Question

Problem 1. 1. Calculate the price of a six-month European put option on a non-dividend-paying stock with an exercise price of $90 when the current stock price is $100, the annualized riskless rate of interest is 3%, and the volatility is 40% per year. 2. Calculate the price of a six-month European call option with an exercise price on this same stock a non-dividend-paying stock with an exercise price of $90. Problem 2. Re-calculate the put and call option prices for problem 1, assuming that a dividend of $1.50 is expected two months from now. Problem 3. Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per year, the volatility is 25% per year, and the time to maturity is four months. 1. What is the price of the option if it is a European call? 2. What is the price of the option if it is an American call? 3. What is the price of the option if it is a European put? 4. Verify that put–call parity holds.

Answer 1

Problem 1. 1. Calculate the price of a six-month European put option on a non-dividend-paying stock with an exercise price of $90 when the current stock price is $100, the annualized riskless rate of interest is 3%, and the volatility is 40% per year.

INPUTS		Outputs	Value
Standard deviation (Annual)	0.40	d1	0.5670
Expiration (in Years)	0.50	d2	0.2841
Risk free rates (annual)	0.03	N(d1)	0.7146
The current market value (S0)	100	N(d2)	0.6118
Exercise price X	90	B/S call value	17.2172
Dividend yield (annual)	0	B/S Put Value	5.8772

The price of a six-month European put option is $5.8772

2. Calculate the price of a six-month European call option with an exercise price on this same stock a non-dividend-paying stock with an exercise price of $90.

INPUTS		Outputs	Value
Standard deviation (Annual)	0.40	d1	0.5670
Expiration (in Years)	0.50	d2	0.2841
Risk free rates (annual)	0.03	N(d1)	0.7146
The current market value (S0)	100	N(d2)	0.6118
Exercise price X	90	B/S call value	17.2172
Dividend yield (annual)	0	B/S Put Value	5.8772

The price of a six-month European call option is $17.2172

Formula used in excel calculation:

Outputs d1 d2 N(d1 N(d2 B/S call value B5 EXP-B7*B3)*D4-B6*EXP(-B4*B3)*D5 B/S Put Value B6*EXP-B4 B3)*(1-D5)-B5*EXP(-B7*B3)*(1-D4 Value 1 INPUTS 2 Standard deviation (Annual0.4 3 Expiration (in Years 4 Risk free rates (annual 5 The current market value (SO) 100 6 Exercise price X 7 Dividend yield (annual) -(LN(B5/B6)+(B4-B7+0.5*B22)*B3)/(B2*SORT(B3)) D2-B2 SQRT(B3) NORMSDIST(D2 NORMSDIST(D3 0.03 90 0

Problem 2. Re-calculate the put and call option prices for problem 1, assuming that a dividend of $1.50 is expected two months from now.

In this case we have to subtract the present value of the dividend from the stock price before using Black Scholes model.

Therefore

S0 = $100 – 1.50 * e^ (0.1667 *0.03) = $98.4925

Now call put calculation:

INPUTS		Outputs	Value
Standard deviation (Annual)	0.40	d1	0.5133
Expiration (in Years)	0.50	d2	0.2304
Risk free rates (annual)	0.03	N(d1)	0.6961
The current market value (S0)	98.4925	N(d2)	0.5911
Exercise price X	90	B/S call value	16.1537
Dividend yield (annual)	0	B/S Put Value	6.3213

Call option price = $16.1537

Put option price = $6.3213

Problem 3. Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per year, the volatility is 25% per year, and the time to maturity is four months.

INPUTS		Outputs	Value
Standard deviation (Annual)	0.25	d1	0.4225
Expiration (in Years)	0.33	d2	0.2782
Risk free rates (annual)	0.05	N(d1)	0.6637
The current market value (S0)	30	N(d2)	0.6096
Exercise price X	29	B/S call value	2.5251
Dividend yield (annual)	0	B/S Put Value	1.0458

Formula used in excel calculation:

Outputs d1 d2 N(d1 N(d2 B/S call value B/S Put Value Value 1 INPUTS 2 Standard deviation (Annual) 0.25 3 Expiration (in Years 4 Risk free rates (annual 5 The current market value (so) 30 6 Exercise price X 7 Dividend yield (annual) (LN(B5/B6)+(B4-B7+0.5 *B2A2) B3)/(B2*SORT(B3)) D2-B2 SQRT(B3) NORMSDIST (D2 NORMSDIST(D3 4/12 0.05 29 0

1. What is the price of the option if it is a European call?

The price of the option if it is a European call is $2.5251

2. What is the price of the option if it is an American call?

The price of the option if it is an American call is the same as the European call price equals to $2.5251

3. What is the price of the option if it is a European put?

The price of the option if it is a European put $1.0458

4. Verify that put–call parity holds

Call-put parity equation can be used in following manner

                C + X* e^ (-r*t) = P + S0

Where,

C = Call premium =$2.5251

X = exercise price of the call/put option =$29

P = Put premium =$1.0458

S0 = Current price of underlying stock =$30

e^ (-r*t) = the present value of the exercise price discounted from the expiration date at risk-free rate

And r =Risk-free rate per annum = 5% and t= time period =0.33 years (4 months)

Therefore,

$2.5251 + $29 *e^ (-5%*0.33) = $1.0458 + $30

Or e^ (-5%*0.33) = 0.9835 it is true value

Therefore the relationship is satisfied