Temp_store

LO 14.4: Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) and lognorm al models.

LO 14.3: Assess the efficacy o f time-dependent volatility models.
Time-dependent volatility models add flexibility to models of future short-term rates. This is particularly useful for pricing multi-period derivatives like interest rate caps and floors. Each cap and floor is made up of single period caplets and floorlets (essentially interest rate calls and puts). The payoff to each caplet or floorlet is based on the strike rate and the current short-term rate over the next period. Hence, the pricing of the cap and floor will depend critically on the forecast of cr(t) at several future dates.
It is impossible to describe the general behavior of the standard deviation over the relevant horizon because it will depend on the deterministic model chosen. However, there are some parallels between Model 3 and the mean-reverting drift (Vasicek) model. Specifically, if the initial volatility for both models is equal and the decay rate is the same as the mean reversion rate, then the standard deviations of the terminal distributions are exactly the same. Similarly, if the time-dependent drift in Model 3 is equal to the average interest rate path in the Vasicek model, then the two terminal distributions are identical, an even stronger observation than having the same terminal standard deviation.
There are still important differences between these models. First, Model 3 will experience a parallel shift in the yield curve from a change in the short-term rate. Second, the purpose of the model drives the choice of the model. If the model is needed to price options on fixed income instruments, then volatility dependent models are preferred to interpolate between
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observed market prices. On the other hand, if the model is needed to value or hedge fixed income securities or options, then there is a rationale for choosing mean reversion models.
One criticism of time-dependent volatility models is that the market forecasts short-term volatility far out into the future, which is not likely. A compromise is to forecast volatility approaching a constant value (in Model 3, the volatility approaches 0). A point in favor of the mean reversion models is the downward-sloping volatility term structure.
C o x -In g e r s o l l -R o s s (CIR) a n d L o g n o r m a l M o d e l s

LO 14.2: Calculate the short-term rate change and determine the behavior o f the standard deviation o f the rate change using a model with tim e dependent volatility.
The relationships between volatility in each period could take on an almost limitless number of combinations. For example, the volatility of the short-term rate in one year, cr(l), could be 220 basis points and the volatility of the short-term rate in two years, cr(2), could be 260 basis points. It is also entirely possible that cr(l) could be 220 basis points and cr(2) could be 160 basis points. To make the analysis more tractable, it is useful to assign a
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specific parameterization of time-dependent volatility. Consider the following model, which is known as Model 3:
dr = \(t)dt + ere atdw
where: a = volatility at t = 0, which decreases exponentially to 0 for a > 0
To illustrate the rate change using Model 3, assume a current short-term rate, rQ, of 3%, a drift, \ , of 0.24%, and, instead of constant volatility, include time-dependent volatility of ae_0-3t (with initial a = 1.30%). If we also assume the dw realization drawn from a normal distribution is 0.2 (with mean = 0 and standard deviation = Vl /12 = 0.2887), the change in the short-term rate after one month is calculated as:
dr = 0.24% x (1/12) + 1.5% x e-0-3(1/12) x 0.2
dr = 0.02% + 0.29% = 0.31%
Therefore, the expected short-term rate of 5% plus the rate change (0.31%) equals 5.31%. Note that this value would be slightly less than the value assuming constant volatility (5.32%). This difference is expected given the exponential decay in the volatility.
M odel 3 Effectiveness

LO 14.1: Describe the short-term rate process under a model with time-dependent volatility.
This topic provides a natural extension to the prior topic on modeling term structure drift by incorporating the volatility of the term structure. Following the notation convention of the previous topic, the generic continuously compounded instantaneous rate is denoted r and will change (over time) according to the following relationship:
dr = \(t)dt + cr(t)dw
It is useful to note how this model augments Model 1 and the Ho-Lee model. The functional form of Model 1 (with no drift), dr = adw, now includes time-dependent drift and time-dependent volatility. The Flo-Lee model, dr = \(t)dt + crdw, now includes non- constant volatility. As in the earlier models, dw is normally distributed with mean 0 and standard deviation V d t.

LO 13.8: Describe the effectiveness o f the Vasicek M odel.
There are some general comments that we can make to compare mean-reverting (Vasicek) models to models without mean reversion. In development of the mean-reverting model, the parameters rQ and 6 were calibrated to match observed market prices. Hence, the mean reversion parameter not only improves the specification of the term structure, but also produces a specific term structure of volatility. Specifically, the Vasicek model will produce a term structure of volatility that is declining. Therefore, short-term volatility is overstated and long-term volatility is understated. In contrast, Model 1 with no drift generates a flat volatility of interest rates across all maturities.
Furthermore, consider an upward shift in the short-term rate. In the mean-reverting model, the short-term rate will be impacted more than long-term rates. Therefore, the Vasicek model does not imply parallel shifts from exogenous liquidity shocks. Another interpretation concerns the nature of the shock. If the shock is based on short-term economic news, then the mean reversion model implies the shock dissipates as it approaches the long-run mean. The larger the mean reversion parameter, the quicker the economic news is incorporated. Similarly, the smaller the mean reversion parameter, the longer it takes for the economic news to be assimilated into security prices. In this case, the economic news is long-lived. In contrast, shocks to short-term rates in models without drift affect all rates equally regardless of maturity (i.e., produce a parallel shift).
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K e y C o n c e p t s
LO 13.1
Model 1 assumes no drift and that interest rates are normally distributed. The continuously compounded instantaneous rate, rf, will change according to:
dr = adw
Model 1 limitations:
The no-drift assumption is not flexible enough to accommodate basic term structure
shapes.
The term structure of volatility is predicted to be flat. There is only one factor, the short-term rate. Any change in the short-term rate would lead to a parallel shift in the yield curve.
Model 2 adds a constant drift: dr = \d t + adw. The new interest rate tree increases each node in the next time period by Adt. The drift combines the expected rate change with a risk premium. The interest rate tree is still recombining, but the middle node rate at time 2 will not equal the initial rate, as was the case with Model 1.
Model 2 limitations:
The calibrated values of drift are often too high. The model requires forecasting different risk premiums for long horizons where reliable
forecasts are unrealistic.
LO 13.2
The interest rate tree for Model 1 is recombining and will increase/decrease each period by the same 30% probability.
LO 13.3
The normality assumption of the terminal interest rates for Model 1 will always have a positive probability of negative interest rates. One solution to eliminate this negative rate problem is to use non-negative distributions, such as the lognormal distribution; however, this may introduce other undesirable features into the model. An alternative solution is to create an adjusted interest rate tree where negative interest rates are replaced with 0%, constraining the data from being negative.
LO 13.4
The Ho-Lee model introduces even more flexibility than Model 2 by allowing the drift term to vary from period to period (i.e., time-dependent drift). The recombined middle node at time 2 = rQ + ( \ l + X2)dt.
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LO 13.3
Arbitrage models are often used to price securities that are illiquid or off-market (e.g., uncommon maturity for a swap). The more liquid security prices are used to develop a consistent pricing model, which in turn is used for illiquid or non-standard securities. Because arbitrage models assume the market price is correct, the models will not be effective if there are short-term imbalances altering bond prices. Similarly, arbitrage-free models cannot be used in relative valuation analysis because the securities being compared are already assumed to be properly priced.
LO 13.6
The Vasicek model assumes mean reversion to a long-run equilibrium rate. The specific functional form of the Vasicek model is as follows:
dr = k(0 – r)dt + crdw
The parameter k measures the speed of the mean reversion adjustment; a high k will produce quicker (larger) adjustments than smaller values of k. Assuming there is a long-run interest rate of
the long-run mean-reverting level is:
0 ri H k
1
The Vasicek model is not recombining. The tree can be approximated as recombining by averaging the unequal two nodes and recalibrating the associated probabilities (i.e., no longer using 30% probabilities for the up and down moves).
LO 13.7
The expected rate in T years can be forecasted assuming exponential decay of the difference between the current level and the mean-reverting level. The half-life, t , can be computed as the time to move halfway between the current level and the mean-reverting level:
(0 – r0)ekT = ‘/2(0 – rQ)
LO 13.8
The Vasicek model not only improves the specification of the term structure, but also produces a downward-sloping term structure of volatility. Model 1, on the other hand, predicts flat volatility of interest rates across all maturities. Model 1 implies parallel shifts from exogenous shocks while the Vasicek model does not. Long- (short-) lived economic shocks have low (high) mean reversion parameters. In contrast, in Model 1, shocks to short- term rates affect all rates equally regardless of maturity.
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C o n c e p t C h e c k e r s
1.
2.
3.
4.
Using Model 1, assume the current short-term interest rate is 3%, annual volatility is 80bps, and dw, a normally distributed random variable with mean 0 and standard deviation Vdt, has an expected value of zero. After one month, the realization of dw is -0.3. What is the change in the spot rate and the new spot rate?
Change in Spot
A. 0.40% B. -0.40% C. 0.80% D. -0.80%
New Spot Rate 5.40% 4.60% 5.80% 4.20%
Using Model 2, assume a current short-term rate of 8%, an annual drift of 50bps, and a short-term rate standard deviation of 2%. In addition, assume the ex-post realization of the dw random variable is 0.3. After constructing a 2-period interest rate tree with annual periods, what is the interest rate in the middle node at the end of year 2? A. 8.0%. B. 8.8%. C. 9.0%. D. 9.6%.
The Bureau of Labor Statistics has just reported an unexpected short-term increase in high-priced luxury automobiles. What is the most likely anticipated impact on a mean-reverting model of interest rates? A. The economic information is long-lived with a low mean-reversion parameter. B. The economic information is short-lived with a low mean-reversion parameter. C. The economic information is long-lived with a high mean-reversion parameter. D. The economic information is short-lived with a high mean-reversion parameter.
Using the Vasicek model, assume a current short-term rate of 6.2% and an annual volatility of the interest rate process of 2.5%. Also assume that the long-run mean- reverting level is 13.2% with a speed of adjustment of 0.4. Within a binomial interest rate tree, what are the upper and lower node rates after the first month?
Upper node
A. 6.67% B. 6.67% C. 7.16% D. 7.16%
Lower node 5.71% 6.24% 6.24% 5.71%
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John Jones, FRM, is discussing the appropriate usage of mean-reverting models relative to no-drift models, models that incorporate drift, and Ho-Lee models. Jones makes the following statements:
Statement 1: Both Model 1 (no drift) and the Vasicek model assume parallel shifts from changes in the short-term rate.
Statement 2: The Vasicek model assumes decreasing volatility of future short-term rates while Model 1 assumes constant volatility of future short-term rates.
Statement 3: The constant drift model (Model 2) is a more flexible model than the Ho-Lee model.
How many of his statements are correct? A. 0. 1. B. C. 2. D. 3.
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C o n c e p t C h e c k e r An s w e r s
1. B Model 1 has a no-drift assumption. Using this model, the change in the interest rate is
predicted as:
dr = adw dr = 0.8% x (-0.5) = -0.4% = 40 basis points
Since the initial rate was 5% and dr = -0.40%, the new spot rate in one month is:
5% – 0.40% = 4.60%
2. C Using Model 2 notation:
current short-term rate, rQ = 8% drift, X = 0.5% standard deviation, a = 2% random variable, dw = 0.3 change in time, dt = 1
Since we are asked to find the interest rate at the second period middle node using Model 2, we know that the tree will recombine to the following rate: rQ + 2\dt.
8 % + 2 x 0 . 5 % x 1 = 9 %
3. D The economic news is most likely short-term in nature. Therefore, the mean reversion parameter is high so the mean reversion adjustment per period will be relatively large.
4. D Using a Vasicek model, the upper and lower nodes for time 1 are computed as follows:
upper node = 6.2% -|—————————– 1—-
j ^ n/ , (0.4)(13.2% 6.2%) , 2.5% Ju lower node = 6.2% + (0.4) (13.2% 6.2%) 2.5% Vl2
12
12
7.16%
5.71%
5. B Only Statement 2 is correct. The Vasicek model implies decreasing volatility and non-parallel
shifts from changes in short-term rates. The Ho-Lee model is actually more general than Model 2 (the no drift and constant drift models are special cases of the Ho-Lee model).
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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
Th e A r t o f T e r m S t r u c t u r e M o d e l s : Vo l a t il it y a n d D i s t r i b u t i o n
Topic 14
E x a m F o c u s
This topic incorporates non-constant volatility into term structure models. The generic time- dependent volatility model is very flexible and particularly useful for valuing multi-period derivatives like interest rate caps and floors. The Cox-Ingersoll-Ross (CIR) mean-reverting model suggests that the term structure of volatility increases with the level of interest rates and does not become negative. The lognormal model also has non-negative interest rates that proportionally increase with the level of the short-term rate. For the exam, you should understand how these models impact the short-term rate process, and be able to identify how a time-dependent volatility model (Model 3) differs from the models discussed in the previous topic. Also, understand the differences between the CIR and the lognormal models, as well as the differences between the lognormal models with drift and mean reversion.
T e r m S t r u c t u r e M o d e l w i t h T i m e -D e p e n d e n t V o l a t i l i t y

LO 13.7: Calculate the Vasicek M odel rate change, standard deviation o f the rate change, expected rate in T years, and half life.
The previous discussion encompassed the rate change in the Vasicek model and the computation of the standard deviation when solving for the parameters in the recombining tree. In this section, we turn our attention to the forecasted rate in T years.
To continue with the previous example, the current short-term rate is 6.2% with the mean-reversion parameter, k, of 0.03. The long-term mean-reverting level will eventually reach 18%, but it will take a long time since the value of k is quite small. Specifically, the current rate of 6.2% is 11.8% from its ultimate natural level and this difference will decay exponentially at the rate of mean reversion (11.8% is calculated as 18% – 6.2%). To forecast the rate in 10 years, we note that 11.8% x e(-0-3xio) _ 8.74%. Therefore, the expected rate in 10 years is 18% – 8.74% = 9.26%.
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In the Vasicek model, the expected rate in T years can be represented as the weighted average between the current short-term rate and its long-run horizon value. The weighting factor for the short-term rate decays exponentially by the speed of the mean-reverting parameter, 9:
rQe
+ 0(1 – e kT)
A more intuitive measure for computing the forecasted rate in T years uses a factors half- life, which measures the number of years to close half the distance between the starting rate and mean-reverting level. Numerically:
(18% – 6.2%)e–03T = V4(18% – 6.2%)
e 0,03t = Vi (cid:31) t = In(2) / 0.03 = 23.1 years
Professor’s Note: A larger mean reversion in a shorter h alf life.
adjustment parameter, k, will result
Vasicek M odel Effectiveness

LO 13.6: Describe the process o f constructing a simple and recombining tree for a short-term rate under the Vasicek M odel with mean reversion.
The Vasicek model assumes a mean-reverting process for short-term interest rates. The underlying assumption is that the economy has an equilibrium level based on economic fundamentals such as long-run monetary supply, technological innovations, and similar factors. Therefore, if the short-term rate is above the long-run equilibrium value, the drift adjustment will be negative to bring the current rate closer to its mean-reverting level. Similarly, short-term rates below the long-run equilibrium will have a positive drift adjustment. Mean reversion is a reasonable assumption but clearly breaks down in periods of extremely high inflation (i.e., hyperinflation) or similar structural breaks.
The formal Vasicek model is as follows:
dr = k(0 – r)dt + crdw
where: k = a parameter that measures the speed of reversion adjustment 0 = long-run value of the short-term rate assuming risk neutrality r = current interest rate level
In this model, k measures the speed of the mean reversion adjustment; a high k will produce quicker (larger) adjustments than smaller values of k. A larger differential between the long- run and current rates will produce a larger adjustment in the current period.
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Similar to the previous discussion, the drift term, A, is a combination of the expected rate change and a risk premium. The risk neutrality assumption of the long-run value of the short-term rate allows 6 to be approximated as:
a X 9 rj H k
where:
= the long-run true rate of interest
Lets consider a numerical example with a reversion adjustment parameter of 0.03, annual standard deviation of 130 basis points, a true long-term interest rate of 6%, a current interest rate of 6.2%, and annual drift of 0.36%. The long-run value of the short-term rate assuming risk neutrality is approximately:
A 9 ~ 6% H———= 18%
0.36% 0.03
It follows that the forecasted change in the short-term rate for the next period is:
0.03(18% – 6.2%)(1/12) = 0.0293%
The volatility for the monthly interval is computed as 1.5% x y/l /12 = 0.43% (43 basis points per month).
The next step is to populate the interest rate tree. Note that this tree will not recombine in the second period because the adjustment in time 2 after a downward movement in interest rates will be larger than the adjustment in time 2 following an upward movement in interest rates (since the lower node rate is further from the long-run value). This can be illustrated directly in the following calculations. Starting with rQ = 6.2%, the interest rate tree over the first period is:
Figure 5: First Period Upper and Lower Node Calculations
(0.03)(18% – 6.200%)
6.200% + —– —————- – +
12
,
= 6.663%
1.5% Vl2
6.200% + ——————— – – -4=r = 5.796%
(0.03)(18%- 6.200%)
12
1.5% yjl2
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If the interest rate evolves upward in the first period, we would turn to the upper node in the second period. The interest rate process can move up to 7.124% or down to 6.238%.
Figure 6: Second Period Upper Node Calculations
6.663% +(Q-Q3)(18%- 6-663%)+ ^ = 7.124%
12
yjil
6.663%
(0.03)(18%-6.6630/0) _ +5% = 6^258%
12
Vl2
If the interest evolves downward in the first period, we would turn to the lower node in the second period. The interest rate process can move up to 6.260% or down to 3.394%.
Figure 7: Second Period Lower Node Calculations
5.796%+ (0-03)(18%-5.796%) L5% = ^
12
Vl2
5.796% + i8 % – 5 7 9 6 % ) _ + 5 % = ^
12
yjl2
Finally, we complete the 2-period interest rate tree with mean reversion. The most interesting observation is that the model is not recombining. The up-down path leads to a 6.258% rate while the down-up path leads to a 6.260% rate. In addition, the down-up path rate is larger than the up-down path rate because the mean reversion adjustment has to be larger for the down path, as the initial interest rate was lower (5.796% versus 6.663%).
Figure 8: 2-Period Interest Rate Tree with Mean Reversion
6.200%
6.663%
5.796%
At this point, the Vasicek model has generated a 2-period non-recombining tree of short- term interest rates. It is possible to modify the methodology so that a recombining tree is the end result. There are several ways to do this, but we will outline one straightforward method. The first step is to take an average of the two middle nodes (6.258% + 6.260%) 12 = 6.259%. Next, we remove the assumption of 50% up and 50% down movements by generically replacing them with {p, 1 p) and (q, 1 q) as shown in Figure 9.
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Figure 9: Recombining the Interest Rate Tree
0.5
6.200%
The final step for recombining the tree is to solve for p and q and ^ and p and q are the respective probabilities of up movements in the trees in the second period after the up and down movements in the first period. yjm an(j yddare the respective interest rates from successive (up, up and down, down) movements in the tree.
We can solve for the unknown values using a system of equations. First, we know that the average of p x ruu and (1 – p) x 6.259% must equal:
6.663% + 0.03(18% – 6.663%)(1/12) = 6.691%
Second, we can use the definition of standard deviation to equate:
Vp(ruu 6.691%)2 + (1 p)(6.259% 6.691%)2 = 1.50% x
We would then repeat the process for the bottom portion of the tree, solving for q and If the tree extends into a third period, the entire process repeats iteratively.

LO 13.3: Describe uses and benefits o f the arbitrage-free models and assess the issue o f fitting models to market prices.
Broadly speaking, there are two types of models: arbitrage-free models and equilibrium models. The key factor in choosing between these two models is based on the need to match
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market prices. Arbitrage models are often used to quote the prices of securities that are illiquid or customized. For example, an arbitrage-free tree is constructed to properly price on-the-run Treasury securities (i.e., the model price must match the market price). Then, the arbitrage-free tree is used to predict off-the-run Treasury securities and is compared to market prices to determine if the bonds are properly valued. These arbitrage models are also commonly used for pricing derivatives based on observable prices of the underlying security (e.g., options on bonds).
There are two potential detractors of arbitrage-free models. First, calibrating to market prices is still subject to the suitability of the original pricing model. For example, if the parallel shift assumption is not appropriate, then a better fitting model (by adding drift) will still be faulty. Second, arbitrage models assume the underlying prices are accurate. This will not be the case if there is an external, temporary, exogenous shock (e.g., oversupply of securities from forced liquidation, which temporarily depresses market prices).
If the purpose of the model is relative analysis (i.e., comparing the value of one security to another), then using arbitrage-free models, which assume both securities are properly priced, is meaningless. Hence, for relative analysis, equilibrium models would be used rather than arbitrage-free models.
V a s i c e k M o d e l

LO 13.4: Construct a short-term rate tree under the Ho-Lee M odel with time- dependent drift.
The Ho-Lee model further generalizes the drift to incorporate time-dependency. That is, the drift in time 1 may be different than the drift in time 2; additionally, each drift does not have to increase and can even be negative. Thus, the model is more flexible than the constant drift model. Once again, the drift is a combination of the risk premium over the period and the expected rate change. The tree in Figure 4 illustrates the interest rate structure and effect of time-dependent drift.
Figure 4: Interest Rate Tree with Time-Dependent Drift
It is clear that if X1 = X2 then the Ho-Lee model reduces to Model 2. Also, it should not be surprising that X j and X2 are estimated from observed market prices. In other words, the observed one-period spot rate. Xj could then be estimated so that the model rate equals the observed two-period market rate. X2 could be calibrated from using observed market rate for a three-period security, and so on.
and X1 and the
is
A r b i t r a g e -F r e e M o d e l s

LO 13.3: Describe methods for addressing the possibility o f negative short-term rates in term structure models.
Note that the terminal nodes in the two-period model generate three possible ending rates: rQ + 2a V dt, rQ, and rQ – 2a V dt. This discrete, finite set of outcomes does not technically represent a normal distribution. However, our knowledge of probability distributions tells us that as the number of steps increases, the terminal distribution at the nodes will approach a continuous normal distribution.
One obvious drawback to Model 1 is that there is always a positive probability that interest rates could become negative. On the surface, negative interest rates do not make much economic sense (i.e., lending $100 and receiving less than $100 back in the future). However, you could plausibly rationalize a small negative interest rate if the safety and/or inconvenience of holding cash were sufficiently high.
The negative interest rate problem will be exacerbated as the investment horizon gets longer, since it is more likely that forecasted interest rates will drop below zero. As an illustration, assume a ten-year horizon and a standard deviation of terminal interest rates of 1.2% x VlO = 3.79%. It is clear that negative interest rates will be well within a two standard deviation confidence interval when centered around a current rate of 6%. Also note that the problem of negative interest rates is greater when the current level of interest rates is low (e.g., 4% instead of the original 6%).
There are two reasonable solutions for negative interest rates. First, the model could use distributions that are always non-negative, such as lognormal or chi-squared distributions. In this way, the interest rate can never be negative, but this action may introduce other non- desirable characteristics such as skewness or inappropriate volatilities. Second, the interest rate tree can force negative interest rates to take a value of zero. In this way, the original interest rate tree is adjusted to constrain the distribution from being below zero. This method may be preferred over the first method because it forces a change in the original distribution only in a very low interest rate environment whereas changing the entire distribution will impact a much wider range of rates.
As a final note, it is ultimately up to the user to decide on the appropriateness of the model. For example, if the purpose of the term structure model is to price coupon-paying bonds, then the valuation is closely tied to the average interest rate over the life of the bond and the possible effect of negative interest rates (small probability of occurring or staying negative for long) is less important. On the other hand, option valuation models that have asymmetric payoffs will be more affected by the negative interest rate problem.
M odel 1 Effectiveness
Given the no-drift assumption of Model 1, we can draw several conclusions regarding the effectiveness of this model for predicting the shape of the term structure:
The no-drift assumption does not give enough flexibility to accurately model basic
term structure shapes. The result is a downward-sloping predicted term structure due to a larger convexity effect. Recall that the convexity effect is the difference between the model par yield using its assumed volatility and the par yield in the structural model with assumed zero volatility.
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Model 1 predicts a flat term structure of volatility, whereas the observed volatility term
structure is hump-shaped, rising and then falling.
Model 1 only has one factor, the short-term rate. Other models that incorporate
additional factors (e.g., drift, time-dependent volatility) form a richer set of predictions. Model 1 implies that any change in the short-term rate would lead to a parallel shift in the yield curve, again, a finding incongruous with observed (non-parallel) yield curve shifts.
T e r m S t r u c t u r e M o d e l w i t h D r i f t (M o d e l 2)
Casual term structure observation typically reveals an upward-sloping yield curve, which is at odds with Model 1, which does not incorporate drift. A natural extension to Model 1 is to add a positive drift term that can be economically interpreted as a positive risk premium associated with longer time horizons. We can augment Model 1 with a constant drift term, which yields Model 2:
dr = \d t + odw
Lets continue with a new example assuming a current short-term interest rate, drift, \ , of 0.24%, and standard deviation, a, of 1.30%. As before, the div realization drawn from a normal distribution (with mean = 0 and standard deviation = 0.2887) is 0.2. Thus, the change in the short-term rate in one month is calculated as:
of 3%,
dr = 0.24% x (1/12) + 1.5% x 0.2 = 0.32%
Hence, the new rate, rv is computed as: 5% + 0.32% = 5.32%. T he monthly drift is 0.24% x 1/12 = 0.02% and the standard deviation of the rate is 1.5% x Vl /12 = 0.43% (i.e., 43 basis points per month). The 2bps drift per month (0.02%) represents any combination of expected changes in the short-term rate (i.e., true drift) and a risk premium. For example, the 2bps observed drift could result from a 1.5bp change in rates coupled with a 0.5bp risk premium.
The interest rate tree for Model 2 will look very similar to Model 1, but the drift term, \dt, will increase by \d t in the next period, 2 \d t in the second period, and so on. This is visually represented in Figure 3. Note that the tree recombines at time 2, but the value at time 2, rQ + 2\dt, is greater than the original rate, rQ, due to the positive drift.
Figure 3: Interest Rate Tree with Constant Drift
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
M odel 2 Effectiveness
As you would expect, Model 2 is more effective than Model 1. Intuitively, the drift term can accommodate the typically observed upward-sloping nature of the term structure. In practice, a researcher is likely to choose Hence, the term structure will fit better. The downside of this approach is that the estimated value of drift could be relatively high, especially if considered as a risk premium only. On the other hand, if the drift is viewed as a combination of the risk premium and the expected rate change, the model suggests that the expected rates in year 10 will be higher than year 9, for example. This view is more appropriate in the short run, since it is more difficult to justify increases in expected rates in the long run.
and X based on the calibration of observed rates.
H o -L e e M o d e l