Time-dependent Volatility: Model 3

•Just as a time-dependent drift may be used to fit many bond or swap rates, a time dependent  volatility function may be used to fit many option prices. A particularly simple model with a  time-dependent volatility function might be written as follows:

𝑑𝑟 = 𝜆 𝑡 𝑑𝑡 + 𝜎 𝑡 𝑑𝑤

•Unlike the models presented in the last chapter, the volatility of the short rate in the above  equation depends on time. If, for example, the function 𝜎(𝑡) were such that 𝜎(1) = 1.26%  and 𝜎(2) = 1.20%, then the volatility of  the short rate in one year is 126 basis points per  year while the volatility of the short rate in two years is 120 basis points per year.

•To illustrate the features of time-dependent volatility, consider the following special case of the  above equation that will be called Model 3:

𝑑𝑟 = 𝜆 𝑡 𝑑𝑡 + 𝜎𝑒^–αt𝑑𝑤

Here, the volatility of the short rate starts at the constant 𝜎 and then exponentially declines to  zero. Volatility could have easily been designed to decline to another constant instead of zero,  but Model 3 serves its pedagogical purpose well enough.

•Setting 𝜎 = 126 basis points and 𝛼 = 0.025, this  figure shows the standard deviation of  the  terminal distribution of the short rate at various  horizons. The standard deviation rises rapidly with  horizon at first but then rises more slowly.

•The behavior of  standard deviation as a function  of horizon in this figure resembles the impact of  mean reversion on horizon standard deviation. In  fact, setting the initial volatility and decay rate in  Model 3 equal to the volatility and mean reversion rate of the numerical example of the Vasicek model, the standard deviations of the terminal  distributions from the two models turn out to be identical. Furthermore, if the time-dependent  drift in Model 3 matches the average path of  rates in the numerical example of  the Vasicek  model, then the two models produce exactly the same terminal distributions.

•While these parameterizations of  the Vasicek Model of  the previous chapter and Model 3 of  this chapter give equivalent terminal distributions, the models remain very different in other  ways. As is the case for any model without mean reversion, Model 3 is a parallel shift model.  Also, the term structure of volatility in Model 3 is flat. Since the volatility in Model 3 changes  over time, the term structure of  volatility is flat at levels that change over time, but it is still  always flat.

•Deterministic volatility functions are popular, particularly among market makers in interest rate  options. Consider the example of  caplets. At expiration, a caplet pays the difference between  the short rate and a strike, if positive, on some notional amount. Furthermore, the value of a  caplet depends on the distribution of the short rate at the caplet’s expiration. Therefore, the  flexibility of the deterministic functions 𝜆 𝑡   and 𝜎 (𝑡) may be used to match the market prices  of caplets expiring on many different dates.

Effectiveness of Time-Dependent Volatility

•The arguments for and against using time-dependent volatility are similar to the arguments for and  against using a time dependent drift. If the purpose of the model is to quote fixed income options  prices that are not easily observable, then a model with time-dependent volatility provides a means  of interpolating from known to unknown option prices. If, however, the purpose of the model is to  value and hedge fixed income securities, including options, then a model with mean reversion might  be preferred for two reasons.

1. First, while mean reversion is based on the economic intuitions outlined earlier, time-dependent  volatility relies on the difficult argument that the market has a forecast of short-term volatility  in the distant future. A modification of the model that addresses this objection, by the way, is to  assume that volatility depends on time in the near future and then settles at a constant.

2. Second, the downward-sloping factor structure and term structure of volatility in mean-  reverting models capture the behavior of interest rate movements better than parallel shifts and  a flat term structure of volatility. It may very well be that the Vasicek model does not capture  the behavior of interest rates sufficiently well to be used for a particular valuation or hedging  purpose. But in that case, it is unlikely that a parallel shift model calibrated to match caplet  prices will be better suited for that purpose.

The Cox-Ingersoll-Ross Model – The Short Rate Process

•The  models  discussed  till  now  assume  that  the  basis-point  volatility  of  the  short  rate  is  independent of the level of the short rate. But this is not true at extreme levels of the short rate.

1. Periods of high inflation and high short-term interest rates are inherently unstable and, as a  result, the basis-point volatility of the short rate tends to be high.

2. Also, when the short-term rate is very low, its basis-point volatility is limited by the fact that  interest rates cannot decline much below zero.

•Hence the basis-point volatility of the short rate is generally taken as an increasing function of  the short rate, which is reflected in the risk-neutral dynamics of the Cox-Ingersoll-Ross (CIR)  model

Here the parameter 𝜎 is constant, but basis-point volatility is not constant. Annualized basis-  point volatility equals 𝜎  𝑟and increases with the level of the short rate.

The Lognormal Model – The Short Rate Process

•Another popular specification is that the basis-point volatility is proportional to rate. In this case  the parameter 𝜎 is often called yield volatility. the simplest lognormal model, to be called Model  4, is

𝑑𝑟 = 𝑎𝑟𝑑𝑡 + 𝜎𝑟𝑑𝑤

Here, yield volatility is constant but basis-point volatility equals or and increases with the level  of the rate.

Comparison of Basis Point Volatility

•This figure graphs the basis-point volatility as a  function of rate for the cases of the constant,  square root, and proportional specifications. For  comparison purposes, 𝜎 is set in all three cases  to be 100 at a short rate of 8%. Mathematically,

𝜎^bp = 0.01 = 1%

•As seen in this figure, the CIR and proportional volatility specifications have basis-point  volatility increasing with rate but at different speeds. Both models have the basis-point volatility  equal to zero at a rate of  zero. The property that basis-point volatility equals zero when the  short rate is zero, combined with the condition that the drift is positive when the rate is zero,  guarantees that the short rate cannot become negative.

Comparison of Terminal Distributions

•This figure graphs terminal distributions of the short rate after 10 years under the  CIR,  normal, and lognormal volatility specifications. The parameters have been chosen so that all of  the distributions have an expected value of 5% and a standard deviation of 2.32%.

1. The figure illustrates the advantage of the CIR and lognormal models with respect to not  allowing negative rates.

2. The figure also indicates that out-of-the money option prices could differ  significantly under the three models.  Even if, as in this case, the mean and  volatility of the  three distributions  are the same, the probability of  outcomes away from the means are  different enough to generate  significantly different options prices.

The Lognormal Model With Deterministic Drift

•After applying mathematical tools (which are not a part of the FRM part 2 curriculum)on Model  4, the lognormal model with deterministic drift is given as

𝑑 𝑙𝑛 𝑟  = 𝑎 𝑡 𝑑𝑡 + 𝜎𝑑𝑤

This equation states says that the natural logarithm of  the short rate is normally distributed.  From FRM Part 1, it is known that a random variable has a lognormal distribution if its natural  logarithm has a normal distribution. Therefore, the above equation implies that the short rate  has a lognormal distribution.

•The above model may  be described as the Ho-Lee model  based on the natural logarithm of the short rate instead of on the short rate itself. Adapting the tree for the Ho-Lee model  accordingly, the tree for the first three dates is  given by this figure.

•To express the previous tree in rate levels, as opposed to the natural logarithm of the rate, each  node is exponentiated to give this tree.

•This tree shows that  the perturbations to the short rate  in a lognormal model are multiplicative  as opposed to the additive perturbations in normal models.

This observation, in turn, reveals why the short rate in this model cannot become negative. Since 𝑒^x is positive for any value of 𝑥, so long as 𝑟₀ is positive every node of the log normal tree  results in a positive rate. The tree also reveals why volatility in a lognormal model is expressed as  a percentage of the rate. For small values of 𝑥, 𝑒^x ≈ 1 + 𝑥. Setting 𝑎₁ = 0 𝑎𝑛𝑑 𝑑𝑡 = 1, for  example, the top node of date 1 may be approximated as 𝑟₀𝑒^σ ≈ 𝑟₀ 1 + 𝜎 . If, for example, 𝜎 = 12.5%, then the short rate in the up-state is 12.5% above the initial short rate. So volatility  is clearly a percentage of the rate in equation.

The Lognormal Model With Mean Reversion

•The final model to be presented in this chapter is a lognormal model with mean reversion called  the Black-Karasinski model. The model allows volatility, mean reversion, and the  central  tendency of the short rate to depend on time, firmly placing the model in the arbitrage free class.  A user may, of course, use or remove as much time dependence as desired. The dynamics of the  model are written in logarithmic form as

𝑑[𝑙𝑛(𝑟)]  = 𝑘(𝑡) (𝑙𝑛(𝜃(𝑡)) — 𝑙𝑛(𝑟))𝑑𝑡 + 𝜎(𝑡)𝑑𝑤

This equation states that that the natural logarithm of the short rate is normally distributed. It reverts to 𝜃(𝑡) at a speed of  𝑘(𝑡) with a volatility of  𝜎 𝑡  Viewed another way, the natural  logarithm of the short rate follows a time dependent version of the Vasicek model.

•The final model to be presented in this chapter is a lognormal model with mean reversion called  the Black-Karasinski model. The model allows volatility, mean reversion, and the  central  tendency of the short rate to depend on time, firmly placing the model in the arbitrage free class.  A user may, of course, use or remove as much time dependence as desired. The dynamics of the  model are written in logarithmic form as

𝑑[𝑙𝑛(𝑟)]  = 𝑘(𝑡) (𝑙𝑛(𝜃(𝑡)) — 𝑙𝑛(𝑟))𝑑𝑡 + 𝜎(𝑡)𝑑𝑤

This equation states that that the natural logarithm  of the short rate is normally distributed. It reverts to 𝜃(𝑡) at a speed of 𝑘(𝑡) with a volatility  of 𝜎 𝑡 . Viewed another way, the natural logarithm of the short rate follows a time-dependent  version of the Vasicek model.

•As in the previous section, the corresponding tree  may be written in terms of the rate or the natural logarithm of the rate. In terms of rate, the process over the first date is given in this figure.

•The variable 𝑟₁ is introduced for readability. The natural logarithms of the rates in the up and  down-states are 𝑙𝑛𝑟₁ + 𝜎 1  𝑑𝑡 and 𝑙𝑛𝑟₁ — 𝜎 1  𝑑𝑡 respectively. A little algebra shows that the tree recombines only if

•This above condition for recombination suggests that the mean reversion speed is completely  determined by the time-dependent volatility function. But it has been observed earlier in this  chapter that the mean reversion controls the term structure of  volatility whereas time-  dependent volatility controls the future volatility of the short-term rate. To create a model  flexible enough to control mean reversion and time-dependent volatility separately, the model  has to construct a recombining tree without imposing this constraint of mean reversion speed  being completely determined by the time-dependent volatility function.. To do so it allows the  length of the time step, 𝑑𝑡, to change over time.

•A little algebra now shows that the tree recombines if

The length of the first time step can be set arbitrarily. The length of  the second time step is set  to satisfy the above equation, allowing the user freedom in choosing the mean reversion and  volatility functions independently.