. Ross, Econometrica, Vol. 53, No. 2 (Mar., 1985), pp. 385-407

[6] STRONG CONVERGENCE RATES FOR COX-INGERSOLL-ROSS PROCESSES - FULL PARAMETER RANGE,MARIO HEFTER & ANDRE HERZWURM, arXiv:1608.00410v1 [math.NA] 1 Aug 2016

[7] TIME-CHANGED CIR DEFAULT INTENSITIES WITH TWO-SIDED MEAN-REVERTING JUMPS, Rafael Mendoza-Arriaga, Vadim Linetsky, arXiv:1403.5402v1 [q-fin.PR] 21 Mar 2014, The Annals of Applied Probability, 2014, Vol. 24, No. 2, 811-856