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More Exam-Style Questions On Ito's Lemma And Brownian Motion

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 69 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here. See Section 49 here. See Section 50 here. See Section 51 here. See Section 52 here. See Section 53 here. See Section 54 here. See Section 55 here. See Section 56 here. See Section 57 here. See Section 58 here. See Section 59 here. See Section 60 here. See Section 61 here. See Section 62 here. See Section 63 here. See Section 64 here. See Section 65 here. See Section 66 here. See Section 67 here. See Section 68 here.

When you are given a geometric Brownian motion {X(t)} in terms of {Z(t)} - some other Brownian motion - and asked to find the drift of this motion, or whether this motion has zero drift, the suggested procedure is to apply Ito's Lemma to find dX(t) and express dX(t) as some Kdt + JdZ(t). If K = 0, then {X(t)} has zero drift. If K is not zero, then {X(t)} has drift Kdt.

By Ito's Lemma, dX(Z, t) = XZdZ + (1/2)XZZ(dZ)2 + Xtdt, but (dZ)2 = dt by our multiplication rules from Section 67, so dX(Z, t) = XZdZ + [(1/2)XZZ + Xt]dt and

dX(t) = [(1/2)XZZ + Xt]dt + XZdZ(t)

Some of 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 MESQILBM1.

Similar to Question 13 from the Society of Actuaries' Sample MFE Questions and Solutions:

C and W are two non-dividend-paying assets, both driven by the Brownian motion {Z(t)}. The following equations hold:

dC(t)/C(t) = 0.23dt + 0.46dZ(t)

dW(t)/W(t) = Adt + BdZ(t), where A and B are constants.

You also know that

(i) d[ln(W(t)] = dt + 0.363dZ(t)

(ii) The annual continuously compounded risk-free interest rate is 0.06.

Find A.

Solution MESQILBM1. This problem involves applications of Ito's Lemma and the Sharpe Ratio.

From Section 65, we know that (1 - r)/1 = (2 - r)/2, where C is Asset 1 and W is Asset 2. It is given that r = 0.06, 1 = 0.23, 1 = 0.46, 2 = A, 2 = B. Thus, (0.23-0.06)/0.46 = (A - 0.06)/B and 0.3695652174 = (A - 0.06)/B.

We note that d(ln[W(t)]) = dt + 0.363dZ(t) must have had Ito's Lemma applied to it. We try to see what that application must have looked like. Let Q(w, t) = ln[W(t)]

By Ito's Lemma, dQ(W, t) = QWdW + (1/2)QWW(dW)2 + Qtdt.

Here, QW = (1/W(t)), QWW = (1/W(t)) = (-1/W(t)2), Qt = 0.

Thus, dQ(W, t) = (1/W(t))dW + (1/2)(-1/W(t)2)(dW)2.

But we are given that dQ(W, t) = dt + 0.363dZ(t)

and that dW(t)/W(t) = Adt + BdZ(t), so dW = W(t)

Now, we make the following substitution for dW:

dQ(W, t) = (1/W(t))W(t) )2)(W(t))2

dQ(W, t) = 2

dQ(W, t) = Adt + BdZ(t) +(-1/2)2dt2 + 2ABdtdZ(t) + B2dZ(t)2]

Here, we will need to apply our multiplication rules. The expression is thereby dramatically simplified:

dQ(W, t) = Adt + BdZ(t) +(-1/2)20 + 2AB*0 + B2dt]

dQ(W, t) = Adt + (-1/2)B2dt + BdZ(t) = dt + 0.363dZ(t)

Thus, because the dZ(t) term is both B and 0.363, we know that B = 0.363. We can use this in the equality we derived from the Sharpe ratios:

0.3695652174 = (A - 0.06)/B

0.3695652174 = (A - 0.06)/0.363

0.3695652174*0.363 + 0.06 = A = 0.1941521739

Note: The reasoning for this solution and for Solution MRESQBMIL5 in Section 67 can save you a lot of intermediate work when encountering problems of this type in the future. Here is a conclusion that we can draw:

Rule 69.1. Whenever we are given

dW(t)/W(t) = Adt + BdZ(t), where A and B are constants.

and d[ln(W(t)] = dt + dZ(t), it is the case that B = . When the value of is given, this is especially convenient, because - when we are also given r and some specific Brownian motion for another asset C, we can use Sharpe ratios, knowing the expression for 2, to arrive at the value of A without needing to apply Ito's Lemma every time. Let us try this approach in the next problem.

Problem MESQILBM2.

Similar to Question 13 from the Society of Actuaries' Sample MFE Questions and Solutions:

X and Y are two non-dividend-paying assets, both driven by the Brownian motion {Z(t)}. The following equations hold:

dX(t)/X(t) = 0.333dt + 0.366dZ(t)

dY(t)/Y(t) = Adt + BdZ(t), where A and B are constants.

You also know that

(i) d[ln(Y(t)] = dt + 0.998dZ(t)

(ii) The annual continuously compounded risk-free interest rate is 0.22.

Find A.

Solution MESQILBM2. By Rule 69.1, we know that B = 0.998.

Now we use Sharpe Ratios.

From Section 65, we know that (1 - r)/1 = (2 - r)/2, where X is Asset 1 and Y is Asset 2. It is given that r = 0.22, 1 = 0.333, 1 = 0.366, 2 = A, 2 = 0.998.

Thus, 2(1 - r)/1 = (2 - r) and 2 = 2(1 - r)/1 + r,

so 2 = A = 0.998(0.333 - 0.22)/0.366 + 0.22 = A = 0.5281256831

Problems 3-5 will prepare you to answer Question 14 of theSociety of Actuaries' Sample MFE Questions and Solutions.

Problem MESQILBM3. Given that {Z(t)} is some Brownian motion, you know that X(t) = 19Z(t) + 96. Find the drift of the Brownian process {X(t)}.

Solution MESQILBM3. We use the form of Ito's Lemma developed in this section to find

dX(t) = [(1/2)XZZ + Xt]dt + XZdZ(t)

Here, XZ = 19, XZZ = 0, and Xt = 0. Thus,

dX(t) = [0XZZ + 0]dt + 19dZ(t), so dX(t) = 19dZ(t), and the dt term is 0.

Thus, {X(t)} has a drift of 0.

Problem MESQILBM4. Given that {Z(t)} is some Brownian motion, you know that Y(t) = 14[Z(t)2] - 8t Find the drift of the Brownian process {Y(t)}.

Solution MESQILBM4. We use the form of Ito's Lemma developed in this section to find

dY(t) = [(1/2)YZZ + Yt]dt + YZdZ(t)

Here, YZ = 28Z(t), YZZ = 28, and Yt = -8. Thus,

dY(t) = [(1/2)28 - 8]dt + 28Z(t)dZ(t) and so the dt term is [(1/2)28 - 8]dt = 6dt, and the drift of {Y(t)} is 6dt.

Problem MESQILBM5. Given that {F(t)} is some Brownian motion, you know that

F(t) = 4t2Z(t) - 100tsZ(s)ds. Find the drift of the Brownian process {F(t)}.

Solution MESQILBM5.

The differential of F(t) can be expressed as the differential of 4t2Z(t) - 100tsZ(s)ds or, by the addition rule for derivatives, as d[4t2Z(t)] - d[100tsZ(s)ds]. Thus,

dF(t) = d[4t2Z(t)] - d[100tsZ(s)ds]. We try to find

d[4t2Z(t)] by Ito's Lemma. Let 4t2Z(t) = Q(t)

Then dQ(t) = [(1/2)QZZ + Qt]dt + QZdZ(t), where

QZ = 4t2, QZZ = 0, and Qt = 8tZ(t)

Thus, dQ(t) = 8tZ(t)dt + 4t2dZ(t)

Now we try to find d[100tsZ(s)ds]. This is the differential of an integral in terms of t, which is simply 10tZ(t)dt. Thus,

dF(t) = d[4t2Z(t)] - d[100tsZ(s)ds] = 4t2dZ(t) + 8tZ(t)dt - 10tZ(t)dt and

dF(t) = 4t2dZ(t) - 2tZ(t)dt. Thus, the drift of {F(t)} is -2tZ(t)dt.

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

By G. Stolyarov II - G. Stolyarov II is a science fiction novelist, independent essayist, poet, amateur mathematician, composer, author, and actuary.

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