### CFA Practice Question

There are 227 practice questions for this study session.

### 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.

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
Put option values at expiration two years from now are:
• p++ = Max (0, 60 - 79.35) = \$0
• p+- = p-+ = Max (0, 60 - 58.65) = \$1.35
• p-- = Max (0, 60 - 43.35) = \$16.65
The option prices at the end of year 1:
p+ = (0.7 x 0 + 0.3 x 1.35)/(1.06) = \$0.3821
p- = (0.7 x 1.35 + 0.3 x 16.65)/(1.06) = \$5.60

The 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.