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)) = 1.22

Down-move factor D = 1/U = 0.82

Probability of up-move   = (e^(0.1*1)-0.82)/(1.22-0.82)

Probability of up-move =0.713

Probability of down -move = = 1-0.5050= 0.287

If the stock moves up after one year, stock price = 100*1.22 = 122

If the stock moves down after one year, stock price = 100*0.82 = 82

If the stock moves up, the payoff on the put option is 0

If the stock moves down, the payoff = (110-82) = 28

Expected value of the payoff today = 0.287*28*e^(-0.1) = 7.27

Price of the put option using binomial model = 7.27

(2)

Put option 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)) = 0.123

d2 = 0.123- (0.2*(1^0.5) = -0.077

N(-0.123) = 0.451054

N(0.077) = 0.530688

p =110*e^(-0.1*1)*0.530688-100*0.451054

p = 7.715

Difference between these two types of methods = 7.715 - 7.27 = 0.445

Hence, the Black-Scholes model estimates a price higher than 0.445 than the Binomial option pricing model