Question

2. (18 points) Consider a European put option where the stock price is 40, the strike price is $42.50, and the risk free rate is 4% per annum. (a) If the life of the option is 6 months, and there are two time steps (3 months each), what is the value of the put if the volatility is 38%? (b) (12 points) What is the value of the put if it is an American option and has a dividend yield of 3% per annum? (Note: the only effect the dividend has is on calculating the value of p, it is not used for discounting purposes).

Answer 1

There are two time steps of 3 months each. We will use the binomial method of valuing options.

Given information:

St = $40

X = 42.5

Rf = 4%

$\sigma$ = 38%

For binomial, we will need to calculate the upside change and the downside change and apply it two times because of two steps. The formula to calculate this is given below:

1 + upside change = u = $e^{\sigma*\sqrt{h}}$

1+ downside change = d = 1/u

probability of upside p =

elr-g)*h - P-n.

where e is the base of natural logarithm = 2.718

h = time interval = 3 months = 3/12 years

r = risk free rate

q = dividend rate

Substitute the given values:

u = = 1.2092

238%* V3/12

d = 1/u = 0.8270

p = = 0.4759

(4%-0)+3/12 - 0.8270 1.2092 - 0.8270

1-p = 0.5211

Create the binomial tree using the calculated values as follows:

Put option is a bearish strategy. If the stock price moves up higher than the exercise price, then the payoff is zero, whereas if the stock price falls below the exercise price then there is a profit of (X-St).

	Step 1 (St1)	option payoff		Step 2 (St2)	option payoff
upside (St*u)	$48.37 (St1 = St*u)	$0.00 (because the stock price is > $42)	upside	$58.49 (St2 = St1*u)	0 (because the stock price is > $42)
			downside	$40.00 (St2 = St1*d)	$2.00 ($42-$40)
downside (St*d)	$33.08 (St1 = St*d)	$8.5 ($42*e(-4%*3/12)- $33.08) The discounting of X is required only for European and not American option.	upside	$40.00 (St2 = St1*u)
			downside	$27.35 (St2 = St1*d)	$14.65 ($42-$27.35)

Now, we will have to calculate the present value of the payoffs and add the payoffs based on the probability.

Put option price = First step + second step

First step = p*0 + (1-p) * 8.5*e^-4%*3/12 = 0 + 4.3869 = 4.3869

Second step= p * p * 0 + p*(1-p)*2*e^-4%*6/12 +(1-p)*(1-p) * 14.65*e^-4%*6/12 .= 0 + 0.4892 +3.8977 = 4.3869

substituting the value of p we get,

European Put option price = 4.3869 + 4.3869 = 8.7739

Part b)

Here we have an American option instead of European and the stock pays dividend.

We will make two changes to the calculations above.

1. We will use the dividend rate to calculate the probability

2. We will not discount the strike price X in Step 1.

p =
elr-g)*h - P-n.
=
(4%–3%)*3/12 - 0.8270 1.2092 - 0.8270
= 0.4592

1-p = 0.5408

Now we calculate the binomial tree as follows

	Step 1 (St1)	option payoff		Step 2 (St2)	option payoff
upside (St*u)	$48.37 (St1 = St*u)	$0.00 (because the stock price is > $42)	upside	$58.49 (St2 = St1*u)	0 (because the stock price is > $42)
			downside	$40.00 (St2 = St1*d)	$2.00 ($42-$40)
downside (St*d)	$33.08 (St1 = St*d)	$8.92 ($42 - $33.08) The discounting of X is not required for American option as one can exercise.	upside	$40.00 (St2 = St1*u)
			downside	$27.35 (St2 = St1*d)	$14.65 ($42-$27.35)

Now, we will have to calculate the present value of the payoffs and add the payoffs based on the probability.

Put option price = First step + second step

First step = p*0 + (1-p) * 8.5*e^-4%*3/12 = 0 + 4.7769 = 4.7769

Second step= p * p * 0 + p*(1-p)*2*e^-4%*6/12 +(1-p)*(1-p) * 14.65*e^-4%*6/12 .= 0 + 0.4868 +4.1986 = 4.6855

substituting the value of p we get,

American Put option price = 4.7769+ 4.6855= 9.4624