Question

In: Finance

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

Solutions

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:

  1. 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

  1. 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:



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