Question

Consider an American put option. Suppose its stock price is $50 and its strike price is $52. If the risk-free rate is 7% per annum with continuous compounding, the volatility is 32%, and the life of the option is 2 years and there are two time steps, what is the value of this American put option? Choose one of the following:

		$7.50
		$5.50
		$2.50
		$8.50
		$6.50

Answer 1

Stock Price Today S₀ = 50

Strike Price K = 52

Risk Free Rate R_f = 7%

Volatility σ = 32%

Life of option T = 2 years

2 step process so ∆t = 1 year

Up Movement u = exp(σ*sqrt(∆t))
Down Movement d = 1/u
where exp is exponential and sqrt is square root function.

u = exp(0.32*sqrt(1)) = exp(0.32) = 1.3771
d = 1/u = 1/1.3771 = 0.7261

a = exp(R_f*∆t) = exp(0.07) = 1.0725

Probability of up movement p = (a-d)/(u-d) = (1.0725 – 0.7261)/(1.3771 – 0.7261) = 0.5321

Probability of down movement = 1 – p = 0.4679

After 1 year -
If price moves up, Stock Price = S₀ * u = 50*1.3771 = 68.86
If price moves down, stock price = S₀*d = 50*0.7261 = 36.31
Pay-off from Put Option if Stock Price = 68.86 à 0
Pay-off from Put Option if stock price = 36.31 à 52 – 36.31 = 15.69

After 2^nd Year
Price was at 68.86 and moved up again, then Stock Price = 68.86*1.3771 = 94.82 à Then payoff = 0
Price was at 36.31 and moved up OR was at 68.86 and moved down, then stock price = 68.86*0.7261 = 36.31*1.3771 = 50 à then pay off = 52 -50 = 2
Price was at 36.31 and moved down again, then stock price = 26.36 à then payoff = 52 – 26.36 = 25.64

Hence, price movement will look like below –

Equivalent Payoff diagram will look like below –

Here is how the payoffs are calculated -

Traverse the binomial tree from right to left. Assigning payoff at right most nodes is easiest. If price is above 52, i.e. strike price, option value will be zero, else it will be (52 – stock price).

Then come to nodes corresponding to 1 year.

Option will not get exercised when price was 68.86 as payoff is 0 there. Using 2^nd year payoffs value of option at 1 year then will be = (p*0 + (1-p)*2)/a = 0.4679*2/1.0725 = 0.87

Note - divided by 'a' to get present value of the payoff

When price is 36.31, if exercised immediately option value is = 15.69
if exercised in 2^nd year through this route, value = (p*(2) + (1-p)*25.64)/a = 13.06/1.0725 = 12.18

As value is higher when exercised immediately, it will be exercised at end of 1 year.

Now consider today, value = (p*0.87 + (1-p)*15.69)/a = 7.28

Hence, value of put option = 7.28