Articles by kenli

LO 15.2: Explain the im plications o f put-call parity on the im plied volatility o f call and put options.
Recall that put-call parity is a no-arbitrage equilibrium relationship that relates European call and put option prices to the underlying assets price and the present value of the options strike price. In its simplest form, put-call parity can be represented by the following relationship:
c – p = S – PV(X)
where: c p S PV(X) = present value of the strike = price of a call = price of a put = price of a call = price of a put = price of the underlying security
PV(X) can be represented in continuous time by:
PV(X) = Xe-rT
where: r = risk-free rate T =
time left to expiration expressed in years
Since put-call parity is a no-arbitrage relationship, it will hold whether or not the underlying asset price distribution is lognormal, as required by the Black-Scholes-Merton (BSM) option pricing model.
2018 Kaplan, Inc.
Page 189
Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
If we simply rearrange put-call parity and denote subscripts for the option prices to indicate whether they are market or Black-Scholes-Merton option prices, the following two equations are generated:
Pmkt + S0e-V = cmkt + PV(X)
P b SM + S 0e qt = CBSM + ^ ( X )
Subtracting the second equation from the first gives us:
Pmkt PBSM = Cmkt CBSM
This relationship shows that, given the same strike price and time to expiration, option market prices that deviate from those dictated by the Black-Scholes-Merton model are going to deviate in the same amount whether they are for calls or puts. Since any deviation in prices will be the same, the implication is that the implied volatility of a call and put will be equal for the same strike price and time to expiration.
V o l a t i l i t y S m i l e s

LO 14.6: Describe lognorm al models with deterministic drift and mean reversion.
Lognorm al M odel with Determ inistic D rift
For this LO, we detail two lognormal models, one with deterministic drift and one with mean reversion. The lognormal model with drift is shown as follows:
d[ln(r)] = a(t)dt + crdw
The natural log of the short-term rate follows a normal distribution and can be used to construct an interest rate tree based on the natural logarithm of the short-term rate. In the spirit of the Flo-Lee model, where drift can vary from period to period, the interest rate tree in Figure 2 is generated using the lognormal model with deterministic drift.
Figure 2: Interest Rate Tree with Lognormal Model (Drift)
0.5
If each node in Figure 2 is exponentiated, the tree will display the interest rates at each node. For example, the adjusted period 1 upper node would be calculated as:
exp(ln r0 + ajdt + aVdt) = roe(ai(lt+ CJ^ )
Figure 3: Lognormal Model Rates at Each Node
In contrast to the Ho-Lee model, where the drift terms are additive, the drift terms in the lognormal model are multiplicative. The implication is that all rates in the tree will always be positive since ex > 0 for all x. Furthermore, since ex 1+ x, and if we assume a1 = 0 and
2018 Kaplan, Inc.
Page 183
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
dt = 1, then: r0eCT r0(l + a). Hence, volatility is a percentage of the rate. For example, if a = 20%, then the rate in the upper node will be 20% above the current short-term rate.
Lognorm al M odel with M ean Reversion
The lognormal distribution combined with a mean-reverting process is known as the Black- Karasinski model. This model is very flexible, allowing for time-varying volatility and mean reversion. In logarithmic terms, the model will appear as:
d[ln(r)] = k(t)[ln0(t) – ln(r)]dt + a(t)dw
Thus, the natural log of the short-term rate follows a normal distribution and will revert to the long-run mean of ln[0(t)] based on adjustment parameter k(t). In addition, volatility is time-dependent, transforming the Vasicek model into a time-varying one. The interest rate tree based on this model is a bit more complicated, but it exhibits the same basic structure as previous models.
Figure 4: Interest Rate Tree with Lognormal Model (Mean Revision)
k(l)(ln0(l) – lnr0)dt +a(l)Vdt
r e k(l)(ln0(l) – lnr0)dt -a(l)Vdt
The notation jq is used to condense the exposition. Therefore, the In (upper node) = lmq + cr(l)Vdt and ln(lower node) = lmq – cr(l) V dt. Following the intuition of the mean- reverting model, the tree will recombine in the second period only if:
q(l) – g(2)
CJr(l)dt
Recall from the previous topic that in the mean-reverting model, the nodes can be forced to recombine by changing the probabilities in the second period to properly value the weighted average of paths in the next period. A similar adjustment can be made in this model. However, this adjustment varies the length of time between periods (i.e., by manipulating the ^variable). After choosing dt^, dt2 is determined with the following equation:
g(2)7dt2 o'(l)N/dtT
Page 184
2018 Kaplan, Inc.
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
K e y C o n c e p t s
LO 14.1
The generic continuously compounded instantaneous rate with time-dependent drift and volatility will evolve over time according to dr = \(t)dt + a(t)dw. Special cases of this model include Model 1 (dr = odw) and the Ho-Lee model (dr = \(t)dt + crdw).
LO 14.2
The relationships between volatility in each period could take on an almost limitless number of combinations. To analyze this factor, it is necessary to assign a specific parameterization of time-dependent volatility such that: dr = \(t)dt + ae_atdw, where a is volatility at t = 0, which decreases exponentially to 0. This model is referred to as Model 3.
LO 14.3
Time-dependent volatility is very useful for pricing interest rate caps and floors that depend critically on the forecast of a(t) on multiple future dates. Under reasonable conditions, Model 3 and the mean-reverting drift (Vasicek) model will have the same standard deviation of the terminal distributions. One criticism of time-dependent volatility models is that the market forecasts short-term volatility far out into the future. A point in favor of the mean- reversion models is the downward-sloping volatility term structure.
LO 14.4
Two common models that avoid negative interest rates are the Cox-Ingersoll-Ross (CIR) model and lognormal model. Although avoiding negative interest rates is attractive, the non-normality of the distributions can lead to derivative mispricings.
LO 14.3
The CIR mean-reverting model has constant that increases at a decreasing rate:
volatility (a) and basis-point volatility (cr Vr)
dr = k(0 – r)dt + a Vr dw
2018 Kaplan, Inc.
Page 185
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
LO 14.6
There are two lognormal models of importance: (1) lognormal with deterministic drift and (2) lognormal with mean reversion.
The lognormal model with drift is:
d [In (r) ] = a(t)dt + adw
This model is very similar in spirit to the Ho-Lee Model with additive drift. The interest rate tree is expressed in rates, as opposed to the natural log of rates, which results in a multiplicative effect for the lognormal model with drift.
The lognormal model with mean reversion is:
d [In (r) ] = k(t)[ln0(t)-ln(r)]dt + cr(t)dw
This model does not produce a naturally recombining interest rate tree. In order to force the tree to recombine, the time steps, dt, must be recalibrated.
Page 186
2018 Kaplan, Inc.
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
C o n c e p t C h e c k e r s
1.
Regarding the validity of time-dependent drift models, which of the following statements is (are) correct? I. Time-dependent drift models are flexible since volatility from period to period
can change. However, volatility must be an increasing function of short-term rate volatilities.
II. Time-dependent volatility functions are useful for pricing interest rate caps and
floors. A. I only. B. II only. C. Both I and II. D. Neither I nor II.
2.
Which of the following choices correctly characterizes basis-point volatility and yield volatility as a function of the level of the rate within the lognormal model?
Basis-point volatilitv A. increases B. increases C. decreases D. decreases
Yield volatilitv constant decreases constant decreases
3.
4.
3.
Which of the following statements is most likely a disadvantage of the CIR model? A. Interest rates are always non-negative. B. Option prices from the CIR distribution may differ significantly from
lognormal or normal distributions.
C. Basis-point volatility increases during periods of high inflation. D. Long-run interest rates hover around a mean-reverting level.
Which of the following statements best characterizes the differences between the Ho-Lee model with drift and the lognormal model with drift? A. In the Ho-Lee model and the lognormal model the drift terms are
multiplicative. In the Ho-Lee model and the lognormal model the drift terms are additive.
B. C. In the Ho-Lee model the drift terms are multiplicative, but in the lognormal
model the drift terms are additive.
D. In the Ho-Lee model the drift terms are additive, but in the lognormal model
the drift terms are multiplicative.
Which of the following statements is true regarding the Black-Karasinski model? A. The model produces an interest rate tree that is recombining by definition. B. The model produces an interest rate tree that is recombining when the dt
variable is manipulated.
C. The model is time-varying and mean-reverting with a slow speed of adjustment. D. The model is time-varying and mean-reverting with a fast speed of adjustment.
2018 Kaplan, Inc.
Page 187
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
C o n c e p t C h e c k e r An s w e r s
1. B Time-dependent volatility models are very flexible and can incorporate increasing,
decreasing, and constant short-term rate volatilities between periods. This flexibility is useful for valuing interest rate caps and floors because there is a potential payout each period, so the flexibility of changing interest rates is more appropriate than applying a constant volatility model.
2. A Choices B and D can be eliminated because yield volatility is constant. Basis-point volatility under the CIR model increases at a decreasing rate, whereas basis-point volatility under the lognormal model increases linearly. Therefore, basis-point volatility is an increasing function for both models.
3. B Choices A and C are advantages of the CIR model. Out-of-the money option prices may
differ with the use of normal or lognormal distributions.
4. D The Ho-Lee model with drift is very flexible, allowing the drift terms each period to vary.
Hence, the cumulative effect is additive. In contrast, the lognormal model with drift allows the drift terms to vary, but the cumulative effect is multiplicative.
5. B A feature of the time-varying, mean-reverting lognormal model is that it will not recombine
naturally. Rather, the time intervals between interest rate changes are recalibrated to force the nodes to recombine. The generic model makes no prediction on the speed of the mean reversion adjustment.
Page 188
2018 Kaplan, Inc.
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:
Vo l a t il it y S m i l e s
E x a m F o c u s
This topic discusses some of the reasons for the existence of volatility smiles, and how volatility affects option pricing as well as other option characteristics. Focus on the explanation of why volatility smiles exist in currency and equity options. Also, understand how volatility smiles impact the Greeks and how to interpret price jumps.
Topic 15
P u t -C a l l Pa r i t y

LO 14.3: Calculate the short-term rate change and describe the basis point volatility using the C IR and lognormal models.
.Another issue with the aforementioned models is that the basis-point volatility of the short- term rate is determined independently of the level of the short-term rate. This is questionable on two fronts. First, imagine a period of extremely high inflation (or even hyperinflation). The associated change in rates over the next period is likely to be larger than when rates are closer to their normal level. Second, if the economy is operating in an extremely low interest rate environment, then it seems natural that the volatility of rates will become smaller, as rates should be bounded below by zero or should be at most small, negative rates. In effect, interest rates of zero provide a downside barrier which dampens volatility.
A common solution to this problem is to apply a model where the basis-point volatility increases with the short-term rate. Whether the basis-point volatility will increase linearly or non-linearly is based on the particular functional form chosen. A popular model where the basis-point volatility (i.e., annualized volatility of dr) increases proportional to the square root of the rate (i.e., (Wr) is the Cox-Ingersoll-Ross (CIR) model where dr increases at a decreasing rate and a is constant. The CIR model is shown as follows:
dr = k(0 r)dt + a Vr dw
As an illustration, lets continue with the example from LO 14.2, given the application of the CIR model. Assume a current short-term rate of 3%, a long-run value of the short-term rate, 6, of 24%, speed of the mean revision adjustment, k, of 0.04, and a volatility, a, of 1.30%. As before, also assume the dw realization drawn from a normal distribution is 0.2. Using the CIR model, the change in the short-term rate after one month is calculated as:
dr = 0.04(24% – 5%)(1/12) + 1.5%
S %x 0.2
dr = 0.063% + 0.067% = 0.13%
Therefore, the expected short-term rate of 5% plus the rate change (0.13%) equals 5.13%.
2018 Kaplan, Inc.
Page 181
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
A second common specification of a model where basis-point volatility increases with the short-term rate is the lognormal model (Model 4). An important property of the lognormal model is that the yield volatility, O’, is constant, but basis-point volatility increases with the level of the short-term rate. Specifically, basis-point volatility is equal to or and the functional form of the model, where a is constant and dr increases at err, is:
dr = ardt + crrdw
For both the CIR and the lognormal models, as long as the short-term rate is not negative then a positive drift implies that the short-term rate cannot become negative. As discussed previously, this is certainly a positive feature of the models, but it actually may not be that important. For example, if a market maker feels that interest rates will be fairly flat and the possibility of negative rates would have only a marginal impact on the price, then the market maker may opt for the simpler constant volatility model rather than the more complex CIR.
The differences between the distributions of the short-term rate for the normal, CIR, and lognormal models are also important to analyze. Figure 1 compares the distributions after ten years, assuming equal means and standard deviations for all three models. As mentioned in Topic 13, the normal distribution will always imply a positive probability of negative interest rates. In addition, the longer the forecast horizon, the greater the likelihood of negative rates occurring. This can be seen directly by the left tail lying above the x-axis for rates below 0%. This is clearly a drawback to assuming a normal distribution.
Figure 1: Terminal Distributions
-5%
0%
5% Rate
10%
15%
——–CIR ——– N o r m a l——-Lognormal
In contrast to the normal distribution, the lognormal and CIR terminal distributions are always non-negative and skewed right. This has important pricing implications particularly for out-of-the money options where the mass of the distributions can differ dramatically.
Page 182
2018 Kaplan, Inc.
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10

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
Page 180
2018 Kaplan, Inc.
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
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
2018 Kaplan, Inc.
Page 179
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
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).
2018 Kaplan, Inc.
Page 173
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.
Page 174
2018 Kaplan, Inc.
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.
2018 Kaplan, Inc.
Page 175
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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%
Page 176
2018 Kaplan, Inc.
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.
2018 Kaplan, Inc.
Page 177
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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).
Page 178
2018 Kaplan, Inc.
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%.
Page 172
2018 Kaplan, Inc.
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.
2018 Kaplan, Inc.
Page 169
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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
Page 170
2018 Kaplan, Inc.
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.
2018 Kaplan, Inc.
Page 171
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
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.