Question

For a 3-month European put option on a stock: (1) The stock's price is 41. (ii) The strike price is 45. (iii) The annual volatility of a prepaid forward on the stock is 0.25. (iv) The stock pays a dividend of 2 at the end of one month. (v) The continuously compounded risk-free interest rate is 0.05. Determine the Black-Scholes premium for the option.

Answer 1

Expected dividend in one month = $2 per share

Current stock price S = $41 per share

Risk free rate r = 0.05 per year or 0.05/12 = 0.0042 per month

Present value of expected dividend = $2/ (1 +0.0042) ^ (1) = $1.9917

Net-of-dividend Or Dividend-adjusted stock price = $41 – $1.9917 = $39.0083

Now put option price calculation:

INPUTS		Outputs	Value
Standard deviation (Annual) σ	25%	d1	-0.9806
Expiration (in Years) T	0.25	d2	-1.1056
Risk free rates (annual) r	5.0%	N(d1)	0.1634
Current stock price (S0)	39.0083	N(d2)	0.1344
Strike price (X)	45.00	B/S call Price	0.3987
Dividend yield (annual)	0	B/S Put Price	5.8314

Black-Scholes premium for this put option is $5.8314

Formulas used in excel calculation:

Outputs di 1 INPUTS 2 Standard deviation (Annual) 3 Expiration (in Years) T 4 Risk free rates (annual) 5 Current stock price (SO) 6 Strike price (x) 7 Dividend yield (annual) 0.25 =3/12 0.05 39.0082987551867 d2 Nd1) Nd2) B/S call Price B/S Put Price Value =(LN(B5/B6)+(B4-B7+0.5*B2^2) *B3)/(B2*SQRT(B3)) =D2-B2*SQRT(B3) =NORMSDIST(D2) =NORMSDIST(D3) =B5*EXP(-B7*B3) *D4-B6*EXP(-B4*B3) *D5 =B6*EXP(-B4*B3)*(1-05)-B5*EXP(-B7*B3)*(1-04)