# P2.T5.23.4 Vasicek term structure model

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Construct and describe the effectiveness of a short-term interest rate tree assuming normally distributed rates, both with and without drift. Calculate the short-term rate change and standard deviation of the rate change using a model with normally distributed rates and no drift ... Describe the process of constructing a simple and recombining tree for a short-term rate under the Vasicek Model with mean reversion. Calculate the Vasicek Model rate change, standard deviation of the rate change, expected rate in T years, and half-life. Describe the effectiveness of the Vasicek Model.

Questions:

23.4.1. Emily is an FRM who has been asked to write a code snippet for her firm's trading unit. Her code implements a simple interest rate term structure that inputs into a trading simulation model. The immediate purpose is scenario analysis, and as this is a first iteration, they specifically want to avoid a complex term structure model. To keep the model simple, they asked Emily to select from the candidate models in Tuckman's Chapter 9 (3rd Edition):
• Model 1, aka Normally Distributed Rates and No Drift
• Model 2, aka Drift and Risk Premium
• Ho-Lee, or
• Vasicek
Among these four candidate term structure models, which be easiest to fit to (a wide variety of) observed prices?

a. Model 1
b. Model 2
c. Ho-Lee
d. Vasicek

23.4.2. An analyst fits a Vasicek model when the current short-term rate is 4.60%. If the theta, θ, parameter is 13.50%, then each of the following statements is true EXCEPT which is false?

a. A key advantage of the Vasicek model is that it is a relatively simple model
b. If the half-life is nine years, then we might expect the short rate to reach ~9.0% in nine years or so
c. The standard deviation of the short rate in ten years is less than it would be without mean reversion
d. Because it is a multi-factor model, the Vasicek is popularly applied to hedge portfolios that contain short-, medium- and long-term bonds

23.4.3. Which of the following is the best interpretation of a Vasicek model with a high mean reversion parameter denoted by (k)?

a. Economic news is short-lived
b. The half-life is a large value, aka., a long time horizon
c. The risk premium must be equal to λ/k because the model implicitly assumes the expectations component equals zero
d. If the short rates' basis point volatility, σ = V, the standard deviation of the short rate's terminal distribution in T years is greater than V*sqrt(T)