Standard Deviation of Returns and Multi-Period Probabilities in the Binomial Model: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 29

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 29 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.

The standard deviation of returns on an asset over time period of length h can be expressed as

σ_h = σ√(h), where σ is the standard deviation over a time period of length 1.

Using the binomial model, stock prices can be modeled as follows:

S_t+h = S_te^{(r- δ)h ± σ√(h)}.

When we take the natural logs of both sides, we get ln(S_t+h/S_t) = (r- δ)h ± σ√(h).

The binomial model is an approximation of the lognormal distribution. According to R. L. McDonald, "The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed."

In a binomial tree, where the risk-neutral probability is p*, the probability of reaching the ith node can be expressed as

p_{ith node} = [n!/((n-i)!i!)](p*)^n-1(1-p*)ⁱ = C(n, i)(p*)^n-1(1-p*)ⁱ

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 354-358.

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

Problem SDRMPPBM1. The standard deviation of returns on Frivolous Co. stock over 1 year is 0.67. Find the standard deviation of returns on Frivolous Co. stock over 12 years.

Solution SDRMPPBM1. We use the formula σ_h = σ√(h), where h = 12, and σ = 0.67. Thus, σ₁₂ = 0.67√(12) = σ₁₂ = 2.320948082