- CFA Exams
- CFA Exam: Level I 2021
- Study Session 16. Derivatives
- Reading 49. Basics of Derivative Pricing and Valuation
- Subject 8. Factors that Affect the Value of an Option

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**CFA Practice Question**

Consider call and put options expiring in 42 days, in which the underlying is at 72 and the risk-free rate is 4.5%. The underlying makes no cash payments during the life of the options. Find the lower bounds for European and American calls and puts with exercise prices of 70 and 75.

Correct Answer: See below

European options

- 70 call: Max [0, 72 - 70/(1.045)
^{0.1151}] = Max[0, 2.35] = 2.35 - 75 call: Max [0, 72 - 75/(1.045)
^{0.1151}] = Max[0, -2.62] = 0 - 70 put: Max [0, 70/(1.045)
^{0.1151}- 72] = Max[0, -2.35] = 0 - 75 put: Max [0, 75/(1.045)
^{0.1151}- 72] = Max[0, 2.62] = 2.62

- 70 call: Max [0, 72 - 70/(1.045)
^{0.1151}] = Max[0, 2.35] = 2.35 - 75 call: Max [0, 72 - 75/(1.045)
^{0.1151}] = Max[0, -2.62] = 0 - 70 put: Max(0, 70 - 72) = 0
- 75 put: Max (0, 75 - 72) = 3

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**User Contributed Comments**
11

User |
Comment |
---|---|

cuong |
Is there anybody please tell me how we can get 0.1151??? |

Pooh |
42 days out of 365 per year = 42/365=.1151 |

Nancyz |
I think the answers for american options calls are wrong. The low bound for American option price is its current intrinsic value: max (0, 72-70)=2, for 75 call =0. any comments? |

Nancyz |
never mind. the answer is correct. the lower bound of the european call holds for american calls. |

stefdunk |
does anyone know why we use the time value of money for american calls, but not for puts? |

johnsk |
Because you can realize the profit right away by exercising the American put option. |

EK65 |
We use time value for american call because otherwise an american option price will be lower than european, which is not possible. Therefore the lower boundary for euro option holds for american. The situatio is different for put, thus we don't need to use the same formula |

0is4eva |
42 days ==> interst rate factor = int = 1/[(1+0.045)^(42/365)] S = 72 Solve for X = 70, 75 for put/call Eur/Am style. Eur 70 call = S - X int = 72 - 70 * int = 2.35 Eur 75 call = S - X int = 72 - 75 * int = neg. --> 0 Eur 70 put = X int - S = 70 * int - 72 = neg. --> 0 Eur 75 put = X int - S = 75 * int - 72 = 2.62 Lower bound for Am options is its intrinsic value or value of Eur option. Am 70 call: S - X = 72 - 70 = 2 but 2.35 for Eur option is greater --> 2.35 Am 75 call: S - X = ... = 0 Am 70 put: X - S = 70 - 72 = neg. --> 0 Am 75 put: X - S = 75 - 72 = 3 Never optimal to exercise Am call on non-dividend-paying stock before expiration date. Strike price paid later (interest earned), no dividend is lost on stock, plus some chance that price of stock will fall below X. (If an overpriced stock it is different). |

Rotigga |
This is an excellent question. It really hammers home the fact that, ceterus paribus, an American option cannot be cheaper than a European option. |

gazza77 |
isnt the convention for interest rate calculations to use 360 days in a year? |

gill15 |
Great Question. Boom got it. It`s the unit summarized. |