Question

There is a European put option on a stock that expires in two months. The stock price is $63 and the standard deviation of the stock returns is 57 percent. The option has a strike price of $73 and the risk-free interest rate is an annual percentage rate of 6.2 percent.

What is the price of the put option today? Use a two-state model with one-month steps. (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

The lattice parameters are

u = exp(s*t^0.5) = exp(0.57*(1/12)^0.5) = 1.17886

d =1/u = 0.84828

So, the stock price lattice is as shown below

		87.55
	74.27	63.00
63.00	53.44	45.33
t=0	t=1	t=2

Risk neutral probability of the options is given as

p= ((1+0.062*1/12)- 0.84828)/(1.17886-0.84828) = 0.4746 (the risk neutral probability of stock going up)

and 1-p = 0.5254 (the risk neutral probability of stock going down)

The payoff (value) of the European put option at maturity = max(K-St,0) where K is the strike price and St is the stock price at maturity and Max function returns the maximum value

So, at t=2.when option matures,

Value (Payoff) of option when St=$87.55 , = max(73-87.55,0) = 0

Value (Payoff) of option when St=$87.55 , = max(73-63,0) = $10

Value (Payoff) of option when St=$87.55 , = max(73- 45.33,0) = $27.67

Now for each preceding node, at t=1 and t=2, the value of the option is calculated as

Value of option = (p*value of option when stock moves up in the next period+ (1-p) * value of option when stock moves down in the next period)/ (1+0.062/12)

Using the above formulas , the option lattice looks like

		0.00
	5.23	10.00
12.49	19.18	27.67
t=0	t=1	t=2

The price of the option today (t=0) is $12.49