Question

There is a European put option on a stock that expires in two months. The stock price is $69 and the standard deviation of the stock returns is 59 percent. The option has a strike price of $78 and the risk-free interest rate is an annual percentage rate of 5.8 percent.

What is the price of the put option today? Use a two-state model with one-month steps.

Answer 1

Sol:

Spot price (S0) = $69

Strike price (k) = $78

Period (t) = 2 months

Risk free rate (r) = 5.8%

Std deviation (sig) = 59%

Steps (n) = 2

Time of each step - dt - t/n = 1 month

To find the possible prices and calculate the price of the put option, we need to find the following values:

Up step - u - exp(sig x sqrt(dt))

u = exp(0.59 x sqrt(1))

u = exp(0.59)

u = 1.804

Down step - d - 1/u

d = 1/u = 1/1.804

d = 0.554

We also need to calculate the probability of each step occuring:

Probability - q - (exp(r x dt)-d)/(u-d)

q = (exp(r x dt)-d)/(u-d)

q = (exp(0.058 x 1) - 0.554)/(1.804 - 0.554)

q = 0.4044

We can now calculate the possible price movements for the stock over the next two months, the up step - u x price and the down step - d x price. This gives us the following:

Two steps Binomial Tree
69	124.4752	224.5518
	38.24858	69
		21.20223

For the end prices, we can now calculate the payoff. The option is only executed in the last two cases. Payoff = K - price.

We then bring it back, with the following formula:

max of ((q x upstep + (1-q) x downstep) x exp(-r x dt)) or K- price of the current stage. We do this because it is an American option, which means it can be executed anytime, even before expiry.

Doing this twice for each month, we get the following table:

Put Option
24.27141	5.05817	0
	39.7514	9
		56.79777

Therefore the price of the put option today is $24.27