Problem 1: Write True or False for the following:
Q1. The profit for a short position in a call option is always negative or zero.
(a) True
(b) False
Q2. As the time to expiration becomes longer, an American call option always becomes more valuable.
(a) True
(b) False
Q3. Consider two European put options with the same expiration dates and the same strike prices. The underlying assets of the two options are different. One option is for stock A whose current price is $50 and has the volatility of 30%. The other is for stock B whose current price is $45 and has the volatility of 25%. Both stock A and B will pay no dividends. The price of put A is always higher than the price of put B.
(a) True
(b) False
Q4. The put-call parity holds only when future stock price changes as described in a binomial tree.
(a) True
(b) False
Q5. To find the option price in a binomial tree, we need an assumption regarding the probability of the increase/decrease in the stock price.
(a) True
(b) False
Q6. Suppose that risk-averse investors expect the return on a stock to be µ per annum and the risk-free rate is r per annum. In a binomial tree, if µ < r, the real probability of an increase in the stock price is lower than the risk-neutral probability of the increase.
(a) True
(b) False
Q7. Consider an American put option on a non-dividend paying stock. The option will expire on date T. On date t(< T ), the option payoff from the immediate exercise is always lower than the value that results from not exercising and holding the contract.
(a) True
(b) False
Q8. In risk-neutral valuation, we recognize that investors are risk-averse and thus modify the probability of an increase in a stock price from the real probability.
(a) True
(b) False
Q9. In the Discounted Cash Flow, the required return on a European option should be higher than the risk-free rate. Otherwise, an arbitrage exists.
(a) True
(b) False
Q10. An investor wants to construct a bull spread using put options with the same expiration dates. The investor needs to long the put with strike price K1 and short the put with strike price K2(> K1).
(a) True
(b) False
Problem 2: Write answers in short:
Q11. Consider a three-year European call option with the strike price of $150. The underlying stock will pay $10-dividend two years later from now. The current stock price is $170. The risk-free rate is 3% per annum. Find the range of the call prices that do not allow any arbitrage.
Q12. An investor wants to construct a bear spread using two put options with the same expiration T. One put has the strike price of $20 and currently sells for $2. The other put has the strike price of $30 and currently sells for $7. Find the range of stock price on T that results in a positive profit for the investor.
Q13. In time 0, an investor takes a calendar spread by selling two-year European call option and buying three-year European call option. These two options have the same strike price of $80 and are for the same stock that pays no dividends. The two-year option sells for $5 and the three-year option sells for $7. Two years later, the stock price turns out to be $90. The risk-free rate is 2% per annum. What is the minimum of the profit from this strategy? (We assume that we sell the longer-term option in year two).
Q14. Consider a European call and a European put on a non-dividend-paying stock. Both the call and the put will expire in one year and have the same strike prices of $120. The stock currently sells for $115. The risk-free rate is 5% per annum. The price of the call is $7 and the price of the put is $5. Is there an arbitrage? If so, show an arbitrage strategy. (To show the arbitrage, present the table listing actions and resulting cash flows).
Q15. Consider a two-year European put on Canadian Dollar (CAD). The strike price of the put is 6.50 HKD (Hong Kong Dollar)/CAD. The risk-free rate is 2% per annum in Hong Kong and 3% per annum in Canada. The current exchange rate is 5.90 HKD/CAD. The put currently sells for $0.4 in Hong Kong. Is there an arbitrage for Hong Kong investors? If so, show an arbitrage strategy. (To show the arbitrage, present the table listing actions and resulting cash flows).
Q16. Suppose that there are two possible states of the economy, A and B, in year T. A stock's price in year T will depend on the economic state as in the table below. A risk-free bond currently sells for $8 and it will pay $10 in year T in every state.
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state A
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state B
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stock
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$150
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$80
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|
bond
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$10
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$10
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In addition, we also see a year-T European call option on the stock above. The call has the strike price of $100 and currently sells for $20. What is the current stock price?
Q17. An one-year European call option has the strike price of $60. The underlying stock pays no dividend and currently sells for $70. One time step is six months long, and the stock price may move up or down by 10% in each step. The risk-free rate is 3% per annum.
(i) What is the risk-neutral probability of an increase in the stock price in each step?
(ii) What is the time-0 current price of the call?
(iii) Find the replicating portfolio that we construct in time 0 to generate the same value as the call six months later.
(iv) Suppose that risk-averse investors require the stock return to be 12% per annum. In the approach of Discounted Cash Flow, what is the discount rate for the call per annum?
Q18. The current term-structure of risk-free rate is as follows.
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Term-structure in year 0
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maturity (years)
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zero-rate (%)
|
|
1
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2.5
|
|
2
|
3
|
A risk-free bond will pay $1,000 two years from now. The price of the bond one year later depends on the term structure then. There are two possible scenarios in year 1:
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Term-structure in year 1
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Scenario A
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Scenario B
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|
maturity (years)
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zero-rate (%)
|
|
maturity (years)
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zero-rate (%)
|
|
1
|
1
|
|
1
|
4
|
|
2
|
2
|
|
2
|
5
|
An investor considers buying a one-year European call option on the bond with the strike price of $970.
(a) What are the payoffs in scenario A and B in year one for a long position in the call?
(b) What is the present value (in year 0) of the call?