The Vasicek and the Hull-White Extension of the Vasicek Model)

PAK Exam Aid Quantitative Finance and Investment Advanced (QFIA) Exam Fall 2015 Edition

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Mock Question Sample

This question pertains to SOA QFIA Objective 1. Each question should be answered independently.

6. (9 points) You are given the following dynamics for the short rate model of interest rate:

The Vasicek process:

The Hull White process:

The price at time t of a zero coupon bond paying $1 at time T is denoted by:

Where we assume the functional form:

The instantaneous forward rate at time t, for a maturity of time T is given by:

You are asked to:

(a) (2 points) Describe the features of the HW model and compare it to the Vasicek model.

For the HW model:

(b) (2 points) Calculate the expression ( )

For both models:

(d) (1 point) Compare the expression for B(t,T).

(e) (2 points) Calculate the volatility of the instantaneous forward rate.

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6. Materials from Interest Rate Models Theory and Practice (The Vasicek and the Hull-White extension of the Vasicek model)

“Note: The notations in the SDE for the short rates have been twisted. Don’t fall on this trap. Whenever, you are given a model, don’t always just go by the nomenclature of the model (Vasicek, CIR,…) and jump to copy/paste the formula from the formula sheet. Always ensure that the dynamics of the equation for the model (mean reversion level and speed, and volatility for short rate) are exactly as given out in the official reading before using the formula in the formula sheet, otherwise, you will need to make a change of variable to use the formula sheet”

(a) Brief comparison of the Vasicek and the Hull-White Model Both models are mean reverting, one factor and normal processes with a constant variance for the short interest rate. The Vasicek is an equilibrium model and the Hull-White is an arbitrage free model. The HW can fit the initial term structure of interest rate and the Vasicek model cannot. The HW model is able to fit a given term structure of volatility, and the Vasicek model cannot. Both models suffer from the possibility of negative interest rates (since they are normal models). Both models are ATS models. Under the models, the entire forward interest curve can be explicitly derived. Both are analytically tractable, and P(t,T) “The price of a ZCB” can be calculated explicitly and analytically without resort to complex MC methods (as in the more advanced LMM models).

(b)

Hull White model We take the integral of both sides of the expression of question (b):

Thus:

Divide both sides by ( ) to get:

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For the Vasicek model: We simply use the fact that and we get the solution to the Vasicek SDE:

Now, if you make a proper change of variable, you can recover the formula on the formula sheet: (3.6) on page 2.

More solutions are available in the PAK Exam Aid.

Note Please note that the mock questions in the PAK Exam Aid are all different from the mock questions in the PAK Test Aid.