Option Prices for Different Strike Prices: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 11

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 11 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.

When the strike price of an option grows at the same rate as the interest rate, the premiums on European calls and puts on a non-dividend-paying stock increase as time to maturity increases.

Let T > t and let both T and t be times to expiration.

For call and put options where x is some time (and the present time is time 0),

K_x = Ke^rx, where r is the annual continuously compounded interest rate. That is, the option's strike price grows at the interest rate.

Then it is always the case that P(T) > P(t) and C(T) > C(t).

Given, two strike prices K₁< K₂, the following relationships always hold with regard to European and American call and put prices C(K₁), C(K₂), P(K₁), and P(K₂), all other things being equal.

C(K₁) ≥ C(K₂)

P(K₂) ≥ P(K₁)

C(K₁) - C(K₂) ≤ K₂ - K₁

P(K₂) - P(K₁) ≤ K₂ - K₁

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 298-300.

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

Problem OPDSP1. Call options on Oblivious Co. are issued with a certain strike price that grows with the interest rate. Which of these times to maturity will result in the highest call premium? (A higher number indicates later time to maturity.)

(a) 2 months
(b) 4 months
(c) 1 year
(d) 2 years
(e) 2 weeks
(f) 366 days

Solution OPDSP1. The greatest time to maturity among these choices is given by (d): 2 years. Because the strike price grows with the interest rate, it is always the case that C(T) > C(t) when T > t. Thus, (d) is the correct answer.

Problem OPDSP2. Obsequious Co. stock options trade on the market, largely at two strike prices: 77 and 89. Which of these statements are true about the following American call and put options? More than one answer may be correct.

(a) P(77) ≤ P(89)
(b) P(89) - P(77) ≥ 12
(c) C(77) ≥ C(89)
(d) P(77) ≥ P(89)
(e) C(89) - C(77) ≥ 12
(f) C(77) - C(89) ≥ 12

Solution OPDSP2. Here, K₁ = 77 and K₂ = 89. (a) accords with the inequality P(K₂) ≥ P(K₁), so (a) is true. (b) violates the inequality P(K₂) - P(K₁) ≤ K₂ - K₁ and so is false. (c) accords with the inequality C(K₁) ≥ C(K₂) and so is true. (d) violates the inequality P(K₂) ≥ P(K₁) and so is false. (e) violates the inequality C(K₁) ≥ C(K₂), since if C(77) ≥ C(89), then C(89) - C(77) can be at most 0 and cannot be ≥ 12. (f) violates with the inequality C(K₁) - C(K₂) ≤ K₂ - K₁ and so is false. So (a) and (c) are correct answers.

Problem OPDSP3. The stock of Impervious LLC trades on the market. Call options on Impervious LLC stock with a strike price of $54 have a premium of $19. Which of these are possible values for call options on Impervious LLC stock with a strike price of $32 and the same time to expiration? More than one answer may be correct.

(a) 16
(b) 20
(c) 34
(d) 25
(e) 45

(f) It is impossible to have call options for this stock with a strike price of $32.

Solution OPDSP3. Here, we consider two constraints, knowing that K₁ = 32 and K₂ = 54. Since C(K₁) ≥ C(K₂), we know that C(32) ≥ 19. Furthermore, C(K₁) - C(K₂) ≤ K₂ - K₁

Implies that C(32) - 19 ≥ 54 - 32, which is equivalent to C(32) ≤ 41. So, answers (b), (c), and (d) are all correct, as they are between 19 and 41. Furthermore, (f) is incorrect, because while a put option's price cannot exceed its strike price, there is no such restriction on call option prices. So (b), (c), and (d) are the correct answers.

Problem OPDSP4. The stock of Impervious LLC trades on the market. Put options on Impervious LLC stock with a strike price of $54 have a premium of $19. Which of these are possible values for put options on Impervious LLC stock with a strike price of $32 and the same time to expiration? More than one answer may be correct.

(a) 16
(b) 20
(c) 34
(d) 25
(e) 45

(f) It is impossible to have put options for this stock with a strike price of $32.

Solution OPDSP4. Here, we consider two constraints, knowing that K₁ = 32 and K₂ = 54. Since P(K₂) ≥ P(K₁), we know that 19 ≥ P(32). Furthermore, P(K₂) - P(K₁) ≤ K₂ - K₁

Implies that 19 - P(32) ≤ 54 - 32, which is equivalent to P(32) ≤ 41. So only the first constraint is binding, and only (a) with a price of 16 fulfills this constraint. (f) is incorrect, because a price of 16 is well below the strike price of 32 and so is not inconsistent with any other boundaries on the possible put price. So (a) is the correct answer.

Problem OPDSP5. Call options on Loquacious, Inc., stock with a strike price of $74 are currently twice as expensive as put options on Loquacious, Inc., stock with a strike price of $74. Put options on Loquacious, Inc., stock with a strike price of $54 are currently worth $21. Which of these are possible prices for call options on Loquacious, Inc., stock with a strike price of $54? More than one answer may be correct.

(a) 51
(b) 23
(c) 103
(d) 34
(e) It is impossible to have a consistent answer given the information in this problem.

Solution OPDSP5. Here, K₁ = 54 and K₂ = 74, and K₂ - K₁ = 20. Thus, we know that

C(54) ≥ C(74) = 2P(74) ≥ 2P(54) = 42. So C(54) ≥ 42 and choices (b) and (d) are ruled out by this constraint. Moreover, we know that P(K₂) - P(K₁) ≤ K₂ - K₁ , which implies that P(74) - 21 ≤ 20 and so P(74) ≤ 41 and so C(74) ≤ 82. We also know that

C(K₁) - C(K₂) ≤ K₂ - K₁, which implies that C(54) - C(74) ≤ 20, so C(54) ≤ 20 + C(74) ≤ 20 + 82. Thus, C(54) ≤ 102, ruling out choice (c). Only choice (a) = 51 is within the constrains 42 ≤ C(54) ≤ 102. Thus, only (a) is the correct answer.

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