Chapter 8 Mean-Reverting Processes and Term Structure Modeling.

Chapter 8 Mean-Reverting Processes and Term Structure Modeling

Short And Long Term Rates

As discussed in Chapters 3 and 4, the yield curve depicts varying spot rates over associated bond terms to maturity. Spot rates are interest rates on loans originated at time zero (e.g., now). The yield curve is typically constructed from the yields of benchmark highly liquid (where possible) default-free fixed income and/or zero coupon instruments.

Bond prices are affected by short-term (technically, instantaneous short rates) and long-term interest rates. Instantaneous rates, which are unobservable because they don't actually exist in reality, might be proxied by overnight rates.

Short And Long Term Rates (Continued)

We will focus on single factor models, where the entire yield curve and its related bond prices are driven by a single factor, being short-term interest rates. These short-term interest rates in turn will be modeled as particular stochastic processes. How one chooses a short-interest rate model is a matter of taste, and there is a variety of models to choose from. We will discuss a number of these models in this chapter.

More generally, understanding the stochastic processes that drive security prices, interest rates and other rates is essential to valuing and hedging the securities, their derivative securities and portfolios related to them. In this chapter, we analyze alternatives to Brownian motion processes for security and rate process, with a particular focus on interest rates and debt instruments.

Ornstein-Uhlenbeck Processes

Unlike returns for stocks, short-term interest rates have a generally accepted empirically verified tendency to drift towards some long-term mean rate µ. This long-term mean interest rate is not the same as the long-term interest rate, but the mean of short-term rates over an extended period of time. For example, when the short-term rate exceeds the long term mean rate ( > µ), the drift might be expected to be negative so that the short-term rate drifts down towards the long-term mean rate. We might say that interest rates are currently high in this scenario, and we expect for them to drop towards the long-term mean. When the short term rate is less than the long term mean rate ( < µ), the drift might be expected to be positive. When the short-term rate equals the long-term mean (or average) rate µ, the drift might be expected to be zero, though random changes might still occur

Ornstein-Uhlenbeck Processes (Continued)

Thus, the short-term rate has a tendency to revert to its long-term mean µ, whose value might be the value justified by economic fundamentals such as capital productivity, long-term monetary policy, etc.

The Ornstein-Uhlenbeck process is a form of Markov process in which the random variables indexed by time have a tendency to revert towards a given constant or long-term mean µ. Define the term 0 < < 1 to be a constant that reflects the speed of the mean-reverting adjustment for the instantaneous short term rate rt towards its constant long-term mean rate µ; that is, is the mean reversion factor, sometimes called a "pullback factor." This pullback factor is typically estimated or calibrated based on a statistical analysis of historical data. Let rdZt represent random shocks or disturbances to rt, where is standard Brownian motion and r is a constant.


The following defines the Ornstein-Uhlenbeck mean reverting process:

The solution to this stochastic differential equation is:

The Ornstein-Uhlenbeck Process applied to modeling short-term interest rates is often referred to as the Vasicek Model.


Conditional on the current instantaneous (short) rate r0, the expected instantaneous rate E[rt] at time t is:

The variance of the instantaneous short rate is

Illustration Of The Vasicek Model

Suppose that instantaneous short rates follows the Vasicek model with a long term mean interest rate of .05, instantaneous short term rate currently .03, pullback factor of .2, and an instantaneous standard deviation of the short rate of .01.

The stochastic differential equation for the short term rate is:

with current value

Illustration Of The Vasicek Model (Continued)

From the solution given earlier, we see that the value of the short term rate at any time t in the future follows the stochastic process:

The expected value of the short term rate at time t is

Observe that at time 0, the expected value of the short term rate is .03, which is the actual short term rate at time 0. This is simply because at time 0 (now), the short term rate is known. As the time t gets larger and larger, the expected value approaches .05. The effect of the initial value of the short term rate (.03) “wears off,” and the long term mean rate of .05 becomes dominant since this is the value towards which the short term interest rate reverts.

Illustration Of The Vasicek Model (Continued)

The variance of the short term rate at time t is

Looking back at the solution for the short term interest rate and referring to the last theorem in section 6.1.5, we see that for each fixed time t, the term is a random variable with a normal distribution with mean 0 and variance given above. Thus, the solution for at time t is a random variable with a normal distribution with mean and variance As the time t gets larger and larger, the normal distributions have the property that their means approach the value .05 and their variances approach the value .00025.

Pricing A Zero-Coupon Bond With Single Factor Interest Rate Models

Valuing a zero coupon bond with face value F in an environment with continuously discounted rates rs following a continuous stochastic process is accomplished with the following very general form:

This implies that the time zero bond value, B0, is the expected value of the time T payment F, where the instantaneous discount rate rs varies over time s. In effect, and in intuitive terms, at each instant, the face value F is discounted at rate rs over time period ds and these discount functions are multiplied together over all time periods from 0 to T. The result is a summation in the exponent, which in the continuous limit becomes an integral.

Pricing A Zero-Coupon Bond With Single Factor Interest Rate Models (Continued)

To obtain the pricing formula for a bond, analogous to how options were priced in chapter 7, one can use either a differential equation approach or a martingale approach. In both approaches, we will assume that the short term interest rate is described by a stochastic differential equation of the form: ,

where and are themselves stochastic processes. We will later consider particular choices for a and b.


In the differential equation approach, we assume we have a complete bond market so that any bond in the market can be replicated with a portfolio of two bonds and Both of these bonds are driven by the same short term interest rate model above. We then create a portfolio of the two bonds so that this portfolio is self-financing; that is:

After applying Itô’s Lemma and going through several steps of calculation, one arrives at:


Here, is the drift term and is the volatility term in the stochastic differential equation for a bond B:

where (from Itô’s Lemma):

and .


Since the expression for dP/P has no volatility term, in an arbitrage free market, dP/P must earn the riskless instantaneous short term rate Thus, we have:

After some algebraic manipulation, one finds that:


Regardless of which two bonds we chose to construct our replicating portfolios, we would always obtain the previous equality. This implies that for any bond B, the ratio will be the same. This ratio, denoted is called the risk premium or Sharpe ratio:

The expected instantaneous return on the bond is then:

This risk premium is common to all bonds in the market that are driven by the same interest rate model. The risk premium is a measure of the market price of risk. It arises from the fact that investors require additional return depending upon the riskiness of the bonds.


If one substitutes the expression for that we stated earlier in the equation above, one obtains the differential equation for the price B of the bond:

There is also the boundary condition that B(T,T)=F where F is the face value of the bond. This differential equation can be solved with the result: = is the original probability space.


One can also derive the solution using the martingale approach. Suppose that the original probability measure is some physical probability measure In the martingale approach to pricing an option, recall that the value of the option at time 0 is the expected value taken at time 0 of its discounted expiration value with respect to the risk-neutral equivalent probability measure . Similarly, the value of the long-term bond at time 0 is the expected value of the bond with respect to the risk-neutral equivalent probability measure is:


Compare the results for the price of the bond from the PDE versus the martingale approach. The comparison suggests that the Radon-Nikodym derivative from the Cameron-Martin-Girsanov Theorem to change from the probability space to the equivalent risk-neutral probability measure should be:

This will enable us to rewrite the PDE solution in the form:


Summary: The two approaches give equivalent pricing for the bond. In a manner analogous to option pricing, the discounted face value of the bond is a martingale with respect to the risk neutral probability space .

The Bond Pricing Formula For The Vasicek Model

The time zero price of a bond that matures at time T driven by interest rates following the Vasicek model and assuming a constant risk premium is:

where and Suppose the price above was obtained in an original

probability space with a constant risk premium with the short term interest following the Vasicek model:

The Bond Pricing Formula For The Vasicek Model (Continued)

Then, it can be shown that this is equivalent to expressing the bond price in the risk neutral probability space where the risk premium now becomes the long term mean becomes , and the short term interest rate now follows the Vasicek model:

where is also standard Brownian motion. This shows that in the case of a constant risk premium, converting to the risk neutral probability space has the effect of eliminating the risk premium while the long term mean is adjusted accordingly.

The Yield Curve

In Chapter 4, the market spot rate of interest, r0,T, implied by the bond's price is given in general by:

In the more specific Vasicek model discussed here, it can be determined from the bond pricing formula that: Note that as T , the spot rate r0,T will asymptotically approach the infinite long-rate As we have seen, the yield curve produced by the Vasicek model starts (intercepts) at the current instantaneous rate r0 and approaches the infinite long-rate

Illustration of Pricing a Bond Using the Vasicek Model

Assume that short term interest rates follow the Vasicek model with a pullback factor of λ = .4, a long term mean of μ = .06, a standard deviation of and a current interest rate of If the face value of a 4 year zero coupon bond is $1,000, find the current price of the bond. Assume a constant risk premium of .2.

Solution: We calculate that and

Illustration of Pricing a Bond Using the Vasicek Model (Continued)

The price of the bond is then: =

Also find the four year yield This is easily found to be:

The Problem with Vasicek Models

The Vasicek yield curve model has a number of desirable characteristics. The model captures the empirical tendency for interest rates to revert towards some sort of mean rate. The model is driven off short term interest rates, much as actual interest rates might be impacted by the Federal Funds rate, the "overnight" bank-to-bank controlled by the central bank (Fed). However, there are a number of problems with the Vasicek model in characterizing the behavior of the yield curve:

1. Empirical tests reveal that the Vasicek model does not characterize observed interest rate structures well.

The Problem with Vasicek Models (Continued)

2. The Vasicek model assumes only a single underlying risk factor when, in fact, there is significant evidence

that there may well be multiple factors. For example, sometimes the yield curve can "twist;" that is, long- and short-term rates can move in opposite directions. Multiple risk factors can often explain such "twisting." 3. The Vasicek model allows for the possibility of negative

interest rates, even for negative real interest rates, a phenomenon that we should expect to observe rarely, if at all.

The Problem with Vasicek Models (Continued)

Why work with an interest rate model that presents all of these difficulties?

As with most other financial models, we simply balance realism and ease of model building. The Vasicek model does capture some of the characteristics of a reasonable interest rate process and it is rather easy to work with, particularly in terms of parameter calibration. In addition, it is useful and sometimes very straightforward to adapt this framework into more realistic alternative depictions of interest rate processes.

Alternative Interest Rate Processes

The Merton ModelThe Cox, Ingersoll and Ross (CIR) Process

The Merton Model

The short-rate dynamics, pricing differential equation and bond pricing model produced by Merton [1973] are given by the next three equations. The short-rate dynamics are characterized by: where μ and σr are constants.

Substituting a=μ and b=σr into the general partial differential equation stated earlier, we obtain the partial differential equation governing the bond price for the Merton model:

The Merton Model (Continued)

Assuming a constant risk premium Θ, the solution to this differential equation for the bond price with face value F at time of maturity T is:

The corresponding term structure (equation for the yield curve) is:

The Merton Model (Continued)

Note :Here that a shift in the short-term rate r0 will result in a parallel shift in the yield curve; that is, the shift in r0,T will be the same for all T. Such a parallel shift is inconsistent with the empirical observation that short-term rates are more volatile than are long-term rates. Note also that the yield curve can never be monotonically increasing; it is either humped (when > r) or downward sloping. However, often when it is humped, the upward sloping portion of the hump can extend beyond the period being analyzed. Thus, the yield curve can be upward sloping over the "relevant" period.

The Cox, Ingersoll and Ross (CIR) Process

A simple "square root correction" in the Cox Ingersoll Ross Model (CIR, Cox, Ingersoll and Ross [1985]) eliminates the potential problem of negative interest rates in the mean reverting Vasicek model:

The price of a bond with face value F that matures in time T is: =where ' is the CIR risk premium, and: , ,

The Cox, Ingersoll and Ross (CIR) Process (Continued)

For the CIR Process, the market spot rate of interest, r0,T , implied by the bond's price is given by:

Note that as T , the spot rate r0,T will asymptotically approach the infinite long-rate r = 2( )/(+'+).

Numerical Illustration with the CIR Process

Suppose short term interest rates follow a CIR Process with pullback factor of .2, a long term mean interest rate of .04, an instantaneous standard deviation of .08, a current short rate of .02, and a risk premium of 0.What stochastic differential equation governs the short term

interest rate? Find the price of a 3 year zero coupon bond with a face value

of $5,000. Find its 3 year spot rate. What rate does the spot rate approach as the maturity date

gets arbitrarily large?

Numerical Illustration with the CIR Process (Continued)

The short term interest rate satisfies the stochastic differential equation:

To calculate the bond price, we first find:

Numerical Illustration with the CIR Process (Continued)

The bond price is: The bond’s 3 year spot rate is: As the maturity date gets arbitrarily large, the spot rate

approaches the limiting value:

Chapter 8 Mean-Reverting Processes and Term Structure Modeling.

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