The Random Walk Model: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 28

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 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.