**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 45 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. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here.**

A **calendar spread** involves selling a call option and buying a call option with the same strike price but a greater time to expiration. A purchased calendar spread takes advantage of time decay and the fact that the sold option will lose its value faster than the purchased option. Once the shorter-lived option expires, the buyer of the spread can make the most profit if the stock price remains unchanged. (Assuming, initially, that the spread consisted of options whose strike price was equal to the stock price when the spread was entered into.)

**Implied volatility** is “the volatility that would explain the observed option price.” That is,

Market Option Price = C(S, K, σimplied, r, T, ∂). So we assume that the option price can be modeled by the Black-Scholes formula, and the other variables – stock price (S), strike price (K), annual continuously compounded risk-free interest rate (r), time to expiration (T), and annual continuously compounded dividend yield (∂) are known. Using these assumptions, we try to find σimplied. There is no way to solve directly for implied volatility in the Black-Scholes formula.

When using the Black-Scholes or the binomial model, it is possible to confine implied volatility within particular boundaries by calculating option prices for two different implied volatilities. If one of these prices is greater than the observed option price and the other is less than the observed option price, then we know that the implied volatility is somewhere between the two values for which calculations were made.

On the actuarial exam, you will be given several ranges within which implied volatility might fall. Test the extreme values of those ranges and see if the observed option price falls somewhere in between the prices calculated by considering each of these extreme values. If it does, then you have obtained the correct value of implied volatility.

**Volatility skew** is a “systematic change in implied volatility across strike prices.”

For European options, calls and puts with the same time to expiration and strike price must have the same implied volatility.

Source: McDonald, R.L., *Derivatives Markets* (Second Edition), Addison Wesley, 2006, Ch. 12,

p. 398-402.

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 CSIV1.** The current price of Stable Co. stock is $60. You purchase a calendar spread on this stock by selling one $60-strike call option with time to expiration of 2 months and θ = -0.05. The premium on this option is $3.45. You also purchase a $60-strike all option with time to expiration of 1 year at θ = -0.04. The premium on this option is $18.88. The annual continuously compounded risk-free interest rate is 0.06. Find your profit 2 months from now, once the sold option has expired, if the stock price remains at $60 and nothing else has changed. Assume that there are 30 days in every month.

**Solution CSIV1.** Two months from now, at a stock price of $60, the written call option will be at-the-money and so will not be exercised. You keep the entire premium on the option – i.e., $3.45, and this value could have accumulated interest over two months. Thus, your profit from selling the 2-month option is 3.45e0.06/6 = 3.484673076.

You also lose some money on your purchased 1-year option because of time decay. Due to time decay, the option price changes by 60θ = 60(-0.04) = -2.4

So your net gain from entering into this spread is 3.484673076 -2.4 = **$1.084673076**

**Problem CSIV2.** You own a calendar spread on a the stock of Ludicrous Co, which you bought when the stock was priced at $22. The spread consists of a written call option with a strike price of $22 and a longer-lived purchased call option with a strike price of $22. Upon the expiration of the shorter-lived option, at which stock price will you make the most money on the calendar spread?

(a) $3

(b) $12

(c) $22

(d) $23

(e) $33

**Solution CSIV2.** The calendar spread will make you the most money if the stock price is identical to the strike price of the options, so the correct answer is **(c): $22.**

**Problem CSIV3.** The stock of Gregarious LLC currently costs $555 per share. In one year, it might increase to $569. The annual continuously compounded risk-free interest rate is 0.02, and the stock pays dividends at an annual continuously compounded yield of 0.01. Find the implied volatility of this stock using a one-period binomial model.

**Solution CSIV3.** In the one-period binomial model, u = e(r-∂)h + σ√(h) = e(0.02 – 0.01)1 + σ√(1) . We know that u = 569/555 = 1.025225225 = e0.01 + σ. Thus, σimplied = ln(1.025225225) – 0.01 =

**σimplied = 0.0149123204**

**Problem CSIV4.** The stock of Imperious LLC currently costs $228 per share. The annual continuously compounded risk-free interest rate is 0.11, and the stock pays dividends at an annual continuously compounded yield of 0.03. The stock price will be $281 next year if it increase. What will the stock price be next year if it decreases? Use a one-period binomial model.

**Solution CSIV4.** In the one-period binomial model, u = e(r-∂)h + σ√(h) = e(0.11 – 0.03)1 + σ√(1) =

e0.08+ σ = 281/228, so σimplied = ln(281/228) – 0.08 = σimplied = 0.1290090404

Thus, d = e(r-∂)h – σ√(h) = e0.08 – 0.1290090404 = d = 0.9521725217.

If the stock price decreases, it will be 228*0.9521725217 = **$217.0953349**

**Problem CSIV5.**

**Similar to Question 23 from the Casualty Actuarial Society’s Fall 2007 Exam 3:**

Lethargic LLC stock currently trades for $300 per share. The annual continuously compounded risk-free interest rate is 0.09, and the stock pays dividends at an annual continuously compounded yield of 0.03. You know that a $300-strike call option on Lethargic LLC stock expiring in one year has a price of $85.61554928. Given that the implied volatility of the stock is greater than 0.06, calculate this implied volatility using a one-period binomial option pricing model.

(a) Between 0.35 and 0.45

(b) Between 0.45 and 0.55

(c) Between 0.55 and 0.65

(d) Between 0.65 and 0.75

(e) Between 0.75 and 0.85

**Solution CSIV5.** We test the option prices that would be generated by volatilities of 0.35, 0.45, 0.55, 0.65, 0.75, and 0.85.

We know that u = e(r-∂)h + σ√(h) and d = e(r-∂)h – σ√(h)

Here, h = 1, σ√(h) = σ and (r – ∂)h = 0.09 – 0.03 = 0.06. Because we know that σ > 0.06, we know that

d 0.06 – 0.06, so d

C = e-rh[p*Cu + (1 – p*)Cd] = e-0.09[p*Cu + (1 – p*)0] = e-0.09[(p*)300u – 300] =

C = e-0.09[300(e0.06- d)e0.06 + σ /(u – d) – 300] =

C(σ) = e-0.09[300(e0.06- e0.06- σ)e0.06 + σ /(e0.06 + σ – e0.06- σ) – 300]

Now we can insert various values of σ to find corresponding values of C(σ).

C(0.35) = 57.44319699

C(0.45) = 71.02295123

C(0.55) = 84.3057477

C(0.65) = 97.23721038

We need not go further, because our desired call price, $85.61554928, is between C(0.55) and C(0.65). Thus, implied volatility is between 0.55 and 0.65, and the correct answer is **(c).**

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