P2.T5.306. Time-dependent interest rate volatility (Model 3)

[Fran]

AIMs: Describe the short-term rate process under a model with time-dependent volatility (Model 3). Calculate the short-term rate change and describe the behavior of the standard deviation of the change of the rate using a model with time dependent volatility. Describe the effectiveness of time-dependent volatility models.

Questions:

306.1. Becky the Analyst wants to apply Tuckman's Model 3 for short-term interest rates. Model 3 captures time-dependent volatility and its process is given by:

(Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011))

She makes the following assumptions:

• The time step is monthly, dt = 1/12
• The initial short-term rate, r(0) = 4.00%
• The annual basis point volatility = 3.00%
• The annual drift, lambda(t), is constant at 130 basis points (+1.30%) per year
• The alpha parameter = 0.380

Her simulation extends over a 10-year horizon. For the 60th month, the random uniform [0,1] value is 0.20 such that (via inverse transformation) the random standard normal = NORM.S.INV(0.20) = -0.8416; and dw = -0.8416*SQRT(1/12) = -0.2430. (note: the exam is unlikely to go into such detail on dw, we show this math simply to remind that dw is not a standard random normal, but rather an already-time scaled random normal)

Which is nearest to the change in the rate, dr, given by Becky's model for the 60th month, dr(5.0)?

a. -1.00%
b. 0.00%
c. +1.00%
d. +2.00%

306.2. Peter wants to utilize Model 3 with time-dependent volatility. His colleague Mary prefers the Vasicek Model in the current environment. Mary makes the following two arguments in favor of the Vasicek Model over Model 3 (Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)) :

I. "Unlike the economic intuitions that attach to mean reversion in Vasicek, Model 3's time-dependent volatility relies on the difficult argument that the market has a forecast of short-term volatility in the distant future."​

II. "The downward-sloping factor structure and term structure of volatility in mean-reverting models capture the behavior of interest rate movements better than parallel shifts and a flat term structure of volatility."​

Which of Mary's argument(s) are good, according to Tuckman?

a. Neither, Mary might consider sitting for the FRM
b. I. only
c. II. only
d. Both, Mary must be a Certified FRM!

306.3. Consider the following five-part summary of the interest rate Models in Tuckman (Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)) :

I. Model 1 is a simple equilibrium model with normally distributed (random) rates but without drift (no drift)​

II. Model 2 adds a constant drift (a.k.a., risk premium) to Model 1 such that Model 2 includes constant drift plus a normally distributed (random) rate; but is also an equilibrium model​

III. The Ho-Lee Model, similar Models 1 and 2 has a normally distributed (random) rate, but unlike them has time-dependent drift and can thusly be considered an arbitrage-free model​

IV. The Vasicek Model assumes a normally distributed (random) interest rate but assumes the rate is characterized by mean reversion toward a long-run value (central tendency)​

V. Unlike the previous Models (Models 1 and 2; Ho-Lee; Vasicek) which assume normally distributed rates, Model 3 introduces time-dependent volatility​

Which of the above are true?

a. None
b. I. and II. only
c. III. and V. only
d. All are true

Answers:

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