The Random Walk Model: Practice Problems and Solutions

This is Section 28 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.

Let a coin be flipped n times and let Y_i denote the outcome of the ith flip. If the coin lands heads on the ith flip, then Y_i = 1. If the coin lands tails on the ith flip, then Y_i = -1. We can obtain the sum of the Y_i's for the n flips (Z_n) as follows.

Z_n = _i=1ⁿ∑Y_i

Furthermore, to find Y_n, the outcome of the nth flip, we use the following formula:

Z_n - Z_n-1 = Y_n

If Y_n is heads: Z_n - Z_n-1 = +1

If Y_n is tails: Z_n - Z_n-1 = -1

The random walk model states that the more times we flip a coin, the likelier it is that we will be farther away from 0.

The random walk model can be applied to stock price movements as well, though the above equations do not suffice to describe such movements. The binomial model is a special case of the random walk model that also incorporates the assumption that "continuously compounded returns are a random walk." According to R. L. McDonald, there are the four properties of continuously compounded returns that the binomial model incorporates (where r = continuously compounded rate of return, S = stock price, and the subscripts denote time periods).

Logarithmic function computes returns from prices: r_t,t+h = ln(S_t+h /S_t)

Exponential function computes prices from returns: S_t+h = S_te^r_(t,t+h)

Continuously compounded returns are additive: r_t,t+nh = _i=1ⁿ∑r_{t+(i-1)h,t+ih}

Continuously compounded returns can be less than - 100%. e^r is always positive, even if r is a large negative number.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 351-353.

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

Problem RWM1. You flip a coin and get the following results: heads, heads, tails, tails, tails, heads, heads, heads, heads, tails. What is the sum Y_i's for all of these flips?

Solution RWM1. We use the formula Z_n = _i=1ⁿ∑Y_i, which in this case can be applied by adding the number of heads (1's) and the number of tails ("-1"'s) resulting from the flips. 10 flips in all were made, 6 of which were heads and 4 of which were tails. Thus, Z_n = 6(1) + 4(-1) = 6 - 4 = Z₁₀ = 2.

Problem RWM2. A coin was flipped 13 times, and you know that Z₁₃ = 6, Y₁₂ = -1, Y₁₁ = -1, Y₁₀ = 1, and Y₉ = 1. Find Z₈.

Solution RWM2. We apply the formula Z_n - Z_n-1 = Y_n multiple times to get.

Z₁₃ - Y₁₂ - Y₁₁- Y₁₀- Y₉ = Z₈ = 6 - (-1) - (-1) -1 -1 = 6 - 0 = Z₈ = 6.

Problem RWM3. The price of Lucrative Co. stock today is $567. Five years ago, it was $320. Find the continuously compounded return on Lucrative Co. stock over the past five years.

Solution RWM3. We use the formula r_t,t+h = ln(S_t+h /S_t) = ln(567/320) =

r_0,5 = 0.5720383079

Problem RWM4. The stock of Despicable Co. earns the following continuously compounded returns:
January 20034 to January 20035: -0.45

January 20035 to January 20036: -0.2

January 20036 to January 20037: 0.43

January 20037 to January 20038: 0.03

January 20038 to January 20039: 23

January 20039 to January 20040: -1.2

January 20040 to January 20041: -32

What is the continuously compounded return on the stock of Despicable Co. from January 20034 to January 20041?

Solution RWM4. We use the formula r_t,t+nh = _i=1ⁿ∑r_{t+(i-1)h,t+ih} and simply add up the returns from each year: r_20034,20041 = -0.45 - 0.2 + 0.43 + 0.03 + 23 - 1.2 - 32 =

r_20034,20041 = -10.39. (If you are alive in 20034, do not invest in this stock!)

Problem RWM5. The stock of Despicable Co. earns the following continuously compounded returns:
January 20034 to January 20035: -0.45

January 20035 to January 20036: -0.2

January 20036 to January 20037: 0.43

January 20037 to January 20038: 0.03

January 20038 to January 20039: 23

January 20039 to January 20040: -1.2

January 20040 to January 20041: -32

Hinjanmin purchases a share of Despicable Co. stock for $56900 in January 20036 and wisely sells it on January 20039. How much money does he get after selling the stock?

Solution RWM5. We note that the three-year rate of return from January 20036 to January 20039 is 0.43 + 0.03 + 23 = 23.46. Thus, using the formula S_t+h = S_te^r_(t,t+h), we get S₂₀₀₃₉ = 56900e^23.46 = $878,336,259,839,150 (making Hinjanmin a multi-trillionaire).

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