- CFA Exams
- CFA Level I Exam
- Topic 7. Derivatives
- Learning Module 34. Valuation of Contingent Claims
- Subject 2. Two-Period Binomial Model

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**CFA Practice Question**

Continue with the previous 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 put option expires in two years and has an exercise price of $60. What is the number of shares needed to construct a risk-free hedge at each point in the binomial tree? Use 10,000 puts.

Correct Answer: See below for explanation.

p

p

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

- p
^{++}= Max (0, 60 - 79.35) = $0 - p
^{+-}= p-+ = Max (0, 60 - 58.65) = $1.35 - p
^{--}= Max (0, 60 - 43.35) = $16.65

p

^{+}= (0.7 x 0 + 0.3 x 1.35)/(1.06) = $0.3821p

^{-}= (0.7 x 1.35 + 0.3 x 16.65)/(1.06) = $5.60The put price today is p = (0.7 x 0.3821 + 0.3 x 5.6)/1.06 = $1.83.

At the current price of $60, n = (p

^{-}- p^{+}) / (S^{+}- S^{-}) = (5.6 - 0.3821) / (69 - 51) = 0.2899.At the end of year 1:

- If the stock price is $69, n
^{+}= (p^{+-}- p^{++}) / (S^{++}- S^{-+}) = (1.35 - 0) / (79.35 - 58.65) = 0.065. - If the stock price $51, n
^{-}= (p^{--}- p^{-+}) / (S^{+-}- S^{--}) = (16.65 - 1.35) / (58.65 - 43.35) = 1. This means that the risk-free hedge would consist of a long position in 10,000 puts and a long position in 10,000 shares.

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**User Contributed Comments**
2

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

Rotigga |
n+ = (1.35 - 0) / (79.35 - 58.65) = 0.06522; therefore 652 shares n- = (16.65 - 1.35) / (58.65 - 43.35) = 1; therefore 10,000 shares n0 = (5.60 - 0.38) / (69 - 51) = 0.29; therefore 2,900 shares |

gregsob2 |
Yep agree with rotigga |