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 annum, the volatility is 25% per annum, and the time to maturity is four months. (a) What is the price of the option if it is a European call? (1 mark) (b) What is the price of the option if it is an American call? (1 mark) (c) What is the price of the option if it is a European put? (1 mark) (d) Verify that put–call parity holds. (1 mark)
Question
Consider an option on a non-dividend-paying stock when the stock price is 29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. (a) What is the price of the option if it is a European call? (1 mark) (b) What is the price of the option if it is an American call? (1 mark) (c) What is the price of the option if it is a European put? (1 mark) (d) Verify that put–call parity holds. (1 mark)
Solution
To solve this problem, we will use the Black-Scholes-Merton model for pricing options.
(a) The price of a European call option can be calculated using the formula:
C = S0 * N(d1) - X * e^(-rT) * N(d2)
where: S0 = current stock price = 29 r = risk-free interest rate = 5% per annum = 0.05 T = time to maturity = 4 months = 4/12 = 0.333 years σ = volatility = 25% per annum = 0.25 N = cumulative distribution function for a standard normal distribution d1 = (ln(S0/X) + (r + σ^2 / 2) * T) / (σ * sqrt(T)) d2 = d1 - σ * sqrt(T)
Substitute the given values into the formula to calculate the price of the European call option.
(b) For an American call option without dividends, it is never optimal to exercise the option early. Therefore, the price of the American call option is the same as the European call option.
(c) The price of a European put option can be calculated using the formula:
P = X * e^(-rT) * N(-d2) - S0 * N(-d1)
Substitute the given values into the formula to calculate the price of the European put option.
(d) To verify the put-call parity, check if the following equation holds:
C + X * e^(-rT) = P + S0
where C is the price of the call option, P is the price of the put option. If the equation holds, then the put-call parity is verified.
Please note that the actual calculations require the use of a standard normal distribution table or a calculator with a built-in standard normal distribution function.
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