LO 14.6: Describe lognorm al models with deterministic drift and mean reversion.

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
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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
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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
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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.
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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.
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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 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.
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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:
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

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