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.
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:
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:
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
Problem 1. 1. Calculate the price of a six-month European put option on a non-dividend-paying stock...
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