Exam-Style Questions on the Vasicek Interest Rate Model

In this section, we will work through multiple instances of one particular highly involved exam-style question regarding the Vasicek interest rate model. This is one of the only examples of prior test questions available, and mastering it will get you a long way toward being able to apply the model.

The recommended way of doing these problems is to see the way either Problem ESQVIRM1 or the exam question on which it is based are solved. Then, using the same methods, try to solve the subsequent problems on your own and check your work. Going through the same procedure multiple times should render it sufficiently firm in your mind.

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem ESQVIRM1.

Similar to Question 16 from the Society of Actuaries' Sample MFE Questions and Solutions:

In a particular Vasicek model, the short-term interest rate is modeled by the following Brownian process: dr(t) = 0.77[b - r(t)]dt + σdZ(t).

For t ≤ T, we can let P[r, t, T] be the price at time t of a zero-coupon bond paying $1 at time T.

The short-term interest rate at time t is r.

The price of every zero-coupon bond in the Vasicek model behaves according to this Ito process:

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You know that α(0.11, 5, 9) = 0.1878

Find α(0.15, 10, 19).

Solution ESQVIRM1. In the Vasicek model, the Sharpe ratio φ is assumed to be constant.

We note that in the geometric Brownian motion

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), α is the drift factor and q is the volatility factor. So the Sharpe ratio can be expressed as

φ = (α[r(t), t, T] - r)/q[r(t), t, T]

In our case, we can express φ as

φ = (α[0.11, 5, 9] - 0.11)/q[0.11, 5, 9] = (α[0.15, 10, 19] - 0.15)/q[0.15, 10, 19]

Furthermore, since α(0.11, 5, 9) = 0.1878,

(0.1878 - 0.11)/q[0.11, 5, 9] = (α[0.15, 10, 19] - 0.15)/q[0.15, 10, 19]

0.0778/q[0.11, 5, 9] = (α[0.15, 10, 19] - 0.15)/q[0.15, 10, 19]

Now we try to find an expression for q[r(t), t, T].

By Ito's Lemma,

dP[r(t), t, T] = P_rdr + (1/2)P_rr(dr)² + P_tdt

For a general Vasicek model, dr(t) = a[b - r(t)]dt + σdZ(t). Thus,

dP[r(t), t, T] = P_rrr(² + P_tdt

dP[r(t), t, T] = a[b - r(t)]P_rdt + P_rσdZ(t) + (1/2)P_rr(σ²)dt + P_tdt by the multiplication rules in Section 67.

dP[r(t), t, T] = (a[b - r(t)]P_r + (1/2)P_rr(σ²) + P_t)dt + P_rσdZ(t)

Fortunately, it is the coefficient of dZ(t) that concerns us here.

q[r(t), t, T] is the coefficient of dZ(t) in the expression for dP[r(t), t, T]/P[r(t), t, T]. To get the value of q[r(t), t, T] using P_rσ, we simply need to divide both sides of the above equation by P[r(t), t, T] in order to get

dP[r(t), t, T]/P[r(t), t, T] = (a[b - r(t)]P_r + (1/2)P_rr(σ²) + P_t)dt/P[r(t), t, T] + (P_rσ/P[r(t), t, T])dZ(t)

Thus, we have q[r(t), t, T] = P_rσ/P[r(t), t, T]

P_r/P[r(t), t, T] is the same as d[lnP[r(t), t, T]]/dr. Try to differentiate the latter expression with respect to r, and you will see that the results are the same.

We recall from Section 70 that in the Vasicek model, P[t, T, r(t)] = A(t, T)e^{-B(t, T)r(t)}.

So lnP[r(t), t, T] = ln^{-B(t, T)r(t)}] = ln

Since, with respect to r, ln

d[lnP[r(t), t, T]]/dr = - B(t, T)

Therefore, q[r(t), t, T] = P_rσ/P[r(t), t, T] = - σB(t, T)

We recall from Section 70 that in the Vasicek model, B(t, T) = (1 - e^-a(T-t))/a

Thus, - σB(t, T) = -σ(1 - e^-a(T-t))/a = q[r(t), t, T]

Please do not go through this derivation every time you solve a problem of this sort! It was intended to help you learn why the answer we will arrive at is correct; it was not intended to be replicated during the exam. Unless you enjoy pain and suffering, you should do with this result the same thing you do with results like the solution for the indefinite integral of sec³(θ)dθ: memorize it! (More precisely, memorize one of the equations 71.1 or 71.2 later in this section.)

In the general case, q[r(t), t, T] = -σ(1 - e^-a(T-t))/a

Because σ and a are the same for all values of t and T, they will cancel out in our equation for the Sharpe ratio. The negative sign will also drop out.

We examine our equation derived above:

0.0778/q[0.11, 5, 9] = (α[0.15, 10, 19] - 0.15)/q[0.15, 10, 19]

0.0778/q[0.11, 5, 9] = (α[0.15, 10, 19] - 0.15)/q[0.15, 10, 19]

We can put the appropriate 1 - e^-a(T-t) in the denominator of each side of the equation.

For the left side, since T = 9 and t = 5, 1 - e^-a(T-t) = 1 - e^-4a

For the left side, since T = 19 and t = 10, 1 - e^-a(T-t) = 1 - e^-9a

Thus, 0.0778/(1 - e^-4a) = (α[0.15, 10, 19] - 0.15)/(1 - e^-9a)

Here, a = 0.77

Thus, 0.0778/(1 - e^-4*0.77) = (α[0.15, 10, 19] - 0.15)/(1 - e^-9*0.77)

0.0778(1 - e^-9*0.77)/(1 - e^-4*0.77) + 0.15 = α[0.15, 10, 19] = 0.231468126

To save you time and mental anguish, here is a procedure for solving problems where you have the following short-term interest rate process in the Vasicek model:

dr(t) = a[b - r(t)]dt + σdZ(t) and

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You are given some α(r₁, t₁, T₁) and are asked to find α(r₂, t₂, T₂).

All you need to do is to solve the following equation:

Equation 71.1

[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) = [α(r₂, t₂, T₂) - r₂]/(1 - exp[-a(T₂ - t₂)])

Better yet,

Equation 71.2

α(r₂, t₂, T₂) = (1 - exp[-a(T₂ - t₂)])[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) + r₂

Now you just need to substitute in the relevant values.

Now that we have learned how to save time, let us proceed to the other four problems of this study guide.

Problem ESQVIRM2.

Similar to Question 16 from the Society of Actuaries' Sample MFE Questions and Solutions:

In a particular Vasicek model, the short-term interest rate is modeled by the following Brownian process: dr(t) = 0.34[b - r(t)]dt + σdZ(t).

For t ≤ T, we can let P[r, t, T] be the price at time t of a zero-coupon bond paying $1 at time T.

The short-term interest rate at time t is r.

The price of every zero-coupon bond in the Vasicek model behaves according to this Ito process:

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You know that α(0.01, 4, 11) = 0.0333

Find α(0.04, 3, 8).

Solution ESQVIRM2.

We use Equation 71.2:

α(r₂, t₂, T₂) = (1 - exp[-a(T₂ - t₂)])[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) + r₂

Here, a = 0.34, t₁ = 4, T₁ = 11, r₁ = 0.01, t₂ = 3, T₂ = 8, r₂ = 0.04.

α(0.04, 3, 8) = (1 - exp[-0.34(8 - 3)])[0.0333 - 0.01]/(1 - exp[-0.34(11 - 4)]) + 0.04

α(0.04, 3, 8) = (1 - exp[-0.34*5])[0.0233]/(1 - exp[-0.34*7]) + 0.04

α(0.04, 3, 8) = 0.0609857138

Problem ESQVIRM3.

Similar to Question 16 from the Society of Actuaries' Sample MFE Questions and Solutions:

In a particular Vasicek model, the short-term interest rate is modeled by the following Brownian process: dr(t) = 0.29[b - r(t)]dt + σdZ(t).

For t ≤ T, we can let P[r, t, T] be the price at time t of a zero-coupon bond paying $1 at time T.

The short-term interest rate at time t is r.

The price of every zero-coupon bond in the Vasicek model behaves according to this Ito process:

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You know that α(0.31, 122, 334) = 0.44

Find α(0.1, 4, 15).

Solution ESQVIRM3.

We use Equation 71.2:

α(r₂, t₂, T₂) = (1 - exp[-a(T₂ - t₂)])[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) + r₂

Here, a = 0.29, t₁ = 122, T₁ = 334, r₁ = 0.31, t₂ = 4, T₂ = 15, r₂ = 0.1.

α(0.1, 4, 15) = (1 - exp[-0.29(15 - 4)])[0.44 - 0.31]/(1 - exp[-0.29(334 - 122)]) + 0.1

α(0.1, 4, 15) = (1 - exp[-0.29*11])[0.13]/(1 - exp[-0.29*212]) + 0.1

α(0.1, 4, 15) = 0.2246476568

Problem ESQVIRM4.

Similar to Question 16 from the Society of Actuaries' Sample MFE Questions and Solutions:

In a particular Vasicek model, the short-term interest rate is modeled by the following Brownian process: dr(t) = 0.86[b - r(t)]dt + σdZ(t).

For t ≤ T, we can let P[r, t, T] be the price at time t of a zero-coupon bond paying $1 at time T.

The short-term interest rate at time t is r.

The price of every zero-coupon bond in the Vasicek model behaves according to this Ito process:

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You know that α(0.04, 0, 15) = 0.09

Find α(0.04, 10, 15).

Solution ESQVIRM4.

We use Equation 71.2:

α(r₂, t₂, T₂) = (1 - exp[-a(T₂ - t₂)])[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) + r₂

Here, a = 0.86, t₁ = 0, T₁ = 15, r₁ = 0.04, t₂ = 10, T₂ = 15, r₂ = 0.04.

α(0.04, 10, 15) = (1 - exp[-0.86(15 - 10)])[0.09 - 0.04]/(1 - exp[-0.86(15 - 0)]) + 0.04

α(0.04, 10, 15) = (1 - exp[-0.86*5])[0.05]/(1 - exp[-0.86*15]) + 0.04

α(0.04, 10, 15)= 0.0893216953

Problem ESQVIRM5.

Similar to Question 16 from the Society of Actuaries' Sample MFE Questions and Solutions:

In a particular Vasicek model, the short-term interest rate is modeled by the following Brownian process: dr(t) = 0.12[b - r(t)]dt + σdZ(t).

For t ≤ T, we can let P[r, t, T] be the price at time t of a zero-coupon bond paying $1 at time T.

The short-term interest rate at time t is r.

The price of every zero-coupon bond in the Vasicek model behaves according to this Ito process:

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t), for t ≤ T.

You know that α(0.19, 0, 30) = 0.23

Find α(0.29, 0, 30).

Solution ESQVIRM5.

We use Equation 71.2:

α(r₂, t₂, T₂) = (1 - exp[-a(T₂ - t₂)])[α(r₁, t₁, T₁) - r₁]/(1 - exp[-a(T₁ - t₁)]) + r₂

Here, a = 0.12, t₁ = 0, T₁ = 30, r₁ = 0.19, t₂ = 0, T₂ = 30, r₂ = 0.29.

α(0.29, 0, 30) = (1 - exp[-0.12(30 - 0)])[0.23 - 0.19]/(1 - exp[-0.12(30 - 0)]) + 0.29

α(0.29, 0, 30) = 0.04 + 0.29 = α(0.29, 0, 30) = 0.33

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.