- CFA Exams
- CFA Level I Exam
- Study Session 14. Derivatives
- Reading 38. Valuation of Contingent Claims
- Subject 2. Two-Period Binomial Model

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**CFA Practice Question**

A stock is worth $60 today. In a year the stock price can rise or fall by 15 percent. The interest rate is 6%. A call option expires in two years and has an exercise price of $55. What is the number of calls that would be sold to construct a risk-free hedge at the end of year 1 if the stock price becomes $69 (115% of the current price)? Use 1,000 shares.

A. 1224

B. 1478

C. 1000

**Explanation:**The risk-neutral probability is π = (1.06 - 0.85) / (1.15 - 0.85) = 0.7, and 1 - π = 0.3.

Stock prices in the binomial tree one and two years from now are:

- S
^{+}= 60 (1.15) = $69 - S
^{-}= 60 (0.85) = $51 - S
^{++}= 60 (1.15) (1.15) = $79.35 - S
^{+-}= S^{-+}= 60 (1.15) (0.85) = $58.65 - S
^{--}= 60 (0.85) (0.85) = $43.35

- c
^{++}= Max (0, 79.35 - 55) = $24.35 - c
^{+-}= c^{-+}= Max (0, 58.65 - 55) = $3.65 - c
^{--}= Max (0, $43.35 - 55) = $0

^{+}= (0.7 x 24.35 + 0.3 x 3.65)/(1.06) = $17.11.

c

^{-}= (0.7 x 3.65 + 0.3 x 0)/(1.06) = $2.41

n

^{+}= (c

^{++}- c

^{+-}) / (S

^{++}- S

^{+-}) = (24.35 - 3.65) / (79.35 - 58.65) = 1

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**User Contributed Comments**
10

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

aero |
But the price of the stock can only be between [79,35,58,65]. the option is in the money. |

hkcfa2 |
true it is in the money. But your situation is that you have the stock and it may go down in two years, to 43.35. You want to sell some in-the-money calls to construct a risk-free portfolio. The fact that the option is in the money or not does not matter here. |

art1997 |
Aero is right, stock price can be in between 58.65 and 79.35 ( +-15%). Stock can not go down to 43.35 as it will be 37% drop ( question said in year year it can go down by 15% only) |

shame |
Yes it will go down by 15% each year and you have to consider a 2-year horizon: that's $43. |

Mdavid2 |
The delta is 1. n+ = (c++ - c+-)/(S++ - S+-). The right answer is 1000. |

ragingrazz |
I agree w/ Mdavid. Hedge ratio =1 for n+ |

irina8 |
I agree w/ you ratio is 1, Q mentions "if the stock price becomes 69"... |

americade |
trick question, this is easier than all that. you don't have to do anything except find delta of option = 24.35/79.35-43.35 = .676. The hedge ratio multiple is the inverse = 1.478437. That means you need to sell 1.478 calls to cover 1 share of stock. ques asks how many CALLS which is the more common question in real world to cover 1000 shares to be hedged. that is just inverse of delta (1.478 X 1000) = 1,478. the questions in the lesson always asked how many shares for 1000 calls would be 676. |

HenryQ |
If the price at the end of year 1 is 69, only the higher part of the tree is relevant (price of 79.35 or 58.65). If so the hedge ratio is 1. |

NIKKIZ |
I don't think you need any maths to do this one. The trick is in the timeline. 'Today' is in one years time at which point the asset price is 69. The strike price is still 55. Since the price can only go up or down by 15% - that means that no matter what happens the option will still be in the money. So whatever happens you need to cover the calls you sell by owning 100% of the underlying otherwise you will have a 100% chance of a loss. |