Question

1a. For a stock trading at $50 with 15% volatility and 2% risk free interest rate, find the prices of a one month put and call options with a strike price of $50. Determine the effect on both the put and call of increasing the strike price to $55 Determine the effect of doubling the time to maturity

Answer 1

a. Following is the European call & put option formula to calculate the value of call option under the Black-Scholes Model where K is the strike price

C = S*N (d1) - N (d2) *K*e ^ (-r*t)                  

P = Ke^–rt * N(–d2) – SN(-d1)

Where,

C = call value

P = Put value

S = current stock price

N = cumulative standard normal probability distribution

t = days until expiration

Standard deviation, SD = σ

K = option strike price

r = risk free interest rate

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ σ^2 /2)* t}/σ *√t

d2 = d1 – σ *√t

Or you can use excel in following manner to calculate the value of call option and put option:

INPUTS		Outputs	Value
Standard deviation (Annual) σ	15.00%	d1	0.0601
Expiration (in Years) T	0.08	d2	0.0168
Risk free rates (annual) r	2.00%	N(d1)	0.5240
Current stock price (S)	$50.00	N(d2)	0.5067
Strike price (K)	$50.00	B/S call Price	0.9052
Dividend yield (annual)	0	B/S Put Price	0.8220

Call Price = $0.9052

Put Price = $0.8220

INPUTS		Outputs	Value
Standard deviation (Annual) σ	15.00%	d1	-2.1410
Expiration (in Years) T	0.08	d2	-2.1843
Risk free rates (annual) r	2.00%	N(d1)	0.0161
Current stock price (S)	$50.00	N(d2)	0.0145
Strike price (K)	$55.00	B/S call Price	0.0123
Dividend yield (annual)	0	B/S Put Price	4.9207

Call Price = $0.0123

Put Price = $4.9207

INPUTS		Outputs	Value
Standard deviation (Annual) σ	15.00%	d1	0.0851
Expiration (in Years) T	0.17	d2	0.0238
Risk free rates (annual) r	2.00%	N(d1)	0.5339
Current stock price (S)	$50.00	N(d2)	0.5095
Strike price (K)	$50.00	B/S call Price	1.3043
Dividend yield (annual)	0	B/S Put Price	1.1379

Call Price = $1.3043

Put Price = $1.1379

Formulas used in excel calculation:

A. Value Outputs 1 INPUTS 2 Standard deviation (Annual) o |=(LN(B5/B6)+(B4-B7+0.5*B2^2) *B3)/(B2*SQRT(B3) ) =D2-B2*SQRT(B3) 0.15 d1 - 1/12 d2 3 Expiration (in Years) T 4 Risk free rates (annual) r 5 Current stock price (S) 6 Strike price (K) 7 Dividend yield (annual) N(d1) N(d2) B/S call Price B/S Put Price 0.02 NORMSDIST(D2) =NORMSDIST(D3) -B5*EXP(-B7*B3)*D4-B6*EXP( -B4*B3)*D5 B6*EXP(-B4*B3)*(1-D5) - B5 * EXP ( - B7*B3) *(1- D4) 50 50 Outputs Value 9 INPUTS =(LN(B13/B14)+(B12-B15+0.5* B10^2) *B11)/(B10*SQRT(B11) -D10 -B10* SQRT(B11) 10 Standard deviation (Annual) o 0.15 d1 -1/12 11 Expiration (in Years) T 12 Risk free rates (annual) r 13 Current stock price (S) 14 Strike price (K) d2 N(d1) N(d2) B/S call Price =NORMSDIST (D10) =NORMSDIST(D11) -B13*EXP(-B15*B11) *D12-B14*EXP(-B12*B11)*D13 B14*EXP(-B12*B11) *(1- D13) - B13*EXP(-B15*B11) *(1- D12) 0.02 50 55 15 Dividend yield (annual) B/S Put Price 0 16 Outputs d1 Value 17 INPUTS |=(LN(B21/B22)+(B20-B23+0.5*B18^2)*B19)/(B18*SQRT(B19)) -D18- B18* SQRT(B19) =NORMSDIST(D18) =NORMSDIST (D19) -B21*EXP(-B23*B19)*D20-B22*EXP(-B20*B19)*D21 B22*EXP(-B20 B19) *(1- D21)- B21 * EXP(-B23*B19) *(1- D20) 18 Standard deviation (Annual) o 0.15 -2/12 19 Expiration (in Years) T 20 Risk free rates (annual)r 21 Current stock price (S) 22 Strike price (K) d2 N(d1) N(d2) B/S call Price B/S Put Price 0.02 50 50 23 Dividend yield (annual) 0