LO 13.8: Describe the effectiveness o f the Vasicek M odel.

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
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
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
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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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.
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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
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