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Midterm Practice Problems
K Foster, Options & Futures, Eco 275, CCNY, Spring 2010 |
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1. Using a binomial tree with a single step, assume that a stock with current price of 17 can either rise to 22 or fall to 12 in the next one month. If I have a put with a strike price of 16, what is its current value? If that put were American, would I exercise it early? Assume that the risk-free rate is 4%.
2. Do the problem above to value a call with strike of 16. Check if put-call parity holds.
3. Do those problems for another month (so two-step models) and value options with 2 months to expiration.
4. Using a two-step binomial tree model, consider pricing a straddle that expires in 2 months. Currently the stock trades at 100; it can move up or down by 10 in each time period (a month). The straddle has both options with strike prices of 100. Assume that the risk-free rate is 4.5%. What is the cost of the straddle?
5. Do the problems above to value bear/bull/box spreads, strangles, strips & straps, butterflies, etc. (Pick interesting strike prices.) What if you had calendar spreads on those?
6. How do above prices change if risk-free rates are just 1% not 4%?
7. A stock currently trades at $5. A call option has a strike price of $7.50 and a put has strike price of $2.50. Both options have maturity of 3 months. If I assume that the riskless interest rate is 5%, what are the upper and lower bounds for the call price? What are the upper and lower bounds for the put price? Using a one-step tree, what prices are implied?
8. Consider a currency call option: currently the Japanese Yen/US Dollar rate is 89 ¥/$; assume that the relevant riskless interest rate is 1%. Assume that after one month the yen will be worth either 85 or 93.
a. What is the implied value of an at-the-money one-month currency call option?
b. What is the implied value of the same option, if the current rate moves up to 91 ¥/$?
9.
Or currency
forwards: the 'zing' trades at 20zing/$.
The zing per transaction?
10. … trying to think of a cross-hedging problem…
11.
Bond valuation
problems continuous time:
a. A bond paying $100,000 in a year is worth $96,079 now. What is the implied zero rate?
b. Another bond from the same issuer pays $10,000 in one year and $10,000 in two years; it is worth $18,747.81. What is implied two-year zero rate? Forward rate from 1 to 2 years?
c. Another bond from the same issuer pays $10,000 in 1, 2, and 3 years. It is worth $27,550. What is the 3-year zero rate? Forward rate from 2-3 years?
d. Another bond from the same issuer pays $10,000 in 1, 2, 3, 4, and 5 years. It is worth $43,995. The 4-year zero rate is 10bps less than the 5-year zero rate. What are these zero rates & forward rates?
12.
Bond valuation
problems discrete semiannual compounding:
a. A bond paying $100,000 in a year is worth $97,066 now. What is the implied zero rate?
b. Another bond from the same issuer pays $10,000 in one year and $10,000 in two years; it is worth $19,082. What is implied two-year zero rate? Forward rate from 1 to 2 years?
c. Another bond from the same issuer pays $10,000 in 1, 2, and 3 years. It is worth 28,094. What is the 3-year zero rate? Forward rate from 2-3 years?
d. Another bond from the same issuer pays $10,000 in 1, 2, 3, 4, and 5 years. It is worth 45,014. The 4-year zero rate is 10bps less than the 5-year zero rate. What are these zero rates & forward rates?
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Past Exams
K Foster, Options & Futures, Eco 275, CCNY, Spring 2010 |
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Midterm Econ 275
Kevin R Foster, CCNY Spring 2009
The questions are worth 75 points. You have 75 minutes to do the exam, one point per minute. All answers should be put into the blue books.
You may refer to your books, notes, calculator, computer, or astrology table. The exam is "open book." However, you must not refer to anyone else, either in person or electronically! You must do all work on your own. Cheating is harshly penalized. Please silence all electronic noisemakers such as mobile phones. Good luck. Stay cool.
1. (20 points) A portfolio of bonds includes the following:
§ Bond ZZZ pays a $4000 coupon every six months, including 6 months from today, 12 months from today, and 18 months from today. It also pays its principal of $200,000 in 18 months at the same time as its last coupon. Its current market value is $201,359.31.
§ Bond YYY pays its $8500 coupon in 6 months and then that coupon again in 12 months along with its $500,000 principal. Its current market value is $499,342.04.
§ Bond VVV pays its principal of $600,000 plus its $8000 semi-annual coupon in 6 months. Its current market value is $595,960.79.
a. Find the six month zero rate (continuously compounded).
b. Find the six-to-twelve month forward rate and the twelve month zero rate (continuously compounded).
c. Find the twelve-to-eighteen month forward rate and the eighteen month zero rate (continuously compounded).
d. Find the par yield for each bond.
e. Find the semiannually-compounded (discrete time every 6 months) zero rates.
f. What is the duration of bond VVV?
2.
(20
points) A stock can be modeled with a one-step discrete-time tree so that after
one month the stock, currently trading at $70 per share, will be worth either
$75 or $60 (note that this is asymmetric).
Assume that the riskfree interest rate is 3%.
a. Find the value of an at-the-money call.
b. Find the value of an at-the-money put.
c. Verify that put-call parity holds.
d. Find the risk-neutral probabilities of up and down movements.
e.
What
value of (shares of the stock) make a riskless portfolio
when combined with one short call? With
one short put?
f.
Why
are these values of different?
3. (20 points) Use the Black-Scholes-Merton formulas, find prices for the same call and put options as above, assuming a volatility of 0.45. Does put-call parity hold now? Explain which model price(s) you would use, if you had to give practical advice in real life.
4. (15 points) A bank has written 1000 calls on stock ABCD. These calls expire in 6 months and have a strike price of 15. ABCD's stock currently trades at 16. The portfolio of calls is currently worth 1956.88.
a. What Black-Scholes volatility is implied by this call price (hint: 100σ is divisible by 5)? (Riskfree rate is 2%.)
b. Puts with the same strike trade at the same implied volatility. What price is this?
c. The bank believes that the stock will be less volatile in the future: five percentage points less volatile than current option prices imply. What strategy of long/short positions in options, riskless bonds, or stock would allow the bank to profit from this knowledge? Explain how risky each strategy is.
Midterm Exam |
Econ 27500 fall 2005 |
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The questions are worth 60 points. You have 60 minutes to do the exam, one point per minute. All answers should be put into the blue books. You may refer to your books, notes, calculator, computer, or astrology table. The exam is open book. However, you must not refer to anyone else! You must do all work on your own. Cheating is harshly penalized. Please silence all electronic noisemakers such as cellphones. Good luck. Stay cool.
1. (5 points) A portfolio consists of two options on the same underlying stock, with the same expiration date in three months. The stock currently trades at $85. One option is a call with strike price of $87. The other option is a put with strike price of $84. Show the payoff graph for the portfolio.
2. (5 points) A bond pays $3000 in six months and then $3000 in twelve months. The current market value is $5740.41. At semi-annual compounding, what is the annual interest rate? (You can solve the quadratic or iterate; assume the interest rate is a whole number less than 10%.)
3. (10 points) A bond with face value of $1,000,000 pays the $1,000,000 of principal in six months. The current market value is $964,399. What is the six-month zero rate (continuously compounded)? What is the bond yield (continuously compounded)?
4. (10 points) A bond with face $10,000,000 pays that principal in one year and has current market value of $9,441,219. What is the one-year zero rate? Based on this answer and the answer from Question 3 above, what is the forward rate for the time period from 6 to 12 months ahead?
5. (10 points) What is the duration of the bond in Question 3 above? What is the duration of the bond in Question 4 above? Calculate how much each bond changes in price when rates rise by 0.0025% after another meeting of Greenspan and the FOMC.
6. (5 points) An insurance company wants to hedge its position against hurricanes. The company believes that a Category-3 hurricane hitting a large city will cost it $500,000,000. It can buy "catastrophe bonds" that each pay out $2,500,000 in the event of a Cat-3 hurricane hitting the city. Each bond costs $10,000. The company would like to hedge at least 50% of its exposure. Should it buy or sell the catastrophe bonds? How many? What is the total cost? Or, the company could hedge using the city's municipal bonds, which would decrease in price if a hurricane struck. Explain the advantages and disadvantages to this alternate hedging strategy.
7. (10 points) I am considering an investment in the country of Cunystan, which uses currency called the "zing". Since my home country offers interest rates of just 1.5%, I am looking to get higher returns. You are my portfolio manager, whose expertise I depend upon. The currency markets allow investors to buy and sell zing at a spot rate of $1 to 30 zing, and buy and sell forward (in one year) at a rate of $1 to 35 zing. Cunystan government bonds pay a riskless interest rate of 5% in one year (that's the continuous-time zero rate). What investment strategy would you recommend, in this situation?
8. (5 points) A stock currently trades at $5. A call option has a strike price of $7.50 and a put has strike price of $2.50. Both options have maturity of 3 months. If I assume that the riskless interest rate is 5%, what are the upper and lower bounds for the call price? What are the upper and lower bounds for the put price?