### CFA Practice Question

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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. Today at time 0, a risk-free hedge consists of a short position in 10,000 calls and a long position in ______ shares of the stock.
A. 5865
B. 7935
C. 8167
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
Call option values at expiration two years from now are:
• c++ = Max (0, 79.35 - 55) = \$24.35
• c+- = c-+ = Max (0, 58.65 - 55) = \$3.65
• c-- = Max (0, \$43.35 - 55) = \$0
The option prices at the end of year 1: c+ = (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

At the current price of \$60, n = (c+ - c-) / (S+ - S-) = (17.11 - 2.41) / (69 - 51) = 0.8167.