Historical Volatility: Practice Problems and Solutions

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

Historical volatility calculations are likely to appear on Exam MFE. Here, a systematic method for finding the historical volatility of a given set of stock price data will be given.

Let S₁, S₂, … S_n+1 be the prices of a stock at times 1 through n+1, with each time period being of length h.

Then the non-annualized continuously compounded return r_j for the jth time period (where j can be anything from 1 to n) are as follows: r_j = ln(S_j+1/S_j)

Then the variance of all of these non-annualized continuously compounded returns r₁ through r_n is σ_h² = (1/(n-1))_j=1ⁿΣ(r_j – m)², where m is the mean of all of the returns r₁ through r_n and σ_h is the standard deviation of returns for a time period of length h.

The annualized historical volatility σ of these returns is then σ = σ_h/√(h)

As you might suspect, the calculation of historical volatility is a step-by-step process where it is important to do the steps in order. The first four problems of this section will illustrate the four steps involved in calculating historical volatility, after which you will be ready to attempt an exam-style question.

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 HV1. The monthly prices S of a certain stock are given as follows:

Month 1: S = 654

Month 2: S = 456

Month 3: S = 679

Month 4: S = 524

Month 5: S = 996

Month 6: S = 1000

Find all of the five non-annualized continuously compounded monthly returns on the stock, from r₁ through r₅.

Solution HV1. We use the following formula: r_j = ln(S_j+1/S_j)

r₁ = ln(S₂/S₁) = ln(456/654) = r₁ = -0.3606145419

r₂ = ln(S₃/S₂) = ln(679/456) = r₂ = 0.398128318

r₃ = ln(S₄/S₃) = ln(524/679) = r₃ = -0.2591294432

r₄ = ln(S₅/S₄) = ln(996/524) = r₄ = 0.6422555733

r₅ = ln(S₆/S₅) = ln(1000/996) = r₅ = 0.0040080214

Problem HV2. The monthly prices S of a certain stock are given as follows:

Month 1: S = 654

Month 2: S = 456

Month 3: S = 679

Month 4: S = 524

Month 5: S = 996

Month 6: S = 1000

Find the mean m of the non-annualized continuously compounded monthly returns on the stock, from r₁ through r₅.

Solution HV2. We already found r₁ through r₅ in Solution HV1. Now we take their arithmetic average: (-0.3606145419 + 0.398128318 – 0.2591294432 + 0.6422555733 + 0.0040080214)/5 =
m = 0.0849295855

Problem HV3. The monthly prices S of a certain stock are given as follows:

Month 1: S = 654

Month 2: S = 456

Month 3: S = 679

Month 4: S = 524

Month 5: S = 996

Month 6: S = 1000

Find the variance of the non-annualized continuously compounded monthly returns on the stock, from r₁ through r₅.

Solution HV3. We use the formula σ_h² = (1/(n-1))_j=1ⁿΣ(r_j – m)². Here, n = 5, m = 0.0849295855 from Solution HV2, and the r_j are known from Solution HV1. Thus,

σ_h² = (1/4)_j=1⁵Σ(r_j – 0.0849295855)² =

(1/4)[(r₁ – 0.0849295855)² + (r₂ – 0.0849295855)² + (r₃ – 0.0849295855)² + (r₄ – 0.0849295855)² + (r₅ – 0.0849295855)²] =

(1/4)[(-0.3606145419 – 0.0849295855)² + (0.398128318 – 0.0849295855)² + (-0.2591294432 – 0.0849295855)² + (0.6422555733 – 0.0849295855)² + (0.0040080214 – 0.0849295855)²] =

σ_h² = 0.1830350467

Problem HV4. The monthly prices S of a certain stock are given as follows:

Month 1: S = 654

Month 2: S = 456

Month 3: S = 679

Month 4: S = 524

Month 5: S = 996

Month 6: S = 1000

Find the annualized historical volatility of the returns on this stock over the given time period.

Solution HV4. We use the formula σ = σ_h/√(h). From Solution HV3, σ_h² = 0.1830350467, so

σ_h = 0.4278259538. Here, since each time period is one month, h = 1/12. Thus, the historical volatility σ = 0.4278259538/√(1/12) = σ = 1.482032578

Problem HV5.

Similar to Question 17 from the Society of Actuaries’ Sample MFE Questions and Solutions:

The monthly prices S of a certain stock are given as follows:

Month 1: S = 135

Month 2: S = 90

Month 3: S = 135

Month 4: S = 90

Month 5: S = 60

Month 6: S = 90

Month 7: S = 135

Find the annualized historical volatility of the returns on this stock over the given time period.

Solution HV5. First, we attempt to find the monthly continuously compounded returns r₁ through r₆, using the formula r_j = ln(S_j+1/S_j).

Conveniently enough, 90/135 = 2/3 and 135/90 = 3/2. Similarly, 60/90 = 2/3 and 90/60 = 3/2.

Thus, each of the r_j is either ln(3/2) or ln(2/3) = -ln(3/2). Furthermore, we note that the stock went up in price three times over this time period and went down in price the other three times. Thus, the average return on this stock m is equal to [3ln(3/2) + 3(-ln(3/2))]/6 = 0.

Now we use the formula σ_h² = (1/(n-1))_j=1ⁿΣ(r_j – m)² to find the variance of these returns. Here, m = 0, n = 6, and each (r_j)² is [-ln(3/2)]² or [ln(3/2)]², both of which are 0.1644019539.

Thus, σ_h² = (1/5)_j=1⁶Σ(0.1644019539) = (1/5)*6*0.1644019539 = σ_h² = 0.1972823447, so

σ_h = 0.444164772 and σ = σ_h/√(h). Here, h = 1/12, since each time period is a month.

Thus, σ = 0.444164772/√(1/12) = σ = 1.538631904

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