Question

Price the following option using the binomial method. Show all work

Stock price = 10

Strike price = 15

Volatility = 0.4

T = 1

r = 0.3

Type : Put

Answer 1

First calculate the Up move and Down move factor:

We will use “e” exponential, value of e is ~ 2.718 to give continuous compounding impact.

U = e^(Volatility x T^(0.5)) = e^(0.4 x 1^(0.5)) = 1.49182469764127 = 1.49

D = 1 / U = 1/1.49 = 0.67

Now, calculate the probability of Up move and Down move:

Probability of Up move = (e^(r x T) – D) / (U – D) = (e^(0.3 x 1) – 0.67) / (1.49 – 0.67) = 0.83

Probability of Down move = 1 – Probability of Up Move = 1 - 0.83 = 0.17

Now, we can calculate the value of stocks using up move factor and down move factor:

Stock price at end of the year for upper node = 10 x 1.49 = 14.9; Hence, value of the put option at up node will be = Strike price – Stock price = 15 – 1.49 = 0.1

Stock price at end of the year for lower node = 10 x 0.67 = 6.7; Hence, value of the put option at down node will be = Strike price – Stock price = 15 – 6.7 = 8.3

We can find value of option applying probability of up move and down move and discounting values with e or exponential or e^(-0.3);

Value of Put = [ Upper node value x Probability of Up move x e^(-0.3) ] + [Lower node value x Probability of down move x e^(-0.3)]

Value of Put option today = [ 0.1 x 0.83 x e^(-0.3) ] + [ 8.3 x 0.17 x e^(-0.3) ]

Value of Put option today = 0.061487912 + 1.045294509 = ~ 1.11

Value of the Put option today = 1.11