Question

QUESTION 16 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%, the volatility is 25% per annum, and the time to maturity is four months. What is the price of the option if it is a European call? QUESTION 17 Use the same information as in the previous question. What is the price of the option if it is an American call? QUESTION 18 Use the same information as before. What is the price of the option if it is a European put?

Please shown in steps，thank you！

Answer 1

Solution:

Given that Stock Price, $S_{0}$ = 30, Exercise Price, $K$ = 29, Risk-free rate, $r$ = 0.05, Volatility,$\sigma$ = 0.25 and Time period, $T$ = 4/12

First, we calculate $d_{1}$

$d_{1} = \frac{ln(S_{0}/X) + (r + \sigma^2/2) \times T }{\sigma\sqrt(T)}$

$d_{1} = \frac{ln(30/29) + (0.05 + 0.25^2/2) \times 4/12 }{0.25\sqrt(4/12)}$

$d_{1}$ = 0.4225

$d_{2} = \frac{ln(S_{0}/X) + (r - \sigma^2/2) \times T }{\sigma\sqrt(T)}$

$d_{2} = \frac{ln(30/29) + (0.05 - 0.25^2/2) \times 4/12 }{0.25\sqrt(4/12)}$

$d_{2}$ = 0.2782

We use normal tables to find

N (0.4225) = 0.6637, N (0.2782) = 0.6096

N (-0.4225) = 0.3363, N (-0.2782) = 0.3904

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a. The European call price is computed using the following equation

$C = S_{0}N(d_{1}) - Xe^{(-rT)}N(d_{{2}})$

C = 30 (0.6637) - 29 $e^{-0.05*4/12}$ (0.6096)

C = $2.52

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b. The American call price is $2.52 which is same as the European call price.

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c. The European put price is computed using the following equation

$P = Xe^{-rT}N(-d_{2})-S_{0}N(-d_{1})$

$P = 29e^{-0.05*4/12}(0.3904)-30(0.3363)$

P = $1.05