Topic 5 tutorial

Question 1

10.24. A European call option and put option on a stock both have a strike price of $20 and an
expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per
annum, the current stock price is $19, and a $1 dividend is expected in one month.
Identify the arbitrage opportunity open to a trader.

Answer 1

Problem 10.24

K=20, T=3/12, C=3, P=3,

r=0.10, S=19, D=1

Using the notation in the chapter, put-call parity [equation (9.7)] gives:

c + Ke^-rT + D e^-rT = p + So

or, p = c + Ke^-rT + D e^-rT - So

In this case:

p = 3 + 20e^-0.1x3/12 + $1e^{-0.1 x 1/12} – 19

= 3 + 19.51 + 0.99 – 19

= $4.50

c = p + S_o- Ke^-rT – D (equation rearranged)

= 3 + 19 – 19.51 - .99

= $1.50

The put is undervalued and the call is overvalued relative to current prices

Therefore, buy a put, sell a call and buy the stock (also known as a covered call)

Net outlay is -3 + 3 – 19 = -19

If the stock price is above $20 in three months, the sold call option will be exercised at $20 (covered by the bought stock) giving a net inflow of $20 or, 20e^{-.10 x 3/12} = $19.51 in present value terms. Receive dividends on the long stock position of $1 or 1e^{-0.1 x 1/12} = $0.99 in present value terms. Therefore there is a profit with a present value of (19.51 – 19) + 0.99 = $1.50.

If the stock price is below $20 in three months, the put option is exercised, and the call option expires worthless. The long put leads to the stock being sold for $20 or, 20e^{-.10 x 5/12} = $19.51 in present value terms. Receive dividends on the long stock position of $1e^{-0.1 x 1/12} = $0.99 in present value terms. Therefore, there is a profit of (19.51 – 19) + 0.99 = $1.50 in present value terms.

Question 2

13.13. What is the price of a European call option on a non-dividend-paying stock when the
stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the
volatility is 30% per annum, and the time to maturity is three months?

Answer 2

Question 3

13.14. What is the price of a European put option on a non-dividend-paying stock when the
stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the
volatility is 35% per annum, and the time to maturity is six months?

Answer 3

Question 4

13.15. A call option on a non-dividend-paying stock has a market price of $2:50. The stock
price is $15, the exercise price is $13, the time to maturity is three months, and the riskfree
interest rate is 5% per annum. What is the implied volatility?

Answer 4

Question 5

13.26. 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?
(b) What is the price of the option if it is an American call?
(c) What is the price of the option if it is a European put?
(d) Verify that put–call parity holds.