Question

1. Please use binomial option pricing model to derive the value of a one-year put option. The current share price is ?0 = 100 and exercise price ? = 110. The T-bill rate is ? = 10% per year and annual standard deviation is 20%.

2. Use the Black-Scholes formula to find the value of the same option in the previous problem and compare the difference between these two types of methods.

Answer 1

1) Up-move factor U = = e^(0.2*(1^0.5)) = 1.2214

Down-move factor D = 1/U = 0.8187

Probability of up-move = = (e^(0.1*1)-0.8187)/(1.2214-0.8187)

Probability of up-move =0.71137

Probability of down -move = = 1-0.71137 = 0.28863

Stock price in case of upmove = 1.2214*100 = $122.14

Stock price in case of down-move = 0.8187*100 = $81.87

Payoff in case of upmove = 0

Payoff in case of down-move = 110-81.87 = $28.13

Put option value = 0.71137*0 + 0.28863*28.13

Put option value = $8.11916

2)

Put opttion value using a Black-Scholes model is given by

d1 = (ln(100/110) + (0.1+ (0.2*0.2/2))*1)/(0.2*(1^0.5)) = 0.123449

d2 = 0.123449 - (0.2*(1^0.5) = -0.07655

p = 110*e^(-0.1*1)*N(0.07655) - 100*N(-0.123449)

N(0.07655) = 0.530509232

N(-0.123449) = 0.450875779

p = 110*e^(-0.1*1)*0.530509232- 100*0.450875779

p = $7.715

Difference between these two types of methods.

Put option price using Binomial model = $8.11916

Put option price using Black-Scholes model = $7.715

Difference = 8.11916-7.715 = $0.40416