Question

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Consider a stock with a price of $120 and a standard deviation of 30 percent. The...

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

Expert Solution

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:

Exercise in 40 days and own the stock; exercise price will be $150 - $5 = $145

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

Do not exercise and hold it like a European option that expires at the original expiration date of 100 days.

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:

jeff jeffy answered 5 months ago

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