Question

There is a European put option on a stock that expires in two months. The stock price is $105 and the standard deviation of the stock returns is 55 percent. The option has a strike price of $115 and the risk-free interest rate is an annual percentage rate of 6.5 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(0.55*(1/12)^0.5) = 1.17207

d =1/u = 0.85319

So, the stock price lattice is as shown below

		144.24
	123.07	105.00
105.00	89.59	76.43
t=0	t=1	t=2

Risk neutral probability of the options is given as

p= ((1+0.065*1/12)- 0.85319)/(1.17207-0.85319) = 0.4774 (the risk neutral probability of stock going up)

and 1-p = 0.5226 (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=$144.24 , = max(115-144.24,0) = 0

Value (Payoff) of option when St=$105 , = max(115-105,0) = $10

Value (Payoff) of option when St=$76.43 , = max(115- 76.43,0) = $38.57

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+0.065/12)

Using the above formulas , the option lattice looks like

		0.00
	5.20	10.00
15.3569	24.80	38.57
t=0	t=1	t=2

The price of the option today (t=0) is $15.36