**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 83 of the Study Guide. See an index of all sections by following the link in this paragraph.**

If S is the price of some underlying asset and Sa is the price of some derivative based on that claim, then it is possible to use Ito’s Lemma to value a claim on Sa, where a is some number. Hereafter, we will call Sa the ** power derivative**of S. (This term is my own invention, and I will use it for the sake of word economy and the ability to concisely express ideas regarding these kinds of derivatives.) Valuation of power derivatives is a new addition to the MFE syllabus (as of Spring 2008) and thus is likely to appear in at least one question.

Let us say we have an asset whose price S follows this geometric Brownian motion.

dS(t)/S(t) = (α – δ)dt + σdZ(t)

Here, α is the expected return on the asset, δ is the asset’s dividend yield, and σ is the asset price volatility.

Then the price of the power derivative, Sa, follows this geometric Brownian motion.

d(Sa)/Sa = (a(α – δ) +(1/2)a(a-1)σ2)dt + aσdZ(t)

It is important to note that *the power derivative will often have a different expected return and a different dividend yield than the underlying asset*! We can call the expected return on the power derivative γ and the dividend yield on the power derivative δ*. We can find these values via the following formulas.

γ = a(α – r) + r, where r is the annual continuously compounded risk-free interest rate.

δ* = r – a(r – δ) – 0.5a(a-1)σ2

We can also neatly express both the forward price F0,T[S(T)a] and the prepaid forward price FP0,T[S(T)a] of a power derivative:

F0,T[S(T)a] = S(0)aexp[(a(r – δ) + 0.5a(a-1)σ2)T]

FP0,T[S(T)a] = e-rTS(0)aexp[(a(r – δ) + 0.5a(a-1)σ2)T]

Note that the only difference between the forward price and the prepaid forward price is that the prepaid forward price has an additional factor of e-rT.

In this section, we will work through practice problems to help you memorize these formulas, and then we will undertake an exam-style question.

**Source:** Actuarial Brew. “Chapter 20 Review Note. Brownian Motion and Ito’s Lemma. Section 20.7, Valuing a Claim on Sa.”

Actuarial Brew also has five excellent practice problems in the aforementioned *freely available* review note – as well as free solutions for all five problems. You are strongly advised to practice on those problems in addition to the five in this section.

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 VCDWPUAPTSP1.** The price S of the stock of Obsequious Co. follows this Brownian motion: dS(t)/S(t) = 0.34dt + 0.23dZ(t)

The stock’s annual continuously compounded dividend yield is 0.05, and the annual continuously compounded risk-free interest rate is 0.13.

Calculate the annual expected return on S5, a power derivative based on this stock.

**Solution VCDWPUAPTSP1.** We use the formula γ = a(α – r) + r. We are given a = 5 and r = 0.13, but what is α? We recall that, in a geometric Brownian motion like the one given, dS(t)/S(t) = (α – δ)dt + σdZ(t). Therefore, (α – δ) = 0.34. Since it is given that δ = 0.05, we find that α = 0.34 + 0.05 = 0.39. Therefore,

γ = 5(0.39 – 0.13) + 0.13 = **γ = 1.43** (Yes, that is an expected return of 143%.)

**Problem VCDWPUAPTSP2.** The price S of the stock of Obsequious Co. follows this Brownian motion: dS(t)/S(t) = 0.34dt + 0.23dZ(t)

The stock’s annual continuously compounded dividend yield is 0.05, and the annual continuously compounded risk-free interest rate is 0.13.

Calculate the annual continuously compounded dividend yield of S5, a power derivative based on this stock.

**Solution VCDWPUAPTSP2.** We use the formula δ* = r – a(r – δ) – 0.5a(a-1)σ2, where a = 5, r = 0.13, δ = 0.05, and σ = 0.23, from the given Brownian motion.

Thus, δ* = 0.13 – 5(0.13 – 0.05) – 0.5*5*4*0.232 = **δ* = -0.799**

Yes, this is a dividend yield of -79.9%. Negative dividend yields on power derivatives will occur frequently when a > 1. The general idea for why this happens is that the expected capital gains on the power derivative are so large (as in Solution VCDWPUAPTSP1, where the expected *annual* return is 143%) that the owner of the derivative has to pay out an effective dividend for the privilege of enjoying such high returns in the future. The party that sold him the derivative would have rather held on to it if it did not receive some form of compensation in terms of dividend payouts.

**Problem VCDWPUAPTSP3.** The price S of the stock of Obsequious Co. follows this Brownian motion: dS(t)/S(t) = 0.34dt + 0.23dZ(t)

The stock’s annual continuously compounded dividend yield is 0.05, and the annual continuously compounded risk-free interest rate is 0.13.

Calculate the forward price of a 4-year forward contract on S5, a power derivative based on this stock. The current stock price is $3 per share.

**Solution VCDWPUAPTSP3.**

We use the formula F0,T[S(T)a] = S(0)aexp[(a(r – δ) + 0.5a(a-1)σ2)T], where

a = 5, r = 0.13, δ = 0.05, T = 4, S(0) = 3, and σ = 0.23.

Thus, F0,4[S(T)5] = 35exp[(5(0.13 – 0.05) + 0.5*5*4*0.232)4] =

F0,4[S(T)5] = 243*exp(3.716) = **F0,4[S(T)5] = $9987.218888**

**Problem VCDWPUAPTSP4.** The price S of the stock of Lucrative Co. follows this Brownian motion: dS(t)/S(t) = 0.24dt + 0.431dZ(t)

The stock’s annual continuously compounded dividend yield is 0.08, and the annual continuously compounded risk-free interest rate is 0.22.

Calculate the price of a 10-year prepaid forward contract on S-3, a power derivative based on this stock. The current stock price is $3.45 per share.

**Solution VCDWPUAPTSP4.** We use the formula

FP0,T[S(T)a] = e-rTS(0)aexp[(a(r – δ) + 0.5a(a-1)σ2)T], where

a = -3, r = 0.22, δ = 0.08, T = 10, S(0) = 3.45, and σ = 0.431.

Thus, FP0,10[S(T)-3] = e-0.22*10*3.45-3exp[(-3(0.22 – 0.08) + 0.5(-3)(-4)*0.4312)10] =

e-2.2*3.45-3exp[6.94566] = FP0,10[S(T)-3] = 115.0837357*3.45-3 =

**FP0,10[S(T)-3] = $2.802571271**

**Problem VCDWPUAPTSP5.**

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

The price of Imperious LLC stock obeys the Black-Scholes framework. The stock pays no dividends. The stock price volatility is 0.46, and the annual continuously-compounded risk-free interest rate is 0.06.

Let S(t) be the price of Imperious LLC stock at some time t ≥ 0.

For time T such that T > t, the power derivative S(T)x, where x is some power, has a prepaid forward price such that the following equality holds:

FP0,T[S(T)x] = S(t)x. There are two solutions to this equation, one of which is x = 1.What is the other solution? Fractional or irrational-number values of x are entirely possible.

**Solution VCDWPUAPTSP5.**

We use the formula

FP0,T[S(T)a] = e-rTS(0)aexp[(a(r – δ) + 0.5a(a-1)σ2)T]. Here, T = T, a = x, r = 0.06, σ = 0.46, and δ = 0, since the stock pays no dividends.

Thus, FP0,T[S(T)x] = e-0.06TS(0)xexp[(x*0.06 + 0.5x(x-1)*0.462)T] =

e-0.06TS(0)xexp[(x*0.06 + 0.1058x(x-1))T]

We know that the following equation holds:

e-0.06TS(0)xexp[(x*0.06 + 0.1058x(x-1))T] = S(t)x.

Fortunately, we know that 1 is a solution for x. Thus, it is true that

e-0.06TS(0)1exp[(1*0.06 + 0.1058*1(1-1))T] = S(t)1 and

e-0.06TS(0)e0.06T = S(t), so S(0) = S(t) and t = 0.

Thus, the following general equation holds:

e-0.06TS(0)xexp[(x*0.06 + 0.1058x(x-1))T] = S(0)x, in which case we can cancel S(0)x on both sides:

e-0.06Texp[(x*0.06 + 0.1058x(x-1))T] = 1 and

exp[(x*0.06 + 0.1058x(x-1))T] = exp[0.06T], so

(x*0.06 + 0.1058x(x-1))T = 0.06T and (x*0.06 + 0.1058x(x-1)) = 0.06.

We expand this expression and get it into the form of a quadratic equation:

0.1058×2 – 0.0458x – 0.06 = 0

Solving this equation, we get x = 1 or x = -300/529. Thus, our desired answer is

**x = -300/529****= -0.5671077505**

If you desire additional practice with valuing claims on power derivatives, remember to take a look at the study note for this material and the five free practice problems available on the website of Actuarial Brew.

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