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.

Use the two-period binomial model to calculate the put option price.

Correct Answer: $1.83

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.