Question

Determine price of a 3-year European put option with a $35 strike price. The annual discount rate is 4% annually compounded, the current price of the stock underlying the put option is $32, and the stock has an estimated up/down price change of 9% per year. Must use the 3 step binomial model.

Answer 1

Solution: Calculation of Put Option Price with the Binomial Option Pricing Model (BOPM) with a Stock Price of 32, an uptick percentage of 9%, a downtick percentage of 9%, a discount rate of 4%, and time = 3.

Since there are 3 periods, our BOPM should contain 23 = 8 items
Each subset term can be written using binary expansion representation starting at 0 through 8 - 1 = 7 where D = Down and U = Up
U = (1 + 0.09) = 1.09 and D = (1 - 0.09) = 0.91

Scenario 1: DDD

Price Adjustment Factor = (0.91)(0.91)(0.91) = 0.753571
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 0.753571 = $24.11
Put Price at time 3 = max(0, 35 - 24.11) = $10.89

Scenario 2: DDU

Price Adjustment Factor = (0.91)(0.91)(1.09) = 0.902629
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 0.902629 = $28.88
Put Price at time 3 = max(0, 35 - 28.88) = $6.12

Scenario 3: DUD

Price Adjustment Factor = (0.91)(1.09)(0.91) = 0.902629
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 0.902629 = $28.88
Put Price at time 3 = max(0, 35 - 28.88) = $6.12

Scenario 4: DUU

Price Adjustment Factor = (0.91)(1.09)(1.09) = 1.081171
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 1.081171 = $34.60
Put Price at time 3 = max(0, 35 - 34.6) = $0.40

Scenario 5: UDD

Price Adjustment Factor = (1.09)(0.91)(0.91) = 0.902629
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 0.902629 = $28.88
Put Price at time 3 = max(0, 35 - 28.88) = $6.12

Scenario 6: UDU

Price Adjustment Factor = (1.09)(0.91)(1.09) = 1.081171
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 1.081171 = $34.60
Put Price at time 3 = max(0, 35 - 34.6) = $0.40

Scenario 7: UUD

Price Adjustment Factor = (1.09)(1.09)(0.91) = 1.081171
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 1.081171 = $34.60
Put Price at time 3 = max(0, 35 - 34.6) = $0.40

Scenario 8: UUU

Price Adjustment Factor = (1.09)(1.09)(1.09) = 1.295029
Stock Price at time 3 = Initial Stock Price x Price Adjustment Factor = 32 x 1.295029 = $41.44
Put Price at time 3 = max(0, 35 - 41.44) = $0.00