Introduction

•The goal of this chapter is to describe the most common building blocks of short-term rate models. Selecting and rearranging these building blocks to create suitable models for the purpose at hand is the art of term structure modeling.

•This chapter begins with an extremely simple model with no drift and normally distributed rates. The next sections add and discuss the implications of alternate specifications of the drift: a constant drift, a time-deterministic shift, and a mean-reverting drift

Notations

•𝑑𝑟 is used to denote the change in the rate over a small time interval, 𝑑𝑡, measured in years;

•𝜎 denotes the annual basis-point volatility of rate changes;

•𝑑𝑤 denotes a normally distributed random variable with mean of  zero and standard   deviation of   𝑑𝑡. Note that 𝑑𝑤 is only a standard random normal when 𝑑𝑡  =  1.0; otherwise, 𝑑𝑤 already scales for time by applying the square root rule.

•The standard deviation of the change in the rate can be referred to as simply the standard deviation of the rate, for convenience.

Model 1:Normally Distributed Rates and No Drift

•The  particularly  simple  model  of  this  section  will  be  called  Model  1.  The  continuously compounded, instantaneous rate 𝑟_t is assumed to evolve according to the following equation:

𝑑𝑟 = 𝜎𝑑𝑤

As stated earlier, the quantity 𝑑𝑟 denotes the change in the rate over a small time interval, 𝑑𝑡, measured in years; 𝜎 denotes the annual basis-point volatility of rate changes; and 𝑑𝑤 denotes a normally distributed random variable with a mean of zero and a standard deviation of  𝑑𝑡.

For example, if the current value of the short-term rate is 6.18%, that volatility equals 113 basis points per year, and that the time interval under consideration is one month or  years. Mathematically, 𝑟₀  =  6.18%; 𝜎 =  1.13%; and 𝑑𝑡 = 1/12. A month passes and the random variable 𝑑𝑤, with its zero mean and its standard deviation of 1/12 or 0.2887, happens to take on a value of 0.15. With these values, the change in the short-term rate is

𝑑𝑟 =  1.13%  × 0.15 = 0.17%  or 17 basis points.

Since the short-term rate started at 6.18%, the short-term rate after a month is 6.35%

•Since the expected value of 𝑑𝑤 is zero, the expected change in the rate, or the drift, is zero. Since the standard deviation of 𝑑𝑤 is  𝑑𝑡, the standard deviation of the change in the rate is 𝜎 𝑑𝑡. For our example, 𝜎  𝑑𝑡 = 1.13% × 1/12 =  0.326% or 32.6 basis points per month.

•A rate tree may be used to approximate the process of Model 1. A tree over dates 0 to 2 takes this form

•In the case of the numerical example, substituting the sample values into the   tree gives the following:

Model 1 Expected Value and Variance

•The change in the interest rate in the up-state is 𝜎  𝑑𝑡 and the change in the down-state is —𝜎  𝑑𝑡. Therefore, with the probabilities given in the tree, the expected change in the rate, often denoted 𝐸[𝑑𝑟], is

𝐸 𝑑𝑟  = 0.5 × 𝜎  𝑑𝑡 + 0.5 × —𝜎  𝑑𝑡 = 0

•The variance of the rate, often denoted 𝑉 [𝑑𝑟], from date 0 to date 1 is computed as follows:

Since the variance is 𝜎²𝑑𝑡, the standard deviation, the square root of the variance, is 𝜎  𝑑𝑡.

Hence the drift and volatility implied by the tree match the drift and volatility of the interest rate process of Model 1. The process and the tree are not identical because the random variable in the process, having a normal distribution, can take on any value while a single step in the tree leads to only two possible values. In the example, when 𝑑𝑤 takes on a value of 0.15, the short rate changes from 6.18% to 6.35%. In the tree, however, the only two possible rates are 6.506% and 5.854%. Nevertheless, after a sufficient number of time steps the branches of the tree used to approximate the process will be numerous enough to approximate a normal distribution.

Model 1 Negative Interest Rate Problem

•This figure shows the distribution of short rates after one year, or the terminal distribution after one year, in Model 1 with 𝑟₀ = 6.18% and 𝜎 = 1.13%. The tick marks on the horizontal axis   are one standard deviation apart from one another. Models in which the terminal distribution of   interest rates has a normal distribution, like Model 1, are called normal or Gaussian models.

•One problem with these models is that the short-term rate can become negative. A negative   short-term rate does not make much economic sense because people would never lend money at   a negative rate when they can hold cash and earn a zero rate instead. The distribution in this   figure, drawn to encompass three standard deviations above and below the mean, shows that   over a horizon of one year the interest rate process will almost certainly not exhibit negative   interest rates.

•But the probability of negative short-term rates in the process, increases with the horizon. Over 10 years, for example, the standard deviation of the terminal distribution in the numerical example is 1.13% ×  10 or 3.573%. Starting with a short-term rate of 6.18%, a random negative shock of only ^6.18%/_3.573% or 1.73 standard deviations would push rates below zero.

Dealing With Negative Interest Rates in Model 1

•The extent to which the possibility of negative rates makes a model unusable depends on the application. For securities whose value depends mostly on the average path of the interest rate, like coupon bonds, the possibility of  negative rates typically does not rule out an otherwise   desirable model. For securities that are asymmetrically sensitive to the probability of low interest   rates, however, using a normal model could be dangerous. Consider the extreme example of a 10-year call option to buy a long-term coupon bond at a yield of 0%. Model 1 would assign that   option much too high a value because the model assigns too much probability to negative rates. The challenge of negative rates for term structure models is much more acute, of course, when   the current level of rates is low. Two solutions can be considered

1)Changing the distribution of  interest rates – For example, lognormally distributed rates cannot become negative. However, building a model around a probability distribution that rules  out  negative  rates  or  makes  them  less  likely  may  result  in  volatilities  that  are unacceptable for the purpose at hand.

2)Another popular method of ruling out negative rates is to construct rate trees with whatever distribution is desired, and then simply set all negative rates to zero. In this methodology, rates in the original tree are called the shadow rates of interest while the rates in the adjusted tree could be called the observed rates of interest. When the observed rate hits zero, it can remain there for a while until the shadow rate crosses back to a positive rate. The economic justification for this framework is that the observed interest rate should be constrained to be positive only because investors have the alternative of investing in cash. But the shadow rate, the result of aggregate savings, investment, and consumption decisions, may very well be negative. The probability distribution of the observed interest rate is not the same as that of the originally postulated shadow rate. The change, however, is localized around zero and negative rates. By contrast, changing the form of  the probability distribution changes dynamics across the entire range of rates.

Model 2:Drift and Risk Premium

•The term structures implied by Model 1 always look relatively flat for early terms and then   downward sloping. But from FRM Part 1, and the previous chapters of Market Risk which have   been covered, it is known that the term structure tends to slope upward and that this behaviour might be explained by the existence of a risk premium. The model of this section, to be called   Model 2, adds a drift to Model 1, interpreted as a risk premium, in order to obtain a richer model   in an economically coherent way.

•The dynamics of the risk-neutral process in Model 2 are written as

𝑑𝑟 = 𝜆𝑑𝑡 + 𝜎𝑑𝑤

This differs from that of Model 1 by adding a drift to the short-term rate equal to 𝜆𝑑𝑡. For this section, consider the values 𝑟₀  = 5.138%, 𝜆 = 0.229%, and  = 1.10%. If the realization of the random variable 𝑑𝑤 is again 0.15 over a month, then the change in rate is

Starting from 5.138%, the new rate is 5.322%.

•The drift of the rate is 0.229% ×  ¹  or 1.9 basis points per month, and the standard deviation is 1.10%  ×  ^1/12  or 31.75 basis points per month. As discussed earlier, the drift in the risk-neutral process is a combination of the true expected change in the interest rate and of a risk premium. A drift of 1.9 basis points per month may arise because the market expects the short-term rate to increase by 1.9 basis points a month, because the short-term rate is expected to increase by one basis point with a risk premium of 0.9 basis points, or because the short-term rate is expected to fall by 0.1 basis points with a risk premium of two basis points.

•The tree approximating this model is shown here. It is easy to verify that the drift and standard deviation of the tree match those of the process of Model 2.

•The constant drift, by raising the mean of the terminal distribution, makes it less likely that the   risk-neutral process will exhibit negative rates.

•Moving from Model 1 with zero drift to Model 2 with a constant drift does not qualitatively   change the term structure of volatility, the magnitude of convexity effects, or the parallel-shift   nature of the model.

•Models 1 and 2 would be called equilibrium models because no effort has been made to match   the initial term structure closely.

•Model 2 can be generalized to get the class of arbitrage-free models.

The Ho-Lee Model: Time Dependent Drift

•The dynamics of the risk-neutral process in the Ho-Lee model are written as

𝑑𝑟 = 𝜆_t𝑑𝑡 + 𝜎𝑑𝑤

•In contrast to Model 2, the drift here depends on time. In other words, the drift of the process may change from date to date. It might be an annualized drift of —20 basis points over the first month, of 20 basis points over the second month, and so on. A drift that varies with time is called a time-dependent drift. Just as with a constant drift, the time-dependent drift over each time period represents some combination of the risk premium and of expected changes in the short-term rate.

•The flexibility of the Ho-Lee model is easily seen from its corresponding tree

•The free parameters 𝜆₁ and 𝜆₂ may be used to match the prices of  securities with fixed cash flows. The procedure may be described as follows.

1. With 𝑑𝑡 =  ^1/12 , set 𝑟₀ equal to the one-month rate.

2. Then find 𝜆1 such that the model produces a two-month spot rate equal to that in the market.

3. Then find 𝜆2 such that the model produces a three-month spot rate equal to that in the market.

4. Continue in this fashion until the tree ends.

The procedure is very much like that used to construct the trees in the previous chapters. The only difference is that earlier, the probabilities were adjusted to match the spot rate curve while rates are adjusted here. The two procedures are equivalent if the step size is small enough.

•The rate curves resulting from this model match all the rates that are input into the model. Just as   adding a constant drift to Model 1 to obtain Model 2 does not affect the shape of the term   structure of volatility nor the parallel-shift characteristic of the model, adding a time-dependent drift does not change these features either.

•To summarize, the most straightforward tree representation of this example takes the following form:

Arbitrage Free Models

•The desirability of matching market prices is the central issue in deciding between arbitrage-free and   equilibrium models. The choice depends on the purpose of building the model.

1)One important use of arbitrage-free models is for quoting the prices of securities that are not actively traded based on the prices of more liquid securities. A customer might ask a swap desk to quote a rate on a swap to a particular date, say three years and four months away, while liquid market prices might be observed only for three- and four-year swaps, or sometimes only for two- and five- year swaps. In this situation, the swap desk may price the odd-maturity swap using an arbitrage-free model essentially as a means of interpolating between observed market prices.

2)Another important use of arbitrage-free models is to value and hedge derivative securities for the purpose of making markets or for proprietary trading. For these purposes, many practitioners wish to assume that some set of underlying securities is priced fairly. For example, when trading an option on a 10-year bond, many practitioners assume that the 10-year bond is itself priced fairly. (An analysis of the fairness of the bond can always be done separately.) Since arbitrage-free models match the prices of many traded securities by construction, these models are ideal for the purpose of pricing derivatives given the prices of underlying securities.

3)  Interpolating by means of arbitrage-free models may very well be superior to other curve-fitting methods, from linear interpolation to more sophisticated approaches. The potential superiority of arbitrage-free models arises from their being based on economic and financial reasoning. In an arbitrage-free model, the expectations and risk premium built into neighbouring swap rates and the convexity implied by the model’s volatility assumptions are used to compute, for example, the three-year and four-month swap rate. In a purely mathematical curve fitting technique, by contrast, the chosen functional form heavily determines the intermediate swap rate. Selecting linear or quadratic interpolation, for example, results in intermediate swap rates with no obvious economic or financial justification. This potential superiority of arbitrage-free models depends crucially on the validity of  the assumptions built into the models. A poor volatility assumption, for example, resulting in a poor estimate of the effect of convexity, might make an arbitrage-free model perform worse than a less financially sophisticated technique.

Desirability Of Fitting To The Term Structure

•If a model matches market prices, this does not necessarily imply that it provides fair values and   accurate hedges for derivative securities. The argument for fitting models to market prices is that a   good deal of information about the future behaviour of interest rates is incorporated into market   prices, and, therefore, a model fitted to those prices captures that interest rate behaviour. While   this is a perfectly reasonable argument, two warnings are appropriate.

1)First, a mediocre or bad model cannot be rescued by calibrating it to match market prices. If, for example, the parallel shift assumption is not a good enough description of reality for the application at hand, adding a time-dependent drift to a parallel shift model so as to match a set of market prices will not make the model any more suitable for that application.

2)Second, the argument for fitting to market prices assumes that those market prices are fair in the context of the model. In many situations, however, particular securities, particular classes of securities, or particular maturity ranges of securities have been distorted due to supply and demand imbalances, taxes, liquidity differences, and other factors unrelated to interest rate models. In these cases, fitting to market prices will make a model worse by attributing these outside factors to the interest rate process.

•Another way to express the problem of fitting the drift to the term structure is to recognize that the drift   of a risk-neutral process arises only from expectations and risk premium. A model that assumes one drift   from years 15 to 16 and another drift from years 16 to 17 implicitly assumes one of two things. First, the   expectation today of the one-year rate in 15 years differs from the expectation today of the one-year rate   in 16 years. Second, the risk premium in 15 years differs in a particular way from the risk premium in 16   years. Since neither of these assumptions is particularly plausible, a fitted drift that changes dramatically   will be erroneous.

•If the purpose of a model is to value bonds or swaps relative to one another, then taking a large number   of bond or swap prices as given is clearly inappropriate: arbitrage-free models, by construction, conclude   that all of these bond or swap prices are fair relative to one another. Investors wanting to choose among   securities, market makers looking to pick up value by strategically selecting hedging securities, or traders   looking to profit from temporary mis-pricings must, therefore, rely on equilibrium models.

•It should be noted that, in practice, there need not be a clear line between the two approaches. A model   might posit a deterministic drift for a few years to reflect relatively short-term interest rate forecasts and posit a constant drift from then on. The proper blending of the arbitrage-free and equilibrium approaches is an important part of the art of term structure modelling.

The Vasicek Model: Mean Reversion

•Assuming that the economy tends toward some equilibrium based on such fundamental factors as   the productivity of  capital, long-term monetary policy, and so on, short-term rates will be   characterized by mean reversion. When the short-term rate is above its long-run equilibrium   value, the drift is negative, driving the rate down toward this long-run value. When the rate is   below its equilibrium value, the drift is positive, driving the rate up toward this value. In addition   to being a reasonable assumption about short rates, mean reversion enables a model to capture   several features of term structure behaviour in an economically intuitive way.

•The risk-neutral dynamics of the Vasicek model are written as

𝑑𝑟 = 𝑘(𝜃 — 𝑟) 𝑑𝑡 + 𝜎𝑑𝑤

•The constant 𝜃 denotes the long-run value or central tendency of the short-term rate in the risk-   neutral process and the positive constant k denotes the speed of mean reversion. The greater the   difference between 𝑟 and 𝜃, the greater the expected change in the short-term rate toward 𝜃.

•Since this process is the risk-neutral process, the drift combines both interest rate expectations   and risk premium. Furthermore, market prices do not depend on how the risk-neutral drift is   divided between the two. Nevertheless, in order to understand whether or not the parameters of a   model make sense, it is useful to make assumptions sufficient to separate the drift and the risk   premium. Assuming, for example, that the true interest rate process exhibits mean reversion to a   long-term value 𝑟_∞ and, as assumed previously, that the risk premium enters into the risk-neutral   process as a constant drift, the Vasicek model takes the following form:

•Note that very many combinations of 𝑟_∞ and 𝜆 give the same 𝜃 and the same market prices   through this risk-neutral process.

•For example, let 𝑘 = 0.025, 𝜎 = 126 basis points per year, 𝑟_∞ = 6.179% , and 𝜆 = .229%.   Then, 𝜃 = 15.339%. With these parameters, the process says that over the next month the   expected change in the short rate is

or 2.13 basis points. The volatility over the next month is 126 ×  ^1/12  or 36.4 basis points.

•Representing this process with a tree is not quite so straightforward as the simpler processes   described previously because the most obvious representation leads to a nonrecombining tree.   Over the first-time step,

•To extend the tree from date 1 to date 2, start from the up state of 5.5060%. The tree branching from there is

while the tree branching from the date 1 down-state of 4.7786% is

•This tree does not recombine since the drift increases with the difference between the short rate   and 𝜃. Since 4.7786% is further from 𝜃 than 5.5060%, the drift from 4.7786% is greater than   the drift from 5.5060%. In this model, the volatility component of an up move followed by a   down move does perfectly cancel the volatility component of a down move followed by an up   move. But since the drift from 4.7786% is greater, the move up from 4.7786% produces a   larger short-term rate than a move down from 5.5060%.

•There are many ways to represent the Vasicek model with a recombining tree. One method is   presented here, but it is beyond the scope of this book to discuss the numerical efficiency of the various possibilities.

•The first time step of the tree may be taken as shown previously:

•Next, fix the centre node of the tree on date 2. Since the expected perturbation due to volatility   over each time step is zero, the drift alone determines the expected value of the process after each   time step. After the first time step, the expected value is

•This tree does not recombine since the drift increases with the difference between the short rate   and 𝜃. Since 4.7786% is further from 𝜃 than 5.5060%, the drift from 4.7786% is greater than   the drift from 5.5060%. In this model, the volatility component of an up move followed by a   down move does perfectly cancel the volatility component of a down move followed by an up   move. But since the drift from 4.7786% is greater, the move up from 4.7786% produces a   larger short-term rate than a move down from 5.5060%.

•There are many ways to represent the Vasicek model with a recombining tree. One method is   presented here, but it is beyond the scope of this book to discuss the numerical efficiency of the various possibilities.

•The first time step of the tree may be taken as shown previously:

•Next, fix the centre node of the tree on date 2. Since the expected perturbation due to volatility   over each time step is zero, the drift alone determines the expected value of the process after each   time step. After the first time step, the expected value is

5.121% + .025 15.339% — 5.121%  × 1/12 = 5.1423%

•After the second time step, the expected value is

5.1423% + .025 15.339% — 5.1423%  × 1/12 = 5.1635%

•Take this value as the center node on date 2 of the recombining tree. The parts of the tree to be solved for, namely, the missing probabilities and interest rate values, are given variable names.

•According to the process and the parameter values set in this section, the expected rate and standard deviation of the rate from 5.5060% are, respectively,

5.5060 + .025 15.339% — 5.5060%  ×  1/12 = 5.5265%

And

For the recombining tree to match this expectation and standard deviation, it must be the case that

Solving the above equations gives 𝑟^uu = 5.8909% and 𝑝  = 0.4990

•The same procedure may be followed to compute 𝑟^dd and 𝑞. The expected rate from 4.7786% is

And

For the recombining tree to match this expectation and standard deviation, it must be the case that

Solving the above equations gives 𝑟^dd = 4.4361% and 𝑞 = 0.5011.

•Putting the results from the up- and down states together, a recombining tree approximating the process with our parameters is

•If the tree extends into a third period, the entire process repeats iteratively.

•This figure illustrates the impact of mean reversion on the terminal, risk-neutral distributions of the short rate at different horizons. The expectation or mean of the short-term rate as a function of horizon gradually rises from its current value of 5.121% toward its limiting value of 𝜃 = 15.339%. Because the mean-reverting parameter 𝑘 = 0.025 is relatively small, the horizon expectation rises very slowly toward 15.339%. While mathematically beyond the scope of this book, it can be shown that the distance between the current value of a factor and its goal decays exponentially at the mean-reverting rate. Since the interest rate is currently 15.339% — 5.121% or 10.218% away from its goal, the distance between the expected rate at a 10-year horizon and the goal is  10.2180%  × 𝑒^–0.025×10 = 7.9578%

Therefore, the expectation of the rate in 10 years is

15.3390% — 7.9578% or 7.3812%.

For completeness, the expectation of the rate in the Vasicek model after 𝑇 years is 𝑟₀𝑒^–kt + 𝜃(1 — 𝑒^–kt)

In words, the expectation is a weighted average of  the current short rate and its long-run value, where the weight on the current short rate decays exponentially at a speed determined by the mean-reverting parameter.

The Vasicek Model-Half Life

•The mean-reverting parameter is not a particularly intuitive way of describing how long it takes a   factor to revert to its long-term goal. A more intuitive quantity is the factor’s half-life, defined as   the time it takes the factor to progress half the distance between the starting rate and mean-   reverting level. In the example of this section, the half-life of the interest rate.r, is given by the   following equation:

(15.339% — 5.121%)𝑒^–0.025c = 1/₂ (15.339% — 5.121%)

Solving the above gives

where 𝑙𝑛(·) is the natural logarithm function. In words, the interest rate factor takes 27.73 years to cover half the distance between its starting value and its goal. This can be seen visually in the last figure where the expected rate 30 years from now is about halfway between its current value and 𝜃. Larger mean-reverting parameters produce shorter half lives.