Tutorial Questions
Question 1 () Consider a European call option with an exercise price of $4 and six months to expiry. The underlying share is currently trading at $5.20. The standard deviation (σ) of the return on the share is 0.90 and the riskfree rate of interest is 10% per annum. a) Use the Black-Scholes formula to calculate the fair value of this call option. Divide the option price into its intrinsic and time value components. b) If the underlying share price fell to $3.80, what is the Black-Scholes price? Before you do the calculation, have a think about whether the price will be higher or lower than in part a. Why does this option have any value at all if it is out-of-the-money? c) Ignore b and assume that share price is $5.20. Calculate the value of a call option with six months to expiry and strike price of $3.00. Before you do the calculation, do you expect the option value to be higher or lower than in part a? d) Ignore b and c (assume that share price is $5.20 and strike price is $4). Assume that the call option has nine months to expiry. Do you expect the option value to be higher or lower than part a? Calculate the value. BFF3751 Derivatives 1 Philip Gray 2020 2 Question 2 ()
Reconsider the data from Question 1. A European put option is also written on the stock. Like the call option, it has a strike price of $4 and six months to expiry. Using your answers to Question 1 and the put-call parity formula:
a) If share price is $5.20, what is the value of the put option? Why does the put option have value when it is out-of-the-money?
b) If the underlying shares were trading at $3.80, what is the put option value?
c) Ignore b. Calculate the value of a put option with six months to expiry and strike price of $3.00. Before you do the calculation, do you expect the option value to be higher or lower than in part a?
d) Ignore b and c. Assume that the put has nine months to expiry. Do you expect the option value to be higher or lower than part a? Calculate the value.
For each part, divide the total put option premium into intrinsic value and time value components.
Question 3 (*)
In each of the following cases, determine if there is an arbitrage opportunity. If there is, explain how you would exploit the opportunity.
a) An American-style call option written on Suncorp has three months to expiry and a strike price of $4.50. It is currently trading at $1.20. Suncorp shares are trading at $6.00. The riskless rate of interest is 5%.
b) An American-style put option on Lend Lease with strike price $7.50 is trading at $0.35. Lend Lease is trading at $6.90. The put option has 4 months before expiry. The riskfree interest rate is 4% pa.
c) A European-style call option on ABC has a strike price of $8, nine months to expiry, and is selling for $2. The riskfree rate is 5% per annum and the standard deviation of ABC’s return is 0.50. ABC shares are trading at $10.
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Question 4
a) A call option is written on XYZ with a $10 strike price. The current price of XYZ shares is $8.50. The call option is selling in the market for a premium of $9. Is there an arbitrage opportunity? If so, explain your trading activity to capture the arbitrage profit. Assume that the riskfree rate is 6% and the option has one year to expiry.
b) An American put option on XYZ has a strike price $5. The current price of XYZ shares is $8.50. The put option is selling in the market for a premium of $5.50. Is there an arbitrage opportunity? If so, explain your trading activity to capture the arbitrage profit. Assume that the riskfree rate is 6% and the option has one year to expiry.
Question 5
a) Call and put options are written on Apple (NASDAQ ticker: AAPL). Assume that these options have 6 months to expiry and a $200 strike price. Also assume that the riskfree rate of interest is 4% per annum and the volatility (σ) of AAPL share returns is 0.20.
Let’s study how changes in share price affect the value of AAPL call and put options. Use the spreadsheet BlackScholes.xlsx to quickly complete the following table.
AAPL share price
Call option value
Put option value
$185
$190
$195
$200
$205
$210
$215
b) Again, assume that AAPL call and put options have 6 months to expiry, the riskfree rate of interest is 4% per annum and the volatility (σ) of AAPL share returns is 0.20.
This time, we’ll see how option prices vary for different strike prices. Assume that the current AAPL share price is $200.
Option strike price
Call option value
Put option value
$185
$190
$195
$200
$205
$210
$215
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c) This time, we’ll look at how the amount of time before expiry influences option values. Assume that AAPL share price is $200 and the option strike price is $200. The riskfree rate of interest is 4% per annum and the volatility (σ) of AAPL share returns is 0.20.
Time to expiry (in years)
Call option value
Put option value
1/12
3/12
6/12
9/12
1 year
1.5 years
2 years
d) Volatility is a very important influence on option prices. Option prices are very sensitive to changes in volatility. Assume that AAPL share price is $200. The options have a strike price of $200 and six months to expiry. The riskfree rate of interest is 4% per annum.
Volatility (σ) of
AAPL returns
Call option
value
Put option
value
0.10
0.15
0.20
0.25
0.30
0.35
0.40
e) Finally, let’s see that option prices are not overly sensitive to changes in the interest rate. Assume that AAPL share price is $200. The options have a strike price of $200 and six months to expiry. The volatility (σ) of AAPL returns is 0.20.
Riskfree rate
of interest
Call option
value
Put option
value
2%
4%
6%
8%
10%
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Key Figure Answers
For calculation questions, I like to give the “key figure answers”. Student always want to know if they answered right or wrong, so key figure answers allow them to check.
If you get it right, great. If you didn’t get the key figure answer, then you did something wrong. The best way to learn and improve is for you to discover by yourself what went wrong. Perhaps check back over your calculations. Re-examine the lecture notes to see if you misunderstood something. If you can work it out for yourself, then you will have a much better understanding of the content.
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