Question

please answer question 7

7. A non-dividend-paying stock has a current price of S30 and a volatility of 20 percent per year. The risk-free rate is 7 % per year.

• a. Use the Black-Scholes equation to value a European call option on the stock with an exercise price of $28 and time to maturity of three months. 

• b. Calculate the price of the put option on this stock with the same exercise price and time to maturity 

• c. Without performing the calculations, state whether the price would be higher if the call were American. Why? 

Answer 1

a) According to Black Scholes Model

$$ \begin{array}{l} c=S_{0} N\left(d_{1}\right)-K e^{-r T} N\left(d_{2}\right) \\ p=K e^{-r T} N\left(-d_{2}\right)-S_{0} N\left(-d_{1}\right) \end{array} $$

where \(d_{1}=\frac{\ln \left(S_{0} / K\right)+\left(r+\sigma^{2} / 2\right) T}{\sigma \sqrt{T}}\)

$$ \begin{array}{c} d_{2}=\frac{\ln \left(S_{0} / K\right)+\left(r-\sigma^{2} / 2\right) T}{\sigma \sqrt{T}}=d_{1}-\sigma \sqrt{T} \\ S_{0}=30, \mathrm{~K}=28, \mathrm{t}=3 / 12, \mathrm{r}=0.07 \\ \sigma=0.2 \\ d_{1}=\frac{\left[\ln \left(\frac{30}{28}\right)+\left(0.07+\frac{0.2^{2}}{2}\right) \times\left(\frac{3}{12}\right)\right]}{0.2 \times \sqrt{\frac{3}{12}}}=0.915 \\ d_{2}=1-d_{1}=1-0.914929=0.085071 \end{array} $$

From normal distribution table, \(\quad N\left(d_{1}\right)=0.8199\)

$$ \begin{array}{l} N\left(d_{2}\right)=0.5339 \\ c=30 \times 0.8199-28 \times e^{-0.07 \times 0.25} \times 0.5339=9.9071 \\ N\left(-d_{1}\right)=0.1801 \\ N\left(-d_{2}\right)=0.4661 \end{array} $$

b) \(p=28 \times e^{-0.07 \times 0.25} \times 0.4661-30 \times 0.1801=7.4214\)

c)

The buyer of an American option can choose to exercise the option until and including the expiration date. However, the buyer of a European option can only exercise the option at the expiration date. By taking advantage of an early favorable price move, the buyer doesn't have to bear the stock's risk of falling back below the strike and having the option expire worthlessly. Having this extra flexibility means the American options generally trade at a premium to European options. This means that all other things being equal, an American call/put option will have a slightly higher purchase price than the same option that has a European exercise style.