In: Finance
Consider a stock with a price of $120 and a standard deviation
of 30 percent. The stock will pay a dividend of $5 in 40 days and a
second dividend of $5 and 140 days. The current risk-free rate is 6
percent per annum. An American call on this stock has an exercise
price of $150 and expires in 100 days.
Price the American call option using the Black-Scholes Model. Show
all calculations
Price the American call option using the Black-Scholes Model.
Current stock price S = $120 per share
Expected dividend in 40 days = $5 per share
Expected dividend in 140 days = $5 per share (this divided need not be adjusted into price of stock as the payout is after expiry of option)
Risk free rate r = 6% per year or 6%/365 per day (assuming 365 days in a year)
Present value of expected dividend = expected dividend / (1+ daily risk free rate) ^ (time period)
= $5/ (1 +6%/365) ^ (40) = $4.9672
Dividend-adjusted stock price = $120 – $4.9672= $115.0328
Now call option price calculation:
At the ex-dividend date, the holder of an American call has two choices: exercise and own the stock or do not exercise and hold it like a European option that expires at the original expiration date.
Now calculate the call option price in both cases:
INPUTS |
Outputs |
Value |
|
Standard deviation (Annual) (σ) |
30.00% |
d1 |
-2.21532 |
Time until Expiration (in Years) (t) |
0.11 |
d2 |
-2.31464 |
Risk free rates (Annual) (r) |
6.00% |
N(d1) |
0.01337 |
Stock Price (S0) |
$ 115.0328 |
N(d2) |
0.01032 |
Strike price (X) |
$145.00 |
B/S call value (C ) |
0.05179 |
Dividend yield |
0 |
B/S Put Value (P) |
29.06872 |
Call price is $0.05179 in this case
INPUTS |
Outputs |
Value |
|
Standard deviation (Annual) (σ) |
30.00% |
d1 |
-1.50707 |
Time until Expiration (in Years) (t) |
0.27 |
d2 |
-1.66410 |
Risk free rates (Annual) (r) |
6.00% |
N(d1) |
0.06590 |
Stock Price (S0) |
$ 115.0328 |
N(d2) |
0.04805 |
Strike price (X) |
$150.00 |
B/S call value (C ) |
0.49077 |
Dividend yield |
0 |
B/S Put Value (P) |
33.01240 |
Call price is $0.49077 in this case
As the call price is higher for - Do not exercise and hold it like a European option that expires at the original expiration date of 100 days. Therefore call holder will choose this option with call price =$0.49077.
Formulas used in excel calculation: